Understanding Map Projections

Understanding Map ProjectionsMelita Kennedy and Steve KoppGIS by ESRI

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ContentsCHAPTER 1: GEOGRAPHIC COORDINATE SYSTEMS ....................1Geographic coordinate systems ............................................................................... 2Spheroids and spheres ............................................................................................. 4Datums ...................................................................................................................... 6North American datums ............................................................................................ 7CHAPTER 2: PROJECTED COORDINATE SYSTEMS ........................ 9Projected coordinate systems ................................................................................. 10What is a map projection? ...................................................................................... 11Projection types ...................................................................................................... 13Other projections .................................................................................................... 19Projection parameters ............................................................................................. 20CHAPTER 3: GEOGRAPHIC TRANSFORMATIONS ........................23Geographic transformation methods ..................................................................... 24Equation-based methods ........................................................................................ 25Grid-based methods ............................................................................................... 27CHAPTER 4: SUPPORTED MAP PROJECTIONS ..............................29List of supported map projections ......................................................................... 30Aitoff ........................................................................................................................ 34Alaska Grid ............................................................................................................. 35Alaska Series E ........................................................................................................ 36Albers Equal Area Conic ........................................................................................ 37Azimuthal Equidistant ............................................................................................. 38Behrmann Equal Area Cylindrical .......................................................................... 39Bipolar Oblique Conformal Conic ......................................................................... 40Bonne ...................................................................................................................... 41Cassini–Soldner ....................................................................................................... 42Chamberlin Trimetric .............................................................................................. 43Craster Parabolic ..................................................................................................... 44Cylindrical Equal Area ............................................................................................ 45Double Stereographic ............................................................................................. 46Eckert I .................................................................................................................... 47

Eckert II ................................................................................................................... 48Eckert III ................................................................................................................. 49Eckert IV ................................................................................................................. 50Eckert V ................................................................................................................... 51Eckert VI ................................................................................................................. 52Equidistant Conic .................................................................................................... 53Equidistant Cylindrical ............................................................................................ 54Equirectangular ....................................................................................................... 55Gall’s Stereographic ................................................................................................ 56Gauss–Krüger .......................................................................................................... 57Geocentric Coordinate System ............................................................................... 58Geographic Coordinate System .............................................................................. 59Gnomonic ............................................................................................................... 60Great Britain National Grid .................................................................................... 61Hammer–Aitoff ........................................................................................................ 62Hotine Oblique Mercator ....................................................................................... 63Krovak ..................................................................................................................... 64Lambert Azimuthal Equal Area............................................................................... 65Lambert Conformal Conic ...................................................................................... 66Local Cartesian Projection ...................................................................................... 67Loximuthal .............................................................................................................. 68McBryde–Thomas Flat-Polar Quartic ..................................................................... 69Mercator .................................................................................................................. 70Miller Cylindrical ..................................................................................................... 71Mollweide ............................................................................................................... 72New Zealand National Grid ................................................................................... 73Orthographic ........................................................................................................... 74Perspective .............................................................................................................. 75Plate Carrée ............................................................................................................. 76Polar Stereographic ................................................................................................. 77Polyconic ................................................................................................................. 78Quartic Authalic ...................................................................................................... 79Rectified Skewed Orthomorphic ............................................................................ 80Robinson ................................................................................................................. 81Simple Conic ........................................................................................................... 82Sinusoidal ................................................................................................................ 83Space Oblique Mercator ......................................................................................... 84iv • Understanding Map Projections

State Plane Coordinate System ............................................................................... 85Stereographic .......................................................................................................... 87Times ....................................................................................................................... 88Transverse Mercator ................................................................................................ 89Two-Point Equidistant ............................................................................................ 91Universal Polar Stereographic ................................................................................ 92Universal Transverse Mercator ............................................................................... 93Van Der Grinten I ................................................................................................... 94Vertical Near-Side Perspective ................................................................................ 95Winkel I................................................................................................................... 96Winkel II ................................................................................................................. 97Winkel Tripel .......................................................................................................... 98SELECTED REFERENCES ...................................................................... 99GLOSSARY ............................................................................................. 101INDEX .....................................................................................................107Contents • v

1GeographiccoordinatesystemsIn this chapter you’ll learn about longitudeand latitude. You’ll also learn about the partsthat comprise a geographic coordinate systemincluding:• Spheres and spheroids• Datums• Prime meridians1

GEOGRAPHIC COORDINATE SYSTEMSA geographic coordinate system (GCS) uses a threedimensionalspherical surface to define locations onthe earth. A GCS is often incorrectly called a datum,but a datum is only one part of a GCS. A GCSincludes an angular unit of measure, a primemeridian, and a datum (based on a spheroid).A point is referenced by its longitude and latitudevalues. Longitude and latitude are angles measuredfrom the earth’s center to a point on the earth’ssurface. The angles often are measured in degrees(or in grads).equal longitude, or meridians. These linesencompass the globe and form a gridded networkcalled a graticule.The line of latitude midway between the poles iscalled the equator. It defines the line of zero latitude.The line of zero longitude is called the primemeridian. For most geographic coordinate systems,the prime meridian is the longitude that passesthrough Greenwich, England. Other countries uselongitude lines that pass through Bern, Bogota, andParis as prime meridians.The origin of the graticule (0,0) is defined by wherethe equator and prime meridian intersect. The globeis then divided into four geographical quadrants thatare based on compass bearings from the origin.North and south are above and below the equator,and west and east are to the left and right of theprime meridian.The world as a globe showing the longitude and latitude values.In the spherical system, ‘horizontal lines’, or east–west lines, are lines of equal latitude, or parallels.‘Vertical lines’, or north–south lines, are lines ofLatitude and longitude values are traditionallymeasured either in decimal degrees or in degrees,minutes, and seconds (DMS). Latitude values aremeasured relative to the equator and range from -90°at the South Pole to +90° at the North Pole.Longitude values are measured relative to the primemeridian. They range from -180° when traveling westto 180° when traveling east. If the prime meridian isat Greenwich, then Australia, which is south of theequator and east of Greenwich, has positivelongitude values and negative latitude values.Although longitude and latitude can locate exactpositions on the surface of the globe, they are notuniform units of measure. Only along the equatorThe parallels and meridians that form a graticule.2 • Understanding Map Projections

does the distance represented by one degree oflongitude approximate the distance represented byone degree of latitude. This is because the equator isthe only parallel as large as a meridian. (Circles withthe same radius as the spherical earth are calledgreat circles. The equator and all meridians are greatcircles.)Above and below the equator, the circles definingthe parallels of latitude get gradually smaller untilthey become a single point at the North and SouthPoles where the meridians converge. As themeridians converge toward the poles, the distancerepresented by one degree of longitude decreases tozero. On the Clarke 1866 spheroid, one degree oflongitude at the equator equals 111.321 km, while at60° latitude it is only 55.802 km. Since degrees oflatitude and longitude don’t have a standard length,you can’t measure distances or areas accurately ordisplay the data easily on a flat map or computerscreen.Geographic coordinate systems • 3

SPHEROIDS AND SPHERESThe shape and size of a geographic coordinatesystem’s surface is defined by a sphere or spheroid.Although the earth is best represented by a spheroid,the earth is sometimes treated as a sphere to makemathematical calculations easier. The assumptionthat the earth is a sphere is possible for small-scalemaps (smaller than 1:5,000,000). At this scale, thedifference between a sphere and a spheroid is notdetectable on a map. However, to maintain accuracyfor larger-scale maps (scales of 1:1,000,000 or larger),a spheroid is necessary to represent the shape of theearth. Between those scales, choosing to use asphere or spheroid will depend on the map’spurpose and the accuracy of the data.A sphere is based on a circle, while a spheroid (orellipsoid) is based on an ellipse. The shape of anellipse is defined by two radii. The longer radius iscalled the semimajor axis, and the shorter radius iscalled the semiminor axis.The semimajor axis and semiminor axis of a spheroid.A spheroid is defined by either the semimajor axis,a, and the semiminor axis, b, or by a and theflattening. The flattening is the difference in lengthbetween the two axes expressed as a fraction or adecimal. The flattening, f, is:f = (a - b) / aThe flattening is a small value, so usually thequantity 1/f is used instead. The spheroid parametersfor the World Geodetic System of 1984 (WGS 1984 orWGS84) are:a = 6378137.0 meters1/f = 298.257223563The flattening ranges from zero to one. A flatteningvalue of zero means the two axes are equal,resulting in a sphere. The flattening of the earth isapproximately 0.003353.Another quantity, that, like the flattening, describesthe shape of a spheroid, is the square of theeccentricity, e 2 . It is represented by:The major and minor axes of an ellipse.Rotating the ellipse around the semiminor axiscreates a spheroid. A spheroid is also known as anoblate ellipsoid of revolution.2e =2 2a − b2aDEFINING DIFFERENT SPHEROIDS FORACCURATE MAPPINGThe earth has been surveyed many times to help usbetter understand its surface features and theirpeculiar irregularities. The surveys have resulted inmany spheroids that represent the earth. Generally, a4 • Understanding Map Projections

spheroid is chosen to fit one country or a particulararea. A spheroid that best fits one region is notnecessarily the same one that fits another region.Until recently, North American data used a spheroiddetermined by Clarke in 1866. The semimajor axis ofthe Clarke 1866 spheroid is 6,378,206.4 meters, andthe semiminor axis is 6,356,583.8 meters.Because of gravitational and surface featurevariations, the earth is neither a perfect sphere nor aperfect spheroid. Satellite technology has revealedseveral elliptical deviations; for example, the SouthPole is closer to the equator than the North Pole.Satellite-determined spheroids are replacing the olderground-measured spheroids. For example, the newstandard spheroid for North America is the GeodeticReference System of 1980 (GRS 1980), whose radiiare 6,378,137.0 and 6,356,752.31414 meters.Because changing a coordinate system’s spheroidchanges all previously measured values, manyorganizations haven’t switched to newer (and moreaccurate) spheroids.Geographic coordinate systems • 5

DATUMSWhile a spheroid approximates the shape of theearth, a datum defines the position of the spheroidrelative to the center of the earth. A datum providesa frame of reference for measuring locations on thesurface of the earth. It defines the origin andorientation of latitude and longitude lines.Whenever you change the datum, or more correctly,the geographic coordinate system, the coordinatevalues of your data will change. Here’s thecoordinates in DMS of a control point in Redlands,California, on the North American Datum of 1983(NAD 1983 or NAD83).-117 12 57.75961 34 01 43.77884Here’s the same point on the North American Datumof 1927 (NAD 1927 or NAD27).surface of the spheroid is matched to a particularposition on the surface of the earth. This point isknown as the origin point of the datum. Thecoordinates of the origin point are fixed, and allother points are calculated from it. The coordinatesystem origin of a local datum is not at the center ofthe earth. The center of the spheroid of a localdatum is offset from the earth’s center. NAD 1927and the European Datum of 1950 (ED 1950) are localdatums. NAD 1927 is designed to fit North Americareasonably well, while ED 1950 was created for usein Europe. Because a local datum aligns its spheroidso closely to a particular area on the earth’s surface,it’s not suitable for use outside the area for which itwas designed.-117 12 54.61539 34 01 43.72995The longitude value differs by about three seconds,while the latitude value differs by about0.05 seconds.In the last 15 years, satellite data has providedgeodesists with new measurements to define thebest earth-fitting spheroid, which relates coordinatesto the earth’s center of mass. An earth-centered, orgeocentric, datum uses the earth’s center of mass asthe origin. The most recently developed and widelyused datum is WGS 1984. It serves as the frameworkfor locational measurement worldwide.A local datum aligns its spheroid to closely fit theearth’s surface in a particular area. A point on the6 • Understanding Map Projections

NORTH A MERICAN DATUMSThe two horizontal datums used almost exclusivelyin North America are NAD 1927 and NAD 1983.NAD 1927NAD 1927 uses the Clarke 1866 spheroid torepresent the shape of the earth. The origin of thisdatum is a point on the earth referred to as MeadesRanch in Kansas. Many NAD 1927 control pointswere calculated from observations taken in the1800s. These calculations were done manually and insections over many years. Therefore, errors variedfrom station to station.NAD 1983Many technological advances in surveying andgeodesy—electronic theodolites, Global PositioningSystem (GPS) satellites, Very Long BaselineInterferometry, and Doppler systems—revealedweaknesses in the existing network of control points.Differences became particularly noticeable whenlinking existing control with newly establishedsurveys. The establishment of a new datum alloweda single datum to cover consistently North Americaand surrounding areas.the High Accuracy Reference Network (HARN), orHigh Precision Geodetic Network (HPGN), is acooperative project between the National GeodeticSurvey and the individual states.Currently all states have been resurveyed, but not allof the data has been released to the public. As ofSeptember 2000, the grids for 44 states and twoterritories have been published.OTHER UNITED STATES DATUMSAlaska, Hawaii, Puerto Rico and the Virgin Islands,and some Alaskan islands have used other datumsbesides NAD 1927. See Chapter 3, ‘Geographictransformations’, for more information. New data isreferenced to NAD 1983.The North American Datum of 1983 is based on bothearth and satellite observations, using the GRS 1980spheroid. The origin for this datum is the earth’scenter of mass. This affects the surface location of alllongitude–latitude values enough to cause locationsof previous control points in North America to shift,sometimes as much as 500 feet. A 10-yearmultinational effort tied together a network ofcontrol points for the United States, Canada, Mexico,Greenland, Central America, and the Caribbean.The GRS 1980 spheroid is almost identical to theWGS 1984 spheroid. The WGS 1984 and NAD 1983coordinate systems are both earth-centered. Becauseboth are so close, NAD 1983 is compatible with GPSdata. The raw GPS data is actually reported in theWGS 1984 coordinate system.HARN OR HPGNThere is an ongoing effort at the state level toreadjust the NAD 1983 datum to a higher level ofaccuracy using state-of-the-art surveying techniquesthat were not widely available when the NAD 1983datum was being developed. This effort, known asGeographic Coordinate Systems • 7

2ProjectedcoordinatesystemsProjected coordinate systems are anycoordinate system designed for a flat surfacesuch as a printed map or a computer screen.Topics in this chapter include:• Characteristics and types of map projection• Different parameter types• Customizing a map projection through itsparameters• Common projected coordinate systems9

PROJECTED COORDINATE SYSTEMSA projected coordinate system is defined on a flat,two-dimensional surface. Unlike a geographiccoordinate system, a projected coordinate system hasconstant lengths, angles, and areas across the twodimensions. A projected coordinate system is alwaysbased on a geographic coordinate system that isbased on a sphere or spheroid.In a projected coordinate system, locations areidentified by x,y coordinates on a grid, with theorigin at the center of the grid. Each position hastwo values that reference it to that central location.One specifies its horizontal position and the other itsvertical position. The two values are called thex-coordinate and y-coordinate. Using this notation,the coordinates at the origin are x = 0 and y = 0.On a gridded network of equally spaced horizontaland vertical lines, the horizontal line in the center iscalled the x-axis and the central vertical line is calledthe y-axis. Units are consistent and equally spacedacross the full range of x and y. Horizontal linesabove the origin and vertical lines to the right of theorigin have positive values; those below or to the lefthave negative values. The four quadrants representthe four possible combinations of positive andnegative x- and y-coordinates.The signs of x,y coordinates in a projected coordinate system.10 • Understanding Map Projections

WHAT IS A MAP PROJECTION?Whether you treat the earth as a sphere or aspheroid, you must transform its three-dimensionalsurface to create a flat map sheet. This mathematicaltransformation is commonly referred to as a mapprojection. One easy way to understand how mapprojections alter spatial properties is to visualizeshining a light through the earth onto a surface,called the projection surface. Imagine the earth’ssurface is clear with the graticule drawn on it. Wrapa piece of paper around the earth. A light at thecenter of the earth will cast the shadows of thegraticule onto the piece of paper. You can nowunwrap the paper and lay it flat. The shape of thegraticule on the flat paper is very different than onthe earth. The map projection has distorted thegraticule.A spheroid can’t be flattened to a plane any easierthan a piece of orange peel can be flattened—it willrip. Representing the earth’s surface in twodimensions causes distortion in the shape, area,distance, or direction of the data.A map projection uses mathematical formulas torelate spherical coordinates on the globe to flat,planar coordinates.Different projections cause different types ofdistortions. Some projections are designed tominimize the distortion of one or two of the data’scharacteristics. A projection could maintain the areaof a feature but alter its shape. In the graphic below,data near the poles is stretched. The diagram on thenext page shows how three-dimensional features arecompressed to fit onto a flat surface.The graticule of a geographic coordinate system is projected onto a cylindrical projection surface.Projected coordinate systems • 11

Map projections are designed for specific purposes.One map projection might be used for large-scaledata in a limited area, while another is used for asmall-scale map of the world. Map projectionsdesigned for small-scale data are usually based onspherical rather than spheroidal geographiccoordinate systems.Conformal projectionsConformal projections preserve local shape. Topreserve individual angles describing the spatialrelationships, a conformal projection must show theperpendicular graticule lines intersecting at 90-degreeangles on the map. A map projection accomplishesthis by maintaining all angles. The drawback is thatthe area enclosed by a series of arcs may be greatlydistorted in the process. No map projection canpreserve shapes of larger regions.Equal area projectionsEqual area projections preserve the area of displayedfeatures. To do this, the other properties—shape,angle, and scale—are distorted. In equal areaprojections, the meridians and parallels may notintersect at right angles. In some instances, especiallymaps of smaller regions, shapes are not obviouslydistorted, and distinguishing an equal area projectionfrom a conformal projection is difficult unlessdocumented or measured.Equidistant projectionsEquidistant maps preserve the distances betweencertain points. Scale is not maintained correctly byany projection throughout an entire map; however,there are, in most cases, one or more lines on a mapalong which scale is maintained correctly. Mostequidistant projections have one or more lines forwhich the length of the line on a map is the samelength (at map scale) as the same line on the globe,regardless of whether it is a great or small circle orstraight or curved. Such distances are said to be true.For example, in the Sinusoidal projection, theequator and all parallels are their true lengths. Inother equidistant projections, the equator and allmeridians are true. Still others (e.g., Two-PointEquidistant) show true scale between one or twopoints and every other point on the map. Keep inmind that no projection is equidistant to and from allpoints on a map.True-direction projectionsThe shortest route between two points on a curvedsurface such as the earth is along the sphericalequivalent of a straight line on a flat surface. That isthe great circle on which the two points lie. Truedirection,or azimuthal, projections maintain someof the great circle arcs, giving the directions orazimuths of all points on the map correctly withrespect to the center. Some true-direction projectionsare also conformal, equal area, or equidistant.12 • Understanding Map Projections

PROJECTION TYPESBecause maps are flat, some of the simplestprojections are made onto geometric shapes that canbe flattened without stretching their surfaces. Theseare called developable surfaces. Some commonexamples are cones, cylinders, and planes. A mapprojection systematically projects locations from thesurface of a spheroid to representative positions on aflat surface using mathematical algorithms.The first step in projecting from one surface toanother is creating one or more points of contact.Each contact is called a point (or line) of tangency.As illustrated in the section about ‘Planar projections’below, a planar projection is tangential to the globeat one point. Tangential cones and cylinders touchthe globe along a line. If the projection surfaceintersects the globe instead of merely touching itssurface, the resulting projection is a secant ratherthan a tangent case. Whether the contact is tangentor secant, the contact points or lines are significantbecause they define locations of zero distortion.Lines of true scale are often referred to as standardlines. In general, distortion increases with thedistance from the point of contact.Many common map projections are classifiedaccording to the projection surface used: conic,cylindrical, or planar.Projected coordinate systems • 13

Conic projectionsThe most simple conic projection is tangent to theglobe along a line of latitude. This line is called thestandard parallel. The meridians are projected ontothe conical surface, meeting at the apex, or point, ofthe cone. Parallel lines of latitude are projected ontothe cone as rings. The cone is then ‘cut’ along anymeridian to produce the final conic projection, whichhas straight converging lines for meridians andconcentric circular arcs for parallels. The meridianopposite the cut line becomes the central meridian.are called secant projections and are defined by twostandard parallels. It is also possible to define asecant projection by one standard parallel and ascale factor. The distortion pattern for secantprojections is different between the standardparallels than beyond them. Generally, a secantprojection has less overall distortion than a tangentprojection. On still more complex conic projections,the axis of the cone does not line up with the polaraxis of the globe. These types of projections arecalled oblique.In general, the further you get from the standardparallel, the more distortion increases. Thus, cuttingoff the top of the cone produces a more accurateprojection. You can accomplish this by not using thepolar region of the projected data. Conic projectionsare used for midlatitude zones that have aneast–west orientation.The representation of geographic features dependson the spacing of the parallels. When equallyspaced, the projection is equidistant north–south butneither conformal nor equal area. An example of thistype of projection is the Equidistant Conic projection.For small areas, the overall distortion is minimal. OnSomewhat more complex conic projections contactthe global surface at two locations. These projectionsthe Lambert Conic Conformal projection, the centralparallels are spaced more closely than the parallelsnear the border, and small geographic shapes are14 • Understanding Map Projections

maintained for both small-scale and large-scalemaps. On the Albers Equal Area Conic projection,the parallels near the northern and southern edgesare closer together than the central parallels, and theprojection displays equivalent areas.Projected coordinate systems • 15

Cylindrical projectionsLike conic projections, cylindrical projections canalso have tangent or secant cases. The Mercatorprojection is one of the most common cylindricalprojections, and the equator is usually its line oftangency. Meridians are geometrically projected ontothe cylindrical surface, and parallels aremathematically projected. This produces graticularangles of 90 degrees. The cylinder is ‘cut’ along anymeridian to produce the final cylindrical projection.The meridians are equally spaced, while the spacingbetween parallel lines of latitude increases towardthe poles. This projection is conformal and displaystrue direction along straight lines. On a Mercatorprojection, rhumb lines, lines of constant bearing,are straight lines, but most great circles are not.In all cylindrical projections, the line of tangency orlines of secancy have no distortion and thus are linesof equidistance. Other geographical properties varyaccording to the specific projection.For more complex cylindrical projections thecylinder is rotated, thus changing the tangent orsecant lines. Transverse cylindrical projections suchas the Transverse Mercator use a meridian as thetangential contact or lines parallel to meridians aslines of secancy. The standard lines then run north–south, along which the scale is true. Obliquecylinders are rotated around a great circle linelocated anywhere between the equator and themeridians. In these more complex projections, mostmeridians and lines of latitude are no longer straight.16 • Understanding Map Projections

Planar projectionsPlanar projections project map data onto a flatsurface touching the globe. A planar projection isalso known as an azimuthal projection or a zenithalprojection. This type of projection is usually tangentto the globe at one point but may be secant, also.The point of contact may be the North Pole, theSouth Pole, a point on the equator, or any point inbetween. This point specifies the aspect and is thefocus of the projection. The focus is identified by acentral longitude and a central latitude. Possibleaspects are polar, equatorial, and oblique.projections. The perspective point may be the centerof the earth, a surface point directly opposite fromthe focus, or a point external to the globe, as if seenfrom a satellite or another planet.Polar aspects are the simplest form. Parallels oflatitude are concentric circles centered on the pole,and meridians are straight lines that intersect withtheir true angles of orientation at the pole. In otheraspects, planar projections will have graticular anglesof 90 degrees at the focus. Directions from the focusare accurate.Great circles passing through the focus arerepresented by straight lines; thus the shortestdistance from the center to any other point on themap is a straight line. Patterns of area and shapedistortion are circular about the focus. For thisreason, azimuthal projections accommodate circularregions better than rectangular regions. Planarprojections are used most often to map polarregions.Some planar projections view surface data from aspecific point in space. The point of view determineshow the spherical data is projected onto the flatsurface. The perspective from which all locations areviewed varies between the different azimuthalProjected coordinate systems • 17

Azimuthal projections are classified in part by thefocus and, if applicable, by the perspective point.The graphic below compares three planarprojections with polar aspects but differentperspectives. The Gnomonic projection views thesurface data from the center of the earth, whereasthe Stereographic projection views it from pole topole. The Orthographic projection views the earthfrom an infinite point, as if from deep space. Notehow the differences in perspective determine theamount of distortion toward the equator.18 • Understanding Map Projections

OTHER PROJECTIONSThe projections discussed previously areconceptually created by projecting from onegeometric shape (a sphere) onto another (a cone,cylinder, or plane). Many projections are not relatedas easily to a cone, cylinder, or plane.Modified projections are altered versions of otherprojections (e.g., the Space Oblique Mercator is amodification of the Mercator projection). Thesemodifications are made to reduce distortion, often byincluding additional standard lines or changing thedistortion pattern.Pseudo projections have some of the characteristicsof another class of projection. For example, theSinusoidal is called a pseudocylindrical projectionbecause all lines of latitude are straight and paralleland all meridians are equally spaced. However, it isnot truly a cylindrical projection because allmeridians except the central meridian are curved.This results in a map of the earth having an ovalshape instead of a rectangular shape.Other projections are assigned to special groupssuch as circular or star.Projected coordinate systems • 19

PROJECTION PARAMETERSA map projection by itself isn’t enough to define aprojected coordinate system. You can state that adataset is in Transverse Mercator, but that’s notenough information. Where is the center of theprojection? Was a scale factor used? Without knowingthe exact values for the projection parameters, thedataset can’t be reprojected.You can also get some idea of the amount ofdistortion the projection has added to the data. Ifyou’re interested in Australia but you know that adataset’s projection is centered at 0,0, the intersectionof the equator and the Greenwich prime meridian,you might want to think about changing the centerof the projection.Each map projection has a set of parameters that youmust define. The parameters specify the origin andcustomize a projection for your area of interest.Angular parameters use the geographic coordinatesystem units, while linear parameters use theprojected coordinate system units.Linear parametersFalse easting—A linear value applied to the origin ofthe x-coordinates.False northing—A linear value applied to the originof the y-coordinates.False easting and northing values are usually appliedto ensure that all x or y values are positive. You canalso use the false easting and northing parameters toreduce the range of the x- or y-coordinate values.For example, if you know all y values are greaterthan five million meters, you could apply a falsenorthing of -5,000,000.Scale factor—A unitless value applied to the centerpoint or line of a map projection.The scale factor is usually slightly less than one. TheUTM coordinate system, which uses the TransverseMercator projection, has a scale factor of 0.9996.Rather than 1.0, the scale along the central meridianof the projection is 0.9996. This creates two almostparallel lines approximately 180 kilometers away,where the scale is 1.0. The scale factor reduces theoverall distortion of the projection in the area ofinterest.Angular parametersAzimuth—Defines the center line of a projection.The rotation angle measures east from north. Usedwith the Azimuth cases of the Hotine ObliqueMercator projection.Central meridian—Defines the origin of thex-coordinates.Longitude of origin—Defines the origin of thex-coordinates. The central meridian and longitude oforigin parameters are synonymous.Central parallel—Defines the origin of they-coordinates.Latitude of origin—Defines the origin of they-coordinates. This parameter may not be located atthe center of the projection. In particular, conicprojections use this parameter to set the origin of they-coordinates below the area of the interest. In thatinstance, you don't need to set a false northingparameter to ensure that all y-coordinates arepositive.Longitude of center—Used with the Hotine ObliqueMercator Center (both Two-Point and Azimuth) casesto define the origin of the x-coordinates. Usuallysynonymous with the longitude of origin and centralmeridian parameters.Latitude of center—Used with the Hotine ObliqueMercator Center (both Two-Point and Azimuth) casesto define the origin of the y-coordinates. It is almostalways the center of the projection.Standard parallel 1 and standard parallel 2—Usedwith conic projections to define the latitude lineswhere the scale is 1.0. When defining a LambertConformal Conic projection with one standardparallel, the first standard parallel defines the originof the y-coordinates.For other conic cases, the y-coordinate origin isdefined by the latitude of origin parameter.Longitude of first pointLatitude of first pointLongitude of second pointLatitude of second point20 • Understanding Map Projections

The four parameters above are used with theTwo-Point Equidistant and Hotine Oblique Mercatorprojections. They specify two geographic points thatdefine the center axis of a projection.Projected coordinate systems • 21

3GeographictransformationsThis chapter discusses the various datumtransformation methods including:• Geocentric Translation• Coordinate Frame and Position Vector• Molodensky and Abridged Molodensky• NADCON and HARN• National Transformation version 2 (NTv2)23

GEOGRAPHIC TRANSFORMATION METHODSMoving your data between coordinate systemssometimes includes transforming between thegeographic coordinate systems.These include the Geocentric Translation,Molodensky, and Coordinate Frame methods.Other methods such as NADCON and NTv2 use agrid of differences and convert the longitude–latitudevalues directly.Because the geographic coordinate systems containdatums that are based on spheroids, a geographictransformation also changes the underlying spheroid.There are several methods, which have differentlevels of accuracy and ranges, for transformingbetween datums. The accuracy of a particulartransformation can range from centimeters to metersdepending on the method and the quality andnumber of control points available to define thetransformation parameters.A geographic transformation always convertsgeographic (longitude–latitude) coordinates. Somemethods convert the geographic coordinates togeocentric (X,Y,Z) coordinates, transform the X,Y,Zcoordinates, and convert the new values back togeographic coordinates.The X,Y,Z coordinate system.24 • Understanding Map Projections

EQUATION-BASED METHODSThree-parameter methodsThe simplest datum transformation method is ageocentric, or three-parameter, transformation. Thegeocentric transformation models the differencesbetween two datums in the X,Y,Z coordinate system.One datum is defined with its center at 0,0,0. Thecenter of the other datum is defined at somedistance (∆X,∆Y,∆Z) in meters away.The rotation values are given in decimal seconds,while the scale factor is in parts per million (ppm).The rotation values are defined in two differentways. It’s possible to define the rotation angles aspositive either clockwise or counterclockwise as youlook toward the origin of the X,Y,Z systems.Usually the transformation parameters are defined asgoing ‘from’ a local datum ‘to’ WGS 1984 or anothergeocentric datum.⎡X⎤ ⎡∆X⎤ ⎡X⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢Y⎥ = ⎢∆Y⎥ + ⎢Y⎥⎣⎢Z⎦⎥ Z Znew ⎣⎢∆⎦⎥⎣⎢⎦⎥originalThe three parameters are linear shifts and are alwaysin meters.Seven-parameter methodsA more complex and accurate datum transformationis possible by adding four more parameters to ageocentric transformation. The seven parameters arethree linear shifts (∆X,∆Y,∆Z), three angular rotationsaround each axis (r x,r y,r z), and scale factor(s).⎡X⎤ ⎡∆X⎤ ⎡ 1 rz− r ⎤y ⎡X⎤⎢ ⎥ ⎢ ⎥ ⎢⎥ ⎢ ⎥⎢Y⎥ = ⎢∆Y⎥ + ( 1 + s)⋅⎢−rz1 rx⎥ ⋅ ⎢Y⎥⎣⎢Z⎦⎥ Zr r Znew ⎣⎢∆⎦⎥ ⎢⎣ y −⎥x 1⎦ ⎣⎢⎦⎥originalThe Coordinate Frame (or Bursa–Wolf) definition of therotation values.The equation in the previous column is how theUnited States and Australia define the equations andis called the Coordinate Frame Rotationtransformation. The rotations are positivecounterclockwise. Europe uses a differentconvention called the Position Vector transformation.Both methods are sometimes referred to as theBursa–Wolf method. In the Projection Engine, theCoordinate Frame and Bursa–Wolf methods are thesame. Both Coordinate Frame and Position Vectormethods are supported, and it is easy to converttransformation values from one method to the othersimply by changing the signs of the three rotationvalues. For example, the parameters to convert fromthe WGS 1972 datum to the WGS 1984 datum withthe Coordinate Frame method are (in the order, ∆X,∆Y,∆Z,r x,r y,r z,s):(0.0, 0.0, 4.5, 0.0, 0.0, -0.554, 0.227)To use the same parameters with the Position Vectormethod, change the sign of the rotation so the newparameters are:(0.0, 0.0, 4.5, 0.0, 0.0, +0.554, 0.227)Geographic transformations • 25

Unless explicitly stated, it’s impossible to tell fromthe parameters alone which convention is beingused. If you use the wrong method, your results canreturn inaccurate coordinates. The only way todetermine how the parameters are defined is bychecking a control point whose coordinates areknown in the two systems.Molodensky methodThe Molodensky method converts directly betweentwo geographic coordinate systems without actuallyconverting to an X,Y,Z system. The Molodenskymethod requires three shifts (∆X,∆Y,∆Z) and thedifferences between the semimajor axes (∆a) and theflattenings (∆f) of the two spheroids. The ProjectionEngine automatically calculates the spheroiddifferences according to the datums involved.( M + h)∆ϕ= − sin ϕ cos λ∆X− sin ϕ sin λ∆Y2e sin ϕ cosϕ+ cosϕ∆Z+2 2 1/(1 − e sin ϕ)+ sin ϕ cosϕ(Ma b+ N ) ∆fb a( N + h) cosϕ∆λ= − sin λ∆X+ cos λ∆Y2∆aM and N are the meridional and prime vertical radiiof curvature, respectively, at a given latitude. Theequations for M and N are:a(1− e )M =2 2(1 − e sin ϕ)aN =2 2( 1 − e sin ϕ)23/ 21/ 2You solve for ∆λ and ∆ϕ. The amounts are addedautomatically by the Projection Engine.Abridged Molodensky methodThe Abridged Molodensky method is a simplifiedversion of the Molodensky method. The equationsare:M ∆ϕ= − sin ϕ cos λ∆X− sin ϕ sin λ∆Y+ cosϕ∆Z+ ( a∆f+ f∆a)⋅ 2sin ϕ cosϕN cosϕ∆λ= − sin λ∆X+ cos λ∆Y∆h= cosϕcos λ∆X+ cosϕsin λ ∆Y+ sin ϕ∆Z+ ( a∆f2+ f∆a) sinϕ −∆a∆h= cosϕcos λ ∆X+ cosϕsin λ ∆Y+ sin ϕ ∆Z− (1 − ea(1− f )+2 2(1 − e sin ϕ)1/ 222sin ϕ)21/ 2sin ϕ ∆f∆ahϕλabfeellipsoid height (meters)latitudelongitudesemimajor axis of the spheroid (meters)semiminor axis of the spheroid (meters)flattening of the spheroideccentricity of the spheroid26 • Understanding Map Projections

GRID-BASED METHODSNADCON and HARN methodsThe United States uses a grid-based method toconvert between geographic coordinate systems.Grid-based methods allow you to model thedifferences between the systems and are potentiallythe most accurate method. The area of interest isdivided into cells. The National Geodetic Survey(NGS) publishes grids to convert between NAD 1927and other older geographic coordinate systems andNAD 1983. We group these transformations into theNADCON method. The main NADCON grid, CONUS,converts the contiguous 48 states. The otherNADCON grids convert older geographic coordinatesystems to NAD 1983 for• Alaska• Hawaiian islands• Puerto Rico and Virgin Islands• St. George, St. Lawrence, and St. Paul Islands inAlaskaThe accuracy is around 0.15 meters for thecontiguous states, 0.50 for Alaska and its islands,0.20 for Hawaii, and 0.05 for Puerto Rico and theVirgin Islands. Accuracies can vary depending onhow good the geodetic data in the area was whenthe grids were computed (NADCON, 1999).The Hawaiian islands were never on NAD 1927.They were mapped using several datums that arecollectively known as the Old Hawaiian datums.New surveying and satellite measuring techniqueshave allowed NGS and the states to update thegeodetic control point networks. As each state isfinished, the NGS publishes a grid that convertsbetween NAD 1983 and the more accurate controlpoint coordinates. Originally, this effort was calledthe High Precision Geodetic Network (HPGN). It isnow called the High Accuracy Reference Network(HARN). More than 40 states have published HARNgrids as of September 2000. HARN transformationshave an accuracy around 0.05 meters (NADCON,2000).The difference values in decimal seconds are storedin two files: one for longitude and the other forlatitude. A bilinear interpolation is used to calculatethe exact difference between the two geographiccoordinate systems at a point. The grids are binaryfiles, but a program, NADGRD, from the NGS allowsyou to convert the grids to an American StandardCode for Information Interchange (ASCII) format.Shown at the bottom of the page is the header andfirst ‘row’ of the CSHPGN.LOA file. This is thelongitude grid for Southern California. The format ofthe first row of numbers is, in order, the number ofcolumns, number of rows, number of z values(always one), minimum longitude, cell size,minimum latitude, cell size, and not used.The next 37 values (in this case) are the longitudeshifts from -122° to -113° at 32° N in 0.25° intervalsin longitude.NADCON EXTRACTED REGIONNADGRD37 21 1 -122.00000 .25000 32.00000 .25000 .00000.007383 .004806 .002222 -.000347 -.002868 -.005296-.007570 -.009609 -.011305 -.012517 -.013093 -.012901-.011867 -.009986 -.007359 -.004301 -.001389 .001164.003282 .004814 .005503 .005361 .004420 .002580.000053 -.002869 -.006091 -.009842 -.014240 -.019217-.025104 -.035027 -.050254 -.072636 -.087238 -.099279-.110968A portion of a HARN grid file.Geographic transformations • 27

National Transformation version 2Like the United States, Canada uses a grid-basedmethod to convert between NAD 1927 and NAD1983. The National Transformation version 2 (NTv2)method is quite similar to NADCON. A set of binaryfiles contains the differences between the twogeographic coordinate systems. A bilinearinterpolation is used to calculate the exact values fora point.Unlike NADCON, which can only use one grid at atime, NTv2 is designed to check multiple grids forthe most accurate shift information. A set of lowdensitybase grids exists for Canada. Certain areassuch as cities have high-density local subgrids thatoverlay portions of the base, or parent, grids. If apoint is within one of the high-density grids, NTv2will use the high-density grid; otherwise, the point‘falls through’ to the low-density grid.Australia of 1994 (GDA 1994). Later, the state gridswill be merged into a countrywide grid. NewZealand has released a countrywide grid to convertbetween New Zealand Geodetic Datum of 1949(NZGD 1949) and NZGD 2000.National Transformation version 1Like NADCON, the National Transformationversion 1 (NTv1) uses a single grid to model thedifferences between NAD 1927 and NAD 1983 inCanada. This version is also known as CNT inArcInfo Workstation. The accuracy is within 0.01 mof the actual difference for 74 percent of the pointsand within 0.5 m for 93 percent of the cases.A high-density subgrid with four cells overlaying a low-densitybase grid, also with four cells.If a point falls in the lower-left part of the abovepicture between the stars, the shifts are calculatedwith the high-density subgrid. A point whosecoordinates are anywhere else will have its shiftscalculated with the low-density base grid. Thesoftware automatically calculates which base orsubgrid to use.The parent grids for Canada have spacings rangingfrom five to 20 minutes. The high-density grids areusually cell sizes of 30 seconds.Unlike NADCON grids, NTv2 grids list the accuracyof each point. Accuracy values can range from a fewcentimeters to around a meter. The high-densitygrids usually have subcentimeter accuracy.Australia and New Zealand adopted the NTv2 formatto convert between datums as well. Australia hasreleased several state-based grids that convertbetween either Australian Geodetic Datum of 1966(AGD 1966) or AGD 1984 and Geodetic Datum of28 • Understanding Map Projections

4SupportedmapprojectionsA map projection converts data from theround earth onto a flat plane. Each mapprojection is designed for a specific purposeand distorts the data differently. This chapterwill describe each projection including:• Method• Linear graticules• Limitations• Uses and applications• Parameters29

LIST OF SUPPORTED MAP PROJECTIONSAitoffAlaska GridAlaska Series EAlbers Equal Area ConicAzimuthal EquidistantBehrmann Equal Area CylindricalBipolar Oblique Conformal ConicBonneCassini–SoldnerChamberlin TrimetricCraster ParabolicCylindrical Equal AreaDouble StereographicEckert IEckert IIEckert IIIEckert IVEckert VEckert VIA compromise projection developed in 1889 and used for worldmaps.This projection was developed to provide a conformal map of Alaskawith less scale distortion than other conformal projections.Developed in 1972 by the United States Geological Survey (USGS) topublish a map of Alaska at 1:2,500,000 scale.This conic projection uses two standard parallels to reduce some ofthe distortion of a projection with one standard parallel. Shape andlinear scale distortion are minimized between the standard parallels.The most significant characteristic of this projection is that bothdistance and direction are accurate from the central point.This projection is an equal-area cylindrical projection suitable forworld mapping.This projection was developed specifically for mapping North andSouth America and maintains conformality.This equal-area projection has true scale along the central meridianand all parallels.This transverse cylindrical projection maintains scale along thecentral meridian and all lines parallel to it. This projection is neitherequal area nor conformal.This projection was developed and used by the National GeographicSociety for continental mapping. The distance from three inputpoints to any other point is approximately correct.This pseudocylindrical equal-area projection is primarily used forthematic maps of the world.Lambert first described this equal-area projection in 1772. It is usedinfrequently.This azimuthal projection is conformal.This pseudocylindrical projection is used primarily as a novelty map.A pseudocylindrical equal-area projection.This pseudocylindrical projection is used primarily for world maps.This equal-area projection is used primarily for world maps.This pseudocylindrical projection is used primarily for world maps.This equal-area projection is used primarily for world maps.30 • Understanding Map Projections

Equidistant ConicEquidistant CylindricalEquirectangularGall’s StereographicGauss–KrügerGeocentric Coordinate SystemGeographic Coordinate SystemGnomonicGreat Britain National GridHammer–AitoffHotine Oblique MercatorKrovakLambert Azimuthal Equal AreaLambert Conformal ConicLocal Cartesian ProjectionThis conic projection can be based on one or two standard parallels.As the name implies, all circular parallels are spaced evenly along themeridians.One of the easiest projections to construct because it forms a grid ofequal rectangles.This projection is very simple to construct because it forms a grid ofequal rectangles.The Gall’s Stereographic projection is a cylindrical projectiondesigned around 1855 with two standard parallels at latitudes 45° Nand 45° S.This projection is similar to the Mercator except that the cylinder istangent along a meridian instead of the equator. The result is aconformal projection that does not maintain true directions.The geocentric coordinate system is not a map projection. The earthis modeled as a sphere or spheroid in a right-handed X,Y,Z system.The geographic coordinate system is not a map projection. The earthis modeled as a sphere or spheroid.This azimuthal projection uses the center of the earth as itsperspective point.This coordinate system uses a Transverse Mercator projected on theAiry spheroid. The central meridian is scaled to 0.9996. The origin is49° N and 2° W.The Hammer–Aitoff projection is a modification of the LambertAzimuthal Equal Area projection.This is an oblique rotation of the Mercator projection. Developed forconformal mapping of areas that do not follow a north–south oreast–west orientation but are obliquely oriented.The Krovak projection is an oblique Lambert conformal conicprojection designed for the former Czechoslovakia.This projection preserves the area of individual polygons whilesimultaneously maintaining true directions from the center.This projection is one of the best for middle latitudes. It is similar tothe Albers Conic Equal Area projection except that the LambertConformal Conic projection portrays shape more accurately thanarea.This is a specialized map projection that does not take into accountthe curvature of the earth.Supported map projections • 31

LoximuthalMcBryde–Thomas Flat-Polar QuarticMercatorMiller CylindricalMollweideNew Zealand National GridOrthographicPerspectivePlate CarréePolar StereographicPolyconicQuartic AuthalicRectified Skewed OrthomorphicRobinsonSimple ConicSinusoidalSpace Oblique MercatorState Plane Coordinate System (SPCS)This projection shows loxodromes, or rhumb lines, as straight lineswith the correct azimuth and scale from the intersection of thecentral meridian and the central parallel.This equal-area projection is primarily used for world maps.Originally created to display accurate compass bearings for seatravel. An additional feature of this projection is that all local shapesare accurate and clearly defined.This projection is similar to the Mercator projection except that thepolar regions are not as areally distorted.Carl B. Mollweide created this pseudocylindrical projection in 1805. Itis an equal-area projection designed for small-scale maps.This is the standard projection for large-scale maps of New Zealand.This perspective projection views the globe from an infinite distance.This gives the illusion of a three-dimensional globe.This projection is similar to the Orthographic projection in that itsperspective is from space. In this projection, the perspective point isnot an infinite distance away; instead, you can specify the distance.This projection is very simple to construct because it forms a grid ofequal rectangles.The projection is equivalent to the polar aspect of the Stereographicprojection on a spheroid. The central point is either the North Poleor the South Pole.The name of this projection translates into ‘many cones’ and refers tothe projection methodology.This pseudocylindrical equal-area projection is primarily used forthematic maps of the world.This oblique cylindrical projection is provided with two options forthe national coordinate systems of Malaysia and Brunei.A compromise projection used for world maps.This conic projection can be based on one or two standard parallels.As a world map, this projection maintains equal area despiteconformal distortion.This projection is nearly conformal and has little scale distortionwithin the sensing range of an orbiting mapping satellite such asLandsat.The State Plane Coordinate System is not a projection. It is acoordinate system that divides the 50 states of the United States,32 •Understanding Map Projections

Puerto Rico, and the U.S. Virgin Islands into more than120 numbered sections, referred to as zones.StereographicTimesTransverse MercatorTwo-Point EquidistantUniversal Polar Stereographic (UPS)Universal Transverse Mercator (UTM)Van Der Grinten IVertical Near-Side PerspectiveWinkel IWinkel IIWinkel TripelThis azimuthal projection is conformal.The Times projection was developed by Moir in 1965 forBartholomew Ltd., a British mapmaking company. It is a modifiedGall’s Stereographic, but the Times has curved meridians.Similar to the Mercator except that the cylinder is tangent along ameridian instead of the equator. The result is a conformal projectionthat does not maintain true directions.This modified planar projection shows the true distance from eitherof two chosen points to any other point on a map.This form of the Polar Stereographic maps areas north of 84° N andsouth of 80° S that are not included in the UTM Coordinate System.The projection is equivalent to the polar aspect of the Stereographicprojection of the spheroid with specific parameters.The Universal Transverse Mercator coordinate system is a specializedapplication of the Transverse Mercator projection. The globe isdivided into 60 zones, each spanning six degrees of longitude.This projection is similar to the Mercator projection except that itportrays the world as a circle with a curved graticule.Unlike the Orthographic projection, this perspective projection viewsthe globe from a finite distance. This perspective gives the overalleffect of the view from a satellite.A pseudocylindrical projection used for world maps that averages thecoordinates from the Equirectangular (Equidistant Cylindrical) andSinusoidal projections.A pseudocylindrical projection that averages the coordinates from theEquirectangular and Mollweide projections.A compromise projection used for world maps that averages thecoordinates from the Equirectangular (Equidistant Cylindrical) andAitoff projections.Supported map projections • 33

AITOFFUSES AND APPLICATIONSDeveloped for use in general world maps.Used for the Winkel Tripel projection.The central meridian is 0°.DESCRIPTIONA compromise projection developed in 1889 and foruse with world maps.PROJECTION METHODModified azimuthal. Meridians are equally spacedand concave toward the central meridian. Thecentral meridian is a straight line and half the lengthof the equator. Parallels are equally spaced curves,concave toward the poles.LINEAR GRATICULESThe equator and the central meridian.PROPERTIESShapeDistortion is moderate.AreaModerate distortion.DirectionGenerally distorted.DistanceThe equator and central meridian are at true scale.LIMITATIONSNeither conformal nor equal area. Useful only forworld maps.34 • Understanding Map Projections

ALASKA GRIDDistanceThe minimum scale factor is 0.997 at approximately62°30' N, 156° W. Scale increases outward from thispoint. Most of Alaska and the Aleutian Islands,excluding the panhandle, are bounded by a line oftrue scale. The scale factor ranges from 0.997 to1.003 for Alaska, which is one-fourth the range for acorresponding conic projection (Snyder, 1987).LIMITATIONSDistortion becomes severe away from Alaska.Parameters are set by the software.DESCRIPTIONThis projection was developed to provide aconformal map of Alaska with less scale distortionthan other conformal projections. A set ofmathematical formulas defines a conformaltransformation between two surfaces (Snyder, 1993).USES AND APPLICATIONSConformal mapping of Alaska as a complete state onthe Clarke 1866 spheroid or NAD27. This projectionis not optimized for use with other datums andspheroids.PROJECTION METHODModified planar. This is a sixth-order equationmodification of an oblique Stereographic conformalprojection on the Clarke 1866 spheroid. The origin isat 64° N, 152° W.POINT OF TANGENCYConceptual point of tangency at 64° N, 152° W.LINEAR GRATICULESNone.PROPERTIESShapePerfectly conformal.AreaVaries about 1.2 percent over Alaska.DirectionLocal angles are correct everywhere.Supported map projections• 35

ALASKA SERIES ELIMITATIONSThis projection is appropriate for mapping Alaska,the Aleutian Islands, and the Bering Sea region only.USES AND APPLICATIONS1972 USGS revision of a 1954 Alaska map that waspublished at 1:2,500,000 scale.1974 map of the Aleutian Islands and the Bering Sea.Parameters are set by the software.DESCRIPTIONThis projection was developed in 1972 by the USGSto publish a map of Alaska at 1:2,500,000 scale.PROJECTION METHODApproximates Equidistant Conic, although it iscommonly referred to as a Modified TransverseMercator.LINES OF CONTACTThe standard parallels at 53°30' N and 66°05'24" N.LINEAR GRATICULESThe meridians are straight lines radiating from acenter point. The parallels closely approximateconcentric circular arcs.PROPERTIESShapeNeither conformal nor equal area.AreaNeither conformal nor equal area.DirectionDistortion increases with distance from the standardparallels.DistanceAccurate along the standard parallels.36 • Understanding Map Projections

ALBERS EQUAL A REA CONICparallels are preserved, but because the scale alongthe lines of longitude does not match the scale alongthe lines of latitude, the final projection is notconformal.AreaAll areas are proportional to the same areas on theearth.DirectionLocally true along the standard parallels.The central meridian is 96° W. The first and second standardparallels are 20° N and 60° N, while the latitude of origin is40° N.DESCRIPTIONThis conic projection uses two standard parallels toreduce some of the distortion of a projection withone standard parallel. Although neither shape norlinear scale is truly correct, the distortion of theseproperties is minimized in the region between thestandard parallels. This projection is best suited forland masses extending in an east-to-west orientationrather than those lying north to south.PROJECTION METHODConic. The meridians are equally spaced straightlines converging to a common point. Poles arerepresented as arcs rather than as single points.Parallels are unequally spaced concentric circleswhose spacing decreases toward the poles.LINES OF CONTACTTwo lines, the standard parallels, defined by degreeslatitude.LINEAR GRATICULESAll meridians.PROPERTIESShapeShape along the standard parallels is accurate andminimally distorted in the region between thestandard parallels and those regions just beyond.The 90 degree angles between meridians andDistanceDistances are most accurate in the middle latitudes.Along parallels, scale is reduced between thestandard parallels and increased beyond them. Alongmeridians, scale follows an opposite pattern.LIMITATIONSBest results for regions predominantly east–west inorientation and located in the middle latitudes. Totalrange in latitude from north to south should notexceed 30–35 degrees. No limitations on theeast–west range.USES AND APPLICATIONSUsed for small regions or countries but not forcontinents.Used for the conterminous United States, normallyusing 29°30' and 45°30' as the two standard parallels.For this projection, the maximum scale distortion forthe 48 states is 1.25 percent.One method to calculate the standard parallels is bydetermining the range in latitude in degrees north tosouth and dividing this range by six. The ‘One-SixthRule’ places the first standard parallel at one-sixththe range above the southern boundary and thesecond standard parallel minus one-sixth the rangebelow the northern limit. There are other possibleapproaches.Supported map projections• 37

AZIMUTHAL EQUIDISTANTAreaDistortion increases outward from the center point.DirectionTrue directions from the center outward.DistanceDistances for all aspects are accurate from the centerpoint outward. For the polar aspect, the distancesalong the meridians are accurate, but there is apattern of increasing distortion along the circles oflatitude, outward from the center.The center of the projection is 0°, 0°.DESCRIPTIONThe most significant characteristic is that bothdistance and direction are accurate from the centralpoint. This projection can accommodate all aspects:equatorial, polar, and oblique.PROJECTION METHODPlanar. The world is projected onto a flat surfacefrom any point on the globe. Although all aspectsare possible, the one used most commonly is thepolar aspect, in which all meridians and parallels aredivided equally to maintain the equidistant property.Oblique aspects centered on a city are also common.POINT OF TANGENCYA single point, usually the North or the South Pole,defined by degrees of latitude and longitude.LINEAR GRATICULESPolar—Straight meridians are divided equally byconcentric circles of latitude.Equatorial—The equator and the projection’s centralmeridian are linear and meet at a 90 degree angle.Oblique—The central meridian is straight, but thereare no 90 degree intersections except along thecentral meridian.PROPERTIESShapeExcept at the center, all shapes are distorted.Distortion increases from the center.LIMITATIONSUsually limited to 90 degrees from the center,although it can project the entire globe. Polar-aspectprojections are best for regions within a 30 degreeradius because there is only minimal distortion.Degrees from center:15 30 45 60 90Scale distortion in percent along parallels:1.2 4.7 11.1 20.9 57USES AND APPLICATIONSRoutes of air and sea navigation. These maps willfocus on an important location as their central pointand use an appropriate aspect.Polar aspect—Polar regions and polar navigation.Equatorial aspect—Locations on or near the equatorsuch as Singapore.Oblique aspect—Locations between the poles andthe equator; for example, large-scale mapping ofMicronesia.If this projection is used on the entire globe, theimmediate hemisphere can be recognized andresembles the Lambert Azimuthal projection. Theouter hemisphere greatly distorts shapes and areas.In the extreme, a polar-aspect projection centered onthe North Pole will represent the South Pole as itslargest outermost circle. The function of this extremeprojection is that, regardless of the conformal andarea distortion, an accurate presentation of distanceand direction from the center point is maintained.38 • Understanding Map Projections

BEHRMANN EQUAL AREA CYLINDRICALUSES AND APPLICATIONSOnly useful for world maps.The central meridian is 0°.DESCRIPTIONThis projection is an equal-area cylindrical projectionsuitable for world mapping.PROJECTION METHODCylindrical. Standard parallels are at 30° N and S. Acase of Cylindrical Equal Area.LINES OF CONTACTThe two parallels at 30° N and S.LINEAR GRATICULESMeridians and parallels are linear.PROPERTIESShapeShape distortion is minimized near the standardparallels. Shapes are distorted north–south betweenthe standard parallels and distorted east–west above30° N and below 30° S.AreaArea is maintained.DirectionDirections are generally distorted.DistanceDirections are generally distorted except along theequator.LIMITATIONSUseful for world maps only.Supported map projections• 39

BIPOLAR OBLIQUE CONFORMAL CONICDESCRIPTIONThis projection was developed specifically formapping North and South America. It maintainsconformality. It is based on the Lambert ConformalConic, using two oblique conic projections side byside.PROJECTION METHODTwo oblique conics are joined with the poles104 degrees apart. A great circle arc 104 degreeslong begins at 20° S and 110° W, cuts throughCentral America, and terminates at 45° N andapproximately 19°59'36" W. The scale of the map isthen increased by approximately 3.5 percent. Theorigin of the coordinates is 17°15' N, 73°02' W(Snyder, 1993).display North America and South America only. Ifhaving problems, check all feature types (particularlyannotation and tics) and remove any features thatare beyond the range of the projection.USES AND APPLICATIONSDeveloped in 1941 by the American GeographicalSociety as a low-error single map of North and SouthAmerica.Conformal mapping of North and South America as acontiguous unit.Used by USGS for geologic mapping of NorthAmerica until it was replaced in 1979 by theTransverse Mercator projection.LINES OF CONTACTThe two oblique cones are each conceptually secant.These standard lines do not follow any singleparallel or meridian.LINEAR GRATICULESOnly from each transformed pole to the nearestactual pole.PROPERTIESShapeConformality is maintained except for a slightdiscrepancy at the juncture of the two conicprojections.AreaMinimal distortion near the standard lines, increasingwith distance.DirectionLocal directions are accurate because ofconformality.DistanceTrue along standard lines.LIMITATIONSSpecialized for displaying North and South Americaonly together. The Bipolar Oblique projection will40 • Understanding Map Projections

BONNEDirectionLocally true along central meridian and standardparallel.DistanceScale is true along the central meridian and eachparallel.LIMITATIONSUsually limited to maps of continents or smallerregions. Distortion pattern makes other equal-areaprojections preferable.The central meridian is 0°.DESCRIPTIONThis equal-area projection has true scale along thecentral meridian and all parallels. Equatorial aspect isa Sinusoidal. Polar aspect is a Werner.USES AND APPLICATIONSUsed during the 19th and early 20th century for atlasmaps of Asia, Australia, Europe, and North America.Replaced with the Lambert Azimuthal Equal Areaprojection for continental mapping by Rand McNally& Co. and Hammond, Inc.Large-scale topographic mapping of France andIreland, along with Morocco and some otherMediterranean countries (Snyder, 1993).PROJECTION METHODPseudoconic. Parallels of latitude are equally spacedconcentric circular arcs, marked true to scale formeridians.POINT OF TANGENCYA single standard parallel with no distortion.LINEAR GRATICULESThe central meridian.PROPERTIESShapeNo distortion along the central meridian andstandard parallel; error increases away from theselines.AreaEqual area.Supported map projections• 41

CASSINI–SOLDNERAreaNo distortion along the central meridian. Distortionincreases with distance from the central meridian.DirectionGenerally distorted.DistanceScale distortion increases with distance from thecentral meridian; however, scale is accurate alongthe central meridian and all lines perpendicular tothe central meridian.The center of the projection is 0°, 0°.DESCRIPTIONThis transverse cylindrical projection maintains scalealong the central meridian and all lines parallel to itand is neither equal area nor conformal. It is mostsuited for large-scale mapping of areaspredominantly north–south in extent. Also calledCassini.PROJECTION METHODA transverse cylinder is projected conceptually ontothe globe and is tangent along the central meridian.Cassini–Soldner is analogous to the Equirectangularprojection in the same way Transverse Mercator is tothe Mercator projection. The name Cassini–Soldnerrefers to the more accurate ellipsoidal version,developed in the 19th century and used in thissoftware.LIMITATIONSUsed primarily for large-scale mapping of areas nearthe central meridian. The extent on a spheroid islimited to five degrees to either side of the centralmeridian. Beyond that range, data projected toCassini–Soldner may not project back to the sameposition. Transverse Mercator often is preferred dueto the difficulty in measuring scale and direction onCassini–Soldner.USES AND APPLICATIONSNormally used for large-scale maps of areaspredominantly north–south in extent.Used for the Ordnance Survey of Great Britain andsome German states in the late 19th century. Alsoused in Cyprus, former Czechoslovakia, Denmark,Malaysia, and the former Federal Republic ofGermany.POINT OF TANGENCYConceptually a line, specified as the central meridian.LINEAR GRATICULESThe equator, central meridian, and meridians90 degrees from the central meridian.PROPERTIESShapeNo distortion along the central meridian. Distortionincreases with distance from the central meridian.42 • Understanding Map Projections

CHAMBERLIN T RIMETRICLIMITATIONSThe three selected input points should be widelyspaced near the edge of the map limits.Chamberlin Trimetric can only be used in ArcInfo asan OUTPUT projection because the inverseequations (Chamberlin Trimetric to geographic) havenot been published.You can’t project an ArcInfo grid or lattice toChamberlin Trimetric because the inverse equationsare required.The three points that define the projection are 120° W, 48° N;98° W, 27° N; and 70° W, 45° N.USES AND APPLICATIONSUsed by the National Geographic Society as thestandard map projection for most continents.DESCRIPTIONThis is the standard projection developed and usedby the National Geographic Society for continentalmapping. The distance from three input points toany other point is approximately correct.PROJECTION METHODModified planar.LINEAR GRATICULESNone.PROPERTIESShapeShape distortion is low throughout if the three pointsare placed near the map limits.AreaAreal distortion is low throughout if the three pointsare placed near the map limits.DirectionLow distortion throughout.DistanceNearly correct representation of distance from threewidely spaced points to any other point.Supported map projections• 43

CRASTER PARABOLICDistanceScale is true along latitudes 36°46' N and S. Scale isalso constant along any given latitude and issymmetrical around the equator.LIMITATIONSUseful only as a world map.USES AND APPLICATIONSThematic world maps.The central meridian is 0°.DESCRIPTIONThis pseudocylindrical equal area projection isprimarily used for thematic maps of the world. Alsoknown as Putnins P4.PROJECTION METHODPseudocylindrical.LINEAR GRATICULESThe central meridian is a straight line half as long asthe equator. Parallels are unequally spaced, straightparallel lines perpendicular to the central meridian.Their spacing decreases very gradually as they moveaway from the equator.PROPERTIESShapeFree of distortion at the central meridian at 36°46' Nand S. Distortion increases with distance from thesepoints and is most severe at the outer meridians andhigh latitudes. Interrupting the projection greatlyreduces this distortion.AreaEqual area.DirectionLocal angles are correct at the intersection of36°46' N and S with the central meridian. Direction isdistorted elsewhere.44 • Understanding Map Projections

CYLINDRICAL EQUAL A READirectionLocal angles are correct along standard parallels orstandard lines. Direction is distorted elsewhere.DistanceScale is true along the equator. Scale distortion issevere near the poles.LIMITATIONSRecommended for narrow areas extending along thecentral line. Severe distortion of shape and scalenear the poles.The central meridian is 0°, and the standard parallel is 40° N.The opposite parallel, 40° S, is also a standard parallel.DESCRIPTIONLambert first described this equal area projection in1772. It has been used infrequently.USES AND APPLICATIONSSuitable for equatorial regions.PROJECTION METHODA normal perspective projection onto a cylindertangent at the equator.POINTS OF INTERSECTIONThe equator.LINEAR GRATICULESIn the normal, or equatorial aspect, all meridians andparallels are perpendicular straight lines. Meridiansare equally spaced and 0.32 times the length of theequator. Parallels are unequally spaced and farthestapart near the equator. Poles are lines of lengthequal to the equator.PROPERTIESShapeShape is true along the standard parallels of thenormal aspect. Distortion is severe near the poles ofthe normal aspect.AreaThere is no area distortion.Supported map projections• 45

DOUBLE STEREOGRAPHICOblique aspect—The central meridian and parallel oflatitude with the opposite sign of the central latitude.PROPERTIESShapeConformal. Local shapes are accurate.AreaTrue scale at center with distortion increasing as youmove away from the center.DirectionDirections are accurate from the center. Local anglesare accurate everywhere.The Rijksdriehoekstelsel coordinate system is used in theNetherlands. The central meridian is 5°23'15.5" E. The latitudeof origin is 52°09'22.178" N. The scale factor is 0.9999079. Thefalse easting is 155,000 meters, and the false northing is463,000 meters.DESCRIPTIONA conformal projection.PROJECTION METHODPlanar perspective projection, viewed from the pointon the globe opposite the point of tangency. Pointsare transformed from the spheroid to a Gaussiansphere before being projected to the plane.DistanceScale increases with distance from the center.LIMITATIONSNormally limited to one hemisphere. Portions of theouter hemisphere may be shown, but with rapidlyincreasing distortion.USES AND APPLICATIONSUsed for large-scale coordinate systems in NewBrunswick and the Netherlands.All meridians and parallels are shown as circular arcsor straight lines. Graticular intersections are90 degrees. In the equatorial aspect, the parallelscurve in opposite directions on either side of theequator. In the oblique case, only the parallel withthe opposite sign to the central latitude is a straightline; other parallels are concave toward the poles oneither side of the straight parallel.POINT OF CONTACTA single point anywhere on the globe.LINEAR GRATICULESPolar aspect—All meridians.Equatorial aspect—The central meridian and theequator.46 • Understanding Map Projections

ECKERT IThe central meridian is 0°.DESCRIPTIONUsed primarily as a novelty map.PROJECTION METHODA pseudocylindrical projection.LINEAR GRATICULESParallels and meridians are equally spaced straightlines. The poles and the central meridian are straightlines half as long as the equator.PROPERTIESShapeShape isn’t preserved.AreaArea isn’t preserved.DirectionDirection is distorted everywhere.DistanceScale is correct along 47°10' N and S.LIMITATIONSDiscontinuities exist at the equator.USES AND APPLICATIONSUseful only as a novelty.Supported map projections• 47

ECKERT IIThe central meridian is 100° W.DESCRIPTIONA pseudocylindrical equal-area projection.PROJECTION METHODA pseudocylindrical projection.Parallels are unequally spaced straight lines.Meridians are equally spaced straight lines. Thepoles and the central meridian are straight lines halfas long as the equator.PROPERTIESShapeShape isn’t preserved.AreaArea is preserved.DirectionDirection is distorted everywhere.DistanceScale is correct along 55°10' N and S.LIMITATIONSDiscontinuities exist at the equator.USES AND APPLICATIONSUseful only as a novelty.48 • Understanding Map Projections

ECKERT IIILIMITATIONSUseful only as a world map.USES AND APPLICATIONSSuitable for thematic mapping of the world.The central meridian is 0°.DESCRIPTIONThis pseudocylindrical projection is used primarilyfor world maps.PROJECTION METHODA pseudocylindrical projection.LINEAR GRATICULESParallels are equally spaced straight lines. Meridiansare equally spaced elliptical curves. The meridians at+/-180° from the central meridian are semicircles.The poles and the central meridian are straight lineshalf as long as the equator.PROPERTIESShapeThis stretching decreases to zero at 37°55' N and S.Nearer the poles, features are compressed in thenorth–south direction.AreaArea isn’t preserved.DirectionThe equator doesn’t have any angular distortion.Direction is distorted elsewhere.DistanceScale is correct only along 37°55' N and S. Nearer thepoles, features are compressed in the north–southdirection.Supported map projections• 49

ECKERT IVparallels. Nearer the poles, features are compressedin the north–south direction.LIMITATIONSUseful only as a world map.USES AND APPLICATIONSThematic maps of the world such as climate.The central meridian is 0°.DESCRIPTIONThis equal area projection is used primarily for worldmaps.PROJECTION METHODA pseudocylindrical equal-area projection.LINEAR GRATICULESParallels are unequally spaced straight lines, closertogether at the poles. Meridians are equally spacedelliptical arcs. The poles and the central meridian arestraight lines half as long as the equator.PROPERTIESShapeShapes are stretched north–south 40 percent alongthe equator, relative to the east–west dimension. Thisstretching decreases to zero at 40°30' N and S at thecentral meridian. Nearer the poles, features arecompressed in the north–south direction.AreaEquivalent.DirectionLocal angles are correct at the intersections of40°30' N and S with the central meridian. Direction isdistorted elsewhere.DistanceScale is distorted north–south 40 percent along theequator relative to the east–west dimension. Thisdistortion decreases to zero at 40°30' N and S at thecentral meridian. Scale is correct only along these50 • Understanding Map Projections

ECKERT VUSES AND APPLICATIONSSuitable for thematic mapping of the world.The central meridian is 89° E.DESCRIPTIONThis pseudocylindrical projection is used primarilyfor world maps.PROJECTION METHODA pseudocylindrical projection.LINEAR GRATICULESParallels are equally spaced straight lines. Meridiansare equally spaced sinusoidal curves. The poles andthe central meridian are straight lines half as long asthe equator.PROPERTIESShapeThis stretching decreases to zero at 37°55' N and S.Nearer the poles, features are compressed in thenorth–south direction.AreaArea isn’t preserved.DirectionThe equator doesn’t have any angular distortion.Direction is distorted elsewhere.DistanceScale is correct only along 37°55' N and S. Nearer thepoles, features are compressed in the north–southdirection.LIMITATIONSUseful only as a world map.Supported map projections• 51

ECKERT VIcentral meridian. Scale is correct only along theseparallels. Nearer the poles, features are compressedin the north–south direction.LIMITATIONSUseful only as a world map.USES AND APPLICATIONSSuitable for thematic mapping of the world.The central meridian is 0°.Used for world distribution maps in the 1937 WorldAtlas by the Soviet Union.DESCRIPTIONThis equal-area projection is used primarily for worldmaps.PROJECTION METHODA pseudocylindrical equal-area projection.LINEAR GRATICULESParallels are unequally spaced straight lines. Theyare closer together at the poles. Meridians areequally spaced sinusoidal curves. The poles and thecentral meridian are straight lines half as long as theequator.PROPERTIESShapeShapes are stretched north–south 29 percent alongthe equator, relative to the east–west dimension. Thisstretching decreases to zero at 49°16' N and S at thecentral meridian. Nearer the poles, features arecompressed in the north–south direction.AreaEquivalent.DirectionLocal angles are correct at the intersection of49°16' N and S with the central meridian. Direction isdistorted elsewhere.DistanceScale is distorted north–south 29 percent along theequator relative to the east–west dimension. Thisdistortion decreases to zero at 49°16' N and S at the52 • Understanding Map Projections

EQUIDISTANT CONICUse the Equirectangular projection if the standardparallel is the equator.LINES OF CONTACTDepends on the number of standard parallels.Tangential projections (Type 1)—One line, indicatedby the standard parallel.Secant projections (Type 2)—Two lines, specified asthe first and second standard parallels.LINEAR GRATICULESAll meridians.PROPERTIESThe central meridian is 60° W. The first and second standardparallels are 5° S and 42° S. The latitude of origin is 32° S.DESCRIPTIONThis conic projection can be based on one or twostandard parallels. As its name implies, all circularparallels are spaced evenly along the meridians. Thisis true whether one or two parallels are used as thestandards.PROJECTION METHODCone is tangential if one standard parallel is specifiedand secant if two standard parallels are specified.Graticules are evenly spaced. Meridian spacing isequal, as is the space between each of theconcentric arcs that describe the lines of latitude.The poles are represented as arcs rather than points.If the pole is given as the single standard parallel,the cone becomes a plane and the resultingprojection is the same as a polar AzimuthalEquidistant.If two standard parallels are placed symmetricallynorth and south of the equator, the resultingprojection is the same as the Equirectangularprojection. In this case, you must use theEquirectangular projection.ShapeLocal shapes are true along the standard parallels.Distortion is constant along any given parallel butincreases with distance from the standard parallels.AreaDistortion is constant along any given parallel butincreases with distance from the standard parallels.DirectionLocally true along the standard parallels.DistanceTrue along the meridians and the standard parallels.Scale is constant along any given parallel butchanges from parallel to parallel.LIMITATIONSRange in latitude should be limited to 30 degrees.USES AND APPLICATIONSRegional mapping of midlatitude areas with apredominantly east–west extent.Common for atlas maps of small countries.Used by the former Soviet Union for mapping theentire country.Supported map projections• 53

EQUIDISTANT CYLINDRICALLINES OF CONTACTTangent at the equator or secant at two parallelssymmetrical about the equator.LINEAR GRATICULESAll meridians and all parallels.PROPERTIESShapeDistortion increases as the distance from thestandard parallels increases.AreaDistortion increases as the distance from thestandard parallels increases.The central meridian is 0°.DESCRIPTIONAlso known as Equirectangular, Simple Cylindrical,Rectangular, or Plate Carrée (if the standard parallelis the equator).This projection is very simple to construct because itforms a grid of equal rectangles. Because of itssimple calculations, its usage was more common inthe past. In this projection, the polar regions are lessdistorted in scale and area than they are in theMercator projection.PROJECTION METHODThis simple cylindrical projection converts the globeinto a Cartesian grid. Each rectangular grid cell hasthe same size, shape, and area. All the graticularintersections are 90 degrees. The central parallel maybe any line, but the traditional Plate Carréeprojection uses the equator. When the equator isused, the grid cells are perfect squares, but if anyother parallel is used, the grids become rectangular.In this projection, the poles are represented asstraight lines across the top and bottom of the grid.DirectionNorth, south, east, and west directions are accurate.General directions are distorted, except locally alongthe standard parallels.DistanceThe scale is correct along the meridians and thestandard parallels.LIMITATIONSNoticeable distortion of all properties away fromstandard parallels.USES AND APPLICATIONSBest used for city maps or other small areas withmap scales large enough to reduce the obviousdistortion.Used for simple portrayals of the world or regionswith minimal geographic data. This makes theprojection useful for index maps.54 • Understanding Map Projections

EQUIRECTANGULARAreaDistortion increases as the distance from thestandard parallels increases.DirectionNorth, south, east, and west directions are accurate.General directions are distorted, except locally alongthe standard parallels.The central meridian is 149° W.DESCRIPTIONAlso known as Simple Cylindrical, EquidistantCylindrical, Rectangular, or Plate Carrée (if thestandard parallel is the equator).This projection is very simple to construct because itforms a grid of equal rectangles. Because of itssimple calculations, its usage was more common inthe past. In this projection, the polar regions are lessdistorted in scale and area than they are in theMercator projection.PROJECTION METHODThis simple cylindrical projection converts the globeinto a Cartesian grid. Each rectangular grid cell hasthe same size, shape, and area. All the graticularintersections are 90 degrees. The central parallel maybe any line, but the traditional Plate Carréeprojection uses the equator. When the equator isused, the grid cells are perfect squares, but if anyother parallel is used, the grids become rectangular.In this projection, the poles are represented asstraight lines across the top and bottom of the grid.DistanceThe scale is correct along the meridians and thestandard parallels.LIMITATIONSNoticeable distortion of all properties away fromstandard parallels.USES AND APPLICATIONSBest used for city maps or other small areas withmap scales large enough to reduce the obviousdistortion.Used for simple portrayals of the world or regionswith minimal geographic data. This makes theprojection useful for index maps.LINES OF CONTACTTangent at the equator or secant at two parallelssymmetrical around the equator.LINEAR GRATICULESAll meridians and all parallels.PROPERTIESShapeDistortion increases as the distance from thestandard parallels increases.Supported map projections• 55

GALL’S STEREOGRAPHICAreaArea is true at latitudes 45° N and S. Distortionslowly increases away from these latitudes andbecomes severe at the poles.DirectionLocally correct at latitudes 45° N and S. Generallydistorted elsewhere.The central meridian is 176° E.DESCRIPTIONGall’s Stereographic was designed around 1855. It isa cylindrical projection with two standard parallels atlatitudes 45° N and S.DistanceScale is true in all directions along latitudes 45° Nand S. Scale is constant along parallels and issymmetrical around the equator. Distances arecompressed between latitudes 45° N and S andexpanded beyond them.LIMITATIONSUsed only for world maps.USES AND APPLICATIONSUsed for world maps in British atlases.PROJECTION METHODCylindrical stereographic projection based on twostandard parallels at 45° N and S. The globe isprojected perspectively onto a secant cylinder fromthe point on the equator opposite a given meridian.Meridians are equally spaced straight lines. Parallelsare straight lines with spacing increasing away fromthe equator. Poles are straight lines.LINES OF CONTACTTwo lines at 45° N and S.LINEAR GRATICULESAll meridians and parallels.PROPERTIESShapeShapes are true at latitudes 45° N and S. Distortionslowly increases away from these latitudes andbecomes severe at the poles.56 • Understanding Map Projections

GAUSS–KRÜGERDESCRIPTIONAlso known as Transverse Mercator.This projection is similar to the Mercator except thatthe cylinder is longitudinal along a meridian insteadof the equator. The result is a conformal projectionthat does not maintain true directions. The centralmeridian is placed on the region to be highlighted.This centering minimizes distortion of all propertiesin that region. This projection is best suited for landmasses that stretch north–south. TheGauss–Krüger (GK) coordinate system is based onthe Gauss–Krüger projection.PROJECTION METHODCylindrical projection with central meridian placed ina particular region.LINES OF CONTACTAny single meridian for the tangent projection. Forthe secant projection, two parallel lines equidistantfrom the central meridian.LINEAR GRATICULESThe equator and the central meridian.LIMITATIONSData on a spheroid or an ellipsoid cannot beprojected beyond 90 degrees from the centralmeridian. In fact, the extent on a spheroid orellipsoid should be limited to 10 to 12 degrees onboth sides of the central meridian. Beyond thatrange, data projected may not project back to thesame position. Data on a sphere does not have theselimitations.USES AND APPLICATIONSGauss–Krüger coordinate system. Gauss–Krügerdivides the world into zones six degrees wide. Eachzone has a scale factor of 1.0 and a false easting of500,000 meters. The central meridian of zone 1 is at3° E. Some places also add the zone number timesone million to the 500,000 false easting value. GKzone 5 could have a false easting value of 500,000 or5,500,000 meters.The UTM system is very similar. The scale factor is0.9996, and the central meridian of UTM zone 1 is at177° W. The false easting value is 500,000 meters,and southern hemisphere zones also have a falsenorthing of 10,000,000.PROPERTIESShapeConformal. Small shapes are maintained. Shapes oflarger regions are increasingly distorted away fromthe central meridian.AreaDistortion increases with distance from the centralmeridian.DirectionLocal angles are accurate everywhere.DistanceAccurate scale along the central meridian if the scalefactor is 1.0. If it is less than 1.0, then there are twostraight lines having an accurate scale, equidistantfrom and on each side of the central meridian.Supported map projections• 57

GEOCENTRIC COORDINATE SYSTEMGeographic coordinates are described as X,Y, and Z values in ageocentric coordinate system.DESCRIPTIONThe geocentric coordinate system is not a mapprojection. The earth is modeled as a sphere orspheroid in a right-handed X,Y,Z system.The X-axis points to the prime meridian, the Y-axispoints 90 degrees away in the equatorial plane, andthe Z-axis points in the direction of the North Pole.USES AND APPLICATIONSThe geocentric coordinate system is used internallyas an interim system for several geographic (datum)transformation methods.58 • Understanding Map Projections

GEOGRAPHIC COORDINATE SYSTEMGeographic coordinates displayed as if the longitude–latitudevalues are linear units. An equivalent projection isEquirectangular with the standard parallel set to the equator.DESCRIPTIONThe geographic coordinate system is not a mapprojection. The earth is modeled as a sphere orspheroid. The sphere is divided into equal partsusually called degrees; some countries use grads. Acircle is 360 degrees or 400 grads. Each degree issubdivided into 60 minutes, with each minutecomposed of 60 seconds.The geographic coordinate system consists oflatitude and longitude lines. Each line of longituderuns north–south and measures the number ofdegrees east or west of the prime meridian. Valuesrange from -180 to +180 degrees. Lines of latituderun east–west and measure the number of degreesnorth or south of the equator. Values range from +90degrees at the North Pole to -90 degrees at the SouthPole.The standard origin is where the Greenwich primemeridian meets the equator. All points north of theequator or east of the prime meridian are positive.USES AND APPLICATIONSMap projections use latitude and longitude values toreference parameters such as the central meridian,the standard parallels, and the latitude of origin.Supported map projections• 59

GNOMONICPROPERTIESShapeIncreasingly distorted from the center; moderatedistortion within 30 degrees of the center point.AreaDistortion increases with distance from the center;moderate distortion within a 30 degree radius of thecenter.DirectionAccurate from the center.DistanceNo line has an accurate scale, and the amount ofdistortion increases with distance from the center.The central meridian is 0°, and the latitude of origin is 90° S.DESCRIPTIONThis azimuthal projection uses the center of the earthas its perspective point. All great circles are straightlines, regardless of the aspect. This is a usefulprojection for navigation because great circleshighlight routes with the shortest distance.PROJECTION METHODThis is a planar perspective projection viewed fromthe center of the globe. The projection can be anyaspect.POINT OF TANGENCYA single point anywhere on the globe.Polar aspect—North Pole or South Pole.Equatorial aspect—Any point along the equator.Oblique aspect—Any other point.LINEAR GRATICULESAll meridians and the equator.Scalar Distortion for Polar AspectDegrees from Center (°) 15.0 30.0 45.0 60.0Meridian Distortion (%) 7.2 33.3 100.0 300.0Latitude Distortion (%) 3.5 15.5 41.4 100.0LIMITATIONSThis projection is limited by its perspective point andcannot project a line that is 90 degrees or more fromthe center point; this means that the equatorialaspect cannot project the poles and the polar aspectscannot project the equator.A radius of 30 degrees produces moderate distortion,as indicated in the table above. This projectionshould not be used more than about 60 degreesfrom the center.USES AND APPLICATIONSAll aspects—Routes of navigation for sea and air.Polar aspect—Navigational maps of polar regions.Equatorial aspect—Navigational maps of Africa andthe tropical region of South America.60 • Understanding Map Projections

GREAT BRITAIN NATIONAL GRIDDirectionLocal directions are accurately maintained.DistanceScale is accurate along the lines of secancy 180 kmfrom the central meridian. Scale is compressedbetween them and expanded beyond them.LIMITATIONSSuitable for Great Britain. Limited in east–westextent.USES AND APPLICATIONSThe national coordinate system for Great Britain;used for large-scale topographic mapping.The central meridian is 2° W, and the latitude of origin is49° N. The scale factor is 0.9996.DESCRIPTIONThis is a Transverse Mercator projected on the Airyspheroid. The central meridian is scaled to 0.9996.The origin is 49° N and 2° W.PROJECTION METHODCylindrical, transverse projection with the centralmeridian centered along a particular region.LINES OF CONTACTTwo lines parallel with and 180 km from the centralmeridian at 2° W.LINEAR GRATICULESThe central meridian.PROPERTIESShapeConformal; therefore, small shapes are maintainedaccurately.AreaDistortion increases beyond Great Britain as thedistance from the central meridian increases.Supported map projections• 61

HAMMER–AITOFFLIMITATIONSUseful only as a world map.USES AND APPLICATIONSThematic maps of the whole world.The central meridian is 0°.DESCRIPTIONThe Hammer–Aitoff projection is a modification ofthe Lambert Azimuthal Equal Area projection.PROJECTION METHODModified azimuthal. The central meridian is a straightline half as long as the equator. The other meridiansare complex curves, concave toward the centralmeridian and unequally spaced along the equator.The equator is a straight line; all other parallels arecomplex curves, concave toward the nearest poleand unequally spaced along the central meridian.POINT OF TANGENCYCentral meridian at the equator.LINEAR GRATICULESThe equator and central meridian are the onlystraight lines.PROPERTIESShapeDistortion increases away from the origin.AreaEqual area.DirectionLocal angles are true only at the center.DistanceScale decreases along the equator and centralmeridian as distance from the origin increases.62 • Understanding Map Projections

HOTINE OBLIQUE MERCATORLINEAR GRATICULESTwo meridians 180 degrees apart.PROPERTIESShapeConformal. Local shapes are true.AreaDistortion increases with distance from the centralline.DirectionLocal angles are correct.The State Plane Coordinate System uses Hotine Azimuth NaturalOrigin for the Alaskan panhandle.DESCRIPTIONAlso known as Oblique Cylindrical Orthomorphic.This is an oblique rotation of the Mercatorprojection. Developed for conformal mapping ofareas that are obliquely oriented and do not follow anorth–south or east–west trend.PROJECTION METHODCylindrical. Oblique aspect of the Mercatorprojection. Oblique Mercator has several differenttypes. You can define the tilt of the projection byeither specifying two points or a point and an anglemeasuring east of north (the azimuth).DistanceTrue along the chosen central line.LIMITATIONSUse should be limited to regions near the centralline. When using an ellipsoid, constant scale alongthe central line and perfect conformality cannot bemaintained simultaneously.USES AND APPLICATIONSIdeal for conformal mapping of regions that have anoblique orientation.Used for large-scale mapping in the Alaskanpanhandle. Switzerland uses a differentimplementation of Oblique Mercator by Rosenmund,while Madagascar uses the Laborde version. Theseimplementations aren’t compatible.By default, the coordinate origin of the projectedcoordinates is located where the central line of theprojection crosses the equator. As an example, if youuse an Oblique Mercator (natural origin) for WestVirginia, while the center of the projection is -80.75,38.5, the natural origin is approximately -112.8253,0.0. You can move the projection origin to the centerof your data by using the Two-Point Center orAzimuth Center cases.LINE OF TANGENCYA single oblique great-circle line or secancy alongtwo oblique small circles parallel to and equidistantfrom the central great circle.Supported map projections• 63

KROVAKDirectionLocal angles are accurate throughout because ofconformality.DistanceMinimal distortion within the boundaries of thecountries.LIMITATIONSDesigned strictly for Czech Republic and Slovakia.This example of the Krovak projection uses a right-handedcoordinate system.USES AND APPLICATIONSUsed for topographic and other mapping in CzechRepublic and Slovakia. The coordinates are usuallypositive to the south and west.DESCRIPTIONThis projection is an oblique case of the Lambertconformal conic projection and was designed in1922 by Josef Krovak. Used in the Czech Republicand Slovakia. Also known as S-JTSK.PROJECTION METHODConic projection based on one standard parallel. Anazimuth parameter tilts the apex of the cone fromthe North Pole to create a new coordinate system. Astandard parallel in the new system, called a pseudostandardparallel, defines the shape of the cone. Ascale factor is applied to the pseudo-standard parallelto create a secant case.LINES OF CONTACTTwo pseudo-standard parallels.LINEAR GRATICULESNone.PROPERTIESShapeSmall shapes are maintained.AreaMinimal distortion within the boundaries of thecountries.64 • Understanding Map Projections

LAMBERT AZIMUTHAL EQUAL AREAPROPERTIESShapeShape is minimally distorted, less than 2 percent,within 15 degrees from the focal point. Beyond that,angular distortion is more significant; small shapesare compressed radially from the center andelongated perpendicularly.AreaEqual area.DirectionTrue direction radiating from the central point.The central meridian is 0°, and the latitude of origin is 90° S.DESCRIPTIONThis projection preserves the area of individualpolygons while simultaneously maintaining a truesense of direction from the center. The generalpattern of distortion is radial. It is best suited forindividual land masses that are symmetricallyproportioned, either round or square.PROJECTION METHODPlanar, projected from any point on the globe. Thisprojection can accommodate all aspects: equatorial,polar, and oblique.POINT OF TANGENCYA single point, located anywhere, specified bylongitude and latitude.LINEAR GRATICULESAll aspects—The central meridian defining the pointof tangency.DistanceTrue at center. Scale decreases with distance fromthe center along the radii and increases from thecenter perpendicularly to the radii.LIMITATIONSThe data must be less than a hemisphere in extent.The software cannot process any area more than90 degrees from the central point.USES AND APPLICATIONSPopulation density (area).Political boundaries (area).Oceanic mapping for energy, minerals, geology, andtectonics (direction).This projection can handle large areas; thus it is usedfor displaying entire continents and polar regions.Equatorial aspect Africa, Southeast Asia, Australia,the Caribbeans, and CentralAmericaOblique aspect North America, Europe, and AsiaEquatorial aspect—The equator.Polar aspect—All meridians.Supported map projections• 65

LAMBERT CONFORMAL CONICAreaMinimal distortion near the standard parallels. Arealscale is reduced between standard parallels andincreased beyond them.DirectionLocal angles are accurate throughout because ofconformality.DistanceCorrect scale along the standard parallels. The scaleis reduced between the parallels and increasedbeyond them.The central meridian is 125° E. The first and second standardparallels are 32° S and 7° N, while the latitude of origin is32° S.DESCRIPTIONThis projection is one of the best for middlelatitudes. It is similar to the Albers Conic Equal Areaprojection except that Lambert Conformal Conicportrays shape more accurately than area. The StatePlane Coordinate System uses this projection for alleast–west zones.PROJECTION METHODConic projection normally based on two standardparallels, making it a secant projection. The latitudespacing increases beyond the standard parallels. Thisis the only common conic projection that representsthe poles as a single point.LIMITATIONSBest for regions predominantly east–west in extentand located in the middle north or south latitudes.Total latitude range should not exceed 35 degrees.USES AND APPLICATIONSSPCS for all east–west zones.USGS 7½-minute quad sheets to match the StatePlane Coordinate System.Used for many new USGS maps created after 1957. Itreplaced the Polyconic projection.Continental United States: standard parallels, 33° and45° N.Entire United States: standard parallels, 37° and65° N.LINES OF CONTACTThe two standard parallels.LINEAR GRATICULESAll meridians.PROPERTIESShapeAll graticular intersections are 90 degrees. Smallshapes are maintained.66 • Understanding Map Projections

LOCAL CARTESIAN PROJECTIONDESCRIPTIONThis is a specialized map projection that does nottake into account the curvature of the earth. It’sdesigned for very large-scale mapping applications.PROJECTION METHODThe coordinates of the center of the area of interestdefine the origin of the local coordinate system. Theplane is tangent to the spheroid at that point, andthe differences in z values are negligible betweencorresponding points on the spheroid and the plane.Because the differences in z values are ignored,distortions will greatly increase beyond roughly onedegree from the origin.USES AND APPLICATIONSLarge-scale mapping. Should not be used for areasgreater than one degree from the origin.Supported map projections• 67

LOXIMUTHALDistanceScale is true along the central meridian. It is constantalong any latitude. The opposite latitude has adifferent scale if the central parallel isn’t the equator.LIMITATIONSUseful only to show loxodromes.USES AND APPLICATIONSSuitable for displaying loxodromes.The central meridian is 100° W. The central parallel is 60° N.DESCRIPTIONKarl Siemon created this pseudocylindrical projectionin 1935. This projection was also presented in 1966by Waldo Tobler. Loxodromes, or rhumb lines, areshown as straight lines with the correct azimuth andscale from the intersection of the central meridianand the central parallel.PROJECTION METHODPseudocylindrical. All parallels are straight lines, andall meridians are equally spaced arcs except thecentral meridian, which is a straight line. The polesare points.LINEAR GRATICULESThe parallels and central meridian.PROPERTIESShapeShape is generally distorted. As the value of thecentral parallel increases from the equator, theoverall shape of the world becomes more distorted.AreaGenerally distorted.DirectionDirections are true only at the intersection of thecentral meridian and central latitude. Direction isdistorted elsewhere.68 • Understanding Map Projections

MCBRYDE–THOMAS FLAT-POLAR QUARTICLIMITATIONSUseful only as a world map.USES AND APPLICATIONSThematic maps of the world.The central meridian is 0°.DESCRIPTIONThis equal-area projection is primarily used for worldmaps.PROJECTION METHODA pseudocylindrical equal-area projection in whichall parallels are straight lines and all meridians,except the straight central meridian, are equallyspaced, fourth-order (quartic) curves.LINEAR GRATICULESAll parallels are unequally spaced straight lines thatare closer together at the poles. The poles arestraight lines one-third as long as the equator. Thecentral meridian is a straight line 0.45 times as longas the equator.PROPERTIESShapeShapes are stretched north–south along the equator,relative to the east–west dimension. This stretchingdecreases to zero at 33°45' N and S at the centralmeridian. Nearer the poles, features are compressedin the north–south direction.AreaEqual area.DirectionDistorted except at the intersection of 33°45' N and Sand the central meridian.DistanceScale is distorted everywhere except along 33°45' Nand S.Supported map projections• 69

MERCATORGreenland is only one-eighth the size of SouthAmerica, Greenland appears to be larger.DirectionAny straight line drawn on this projection representsan actual compass bearing. These true direction linesare rhumb lines and generally do not describe theshortest distance between points.DistanceScale is true along the equator or along the secantlatitudes.The central meridian is 0°.DESCRIPTIONOriginally created to display accurate compassbearings for sea travel. An additional feature of thisprojection is that all local shapes are accurate andclearly defined.PROJECTION METHODCylindrical projection. Meridians are parallel to eachother and equally spaced. The lines of latitude arealso parallel but become farther apart toward thepoles. The poles cannot be shown.LINES OF CONTACTThe equator or two latitudes symmetrical around theequator.LIMITATIONSThe poles cannot be represented on the Mercatorprojection. All meridians can be projected, but theupper and lower limits of latitude are approximately80° N and S. Large area distortion makes theMercator projection unsuitable for generalgeographic world maps.USES AND APPLICATIONSStandard sea navigation charts (direction).Other directional uses: air travel, wind direction,ocean currents.Conformal world maps.The best use of this projection’s conformal propertiesapplies to regions near the equator such asIndonesia and parts of the Pacific Ocean.LINEAR GRATICULESAll meridians and all parallels.PROPERTIESShapeConformal. Small shapes are well representedbecause this projection maintains the local angularrelationships.AreaIncreasingly distorted toward the polar regions. Forexample, in the Mercator projection, although70 • Understanding Map Projections

MILLER CYLINDRICALDirectionLocal angles are correct only along the equator.DistanceCorrect distance is measured along the equator.LIMITATIONSUseful only as a world map.USES AND APPLICATIONSGeneral-purpose world maps.The central meridian is 118° W.DESCRIPTIONThis projection is similar to the Mercator projectionexcept that the polar regions are not as areallydistorted. Spacing between lines of latitude as theyapproach the poles is less than in the Mercatorprojection. It decreases the distortion in area, but thecompromise introduces distortion in local shape anddirection.PROJECTION METHODCylindrical projection. Meridians are parallel andequally spaced, lines of latitude are parallel, and thedistance between them increases toward the poles.Both poles are represented as straight lines.LINE OF CONTACTThe equator.LINEAR GRATICULESAll meridians and all parallels.PROPERTIESShapeMinimally distorted between 45th parallels,increasingly toward the poles. Land masses arestretched more east–west than they are north–south.AreaDistortion increases from the equator toward thepoles.Supported map projections• 71

MOLLWEIDEDistanceScale is true along latitudes 40°44' N and S.Distortion increases with distance from these linesand becomes severe at the edges of the projection.LIMITATIONSUseful only as a world map.The central meridian is 65° E.DESCRIPTIONAlso called Babinet, Elliptical, Homolographic, orHomalographic.USES AND APPLICATIONSSuitable for thematic or distribution mapping of theentire world, frequently in interrupted form.Combined with the Sinusoidal to create Goode’sHomolosine and Boggs.Carl B. Mollweide created this pseudocylindricalprojection in 1805. It is an equal-area projectiondesigned for small-scale maps.PROJECTION METHODPseudocylindrical equal-area projection. All parallelsare straight lines, and all meridians are equallyspaced elliptical arcs. The exception is the centralmeridian, which is a straight line. The poles arepoints.LINEAR GRATICULESThe equator and central meridian.PROPERTIESShapeShape is not distorted at the intersection of thecentral meridian and latitudes 40°44' N and S.Distortion increases outward from these points andbecomes severe at the edges of the projection.AreaEqual area.DirectionLocal angles are true only at the intersection of thecentral meridian and latitudes 40°44' N and S.Direction is distorted elsewhere.72 • Understanding Map Projections

NEW ZEALAND NATIONAL GRIDAreaDistortion is less than 0.04 percent for New Zealand.DirectionMinimal distortion within New Zealand.DistanceScale is within 0.02 percent of true scale for NewZealand.LIMITATIONSNot useful for areas outside New Zealand.USES AND APPLICATIONSUsed for large-scale maps of New Zealand.The central meridian is 173° E, and the latitude of origin is41° S. The false easting is 2,510,000 meters, and the falsenorthing is 6,023,150 meters.DESCRIPTIONThis is the standard projection for large-scale mapsof New Zealand.PROJECTION METHODModified cylindrical. A sixth-order conformalmodification of the Mercator projection using theInternational spheroid.POINT OF TANGENCY173° E, 41° S.LINEAR GRATICULESNone.PROPERTIESShapeConformal. Local shapes are correct.Supported map projections• 73

ORTHOGRAPHICPROPERTIESShapeMinimal distortion near the center; maximaldistortion near the edge.AreaThe areal scale decreases with distance from thecenter. Areal scale is zero at the edge of thehemisphere.DirectionTrue direction from the central point.Central meridian is 0°, and latitude of origin is 90° S.DESCRIPTIONThis perspective projection views the globe from aninfinite distance. This gives the illusion of a threedimensionalglobe. Distortion in size and area nearthe projection limit appears more realistic to our eyethan almost any other projection, except the VerticalNear-Side Perspective.DistanceThe radial scale decreases with distance from thecenter and becomes zero on the edges. The scaleperpendicular to the radii, along the parallels of thepolar aspect, is accurate.LIMITATIONSLimited to a view 90 degrees from the central point,a global hemisphere.USES AND APPLICATIONSUses of this projection are aesthetic more thantechnical. The most commonly used aspect for thispurpose is the oblique.PROJECTION METHODPlanar perspective projection, viewed from infinity.On the polar aspect, meridians are straight linesradiating from the center, and the lines of latitude areprojected as concentric circles that become closertoward the edge of the globe. Only one hemispherecan be shown without overlapping.POINT OF CONTACTA single point located anywhere on the globe.LINEAR GRATICULESAll aspects—The central meridian of the projection.Equatorial aspect—All lines of latitude.Polar aspect—All meridians.74 • Understanding Map Projections

PERSPECTIVEPolar aspect—All meridians.Equatorial aspect—The equator.PROPERTIESShapeMinimally distorted near the center, increasingtoward the edge.AreaMinimally distorted near the center; the area scalethen decreases to zero on the edge or horizon.DirectionTrue directions from the point of tangency.The central meridian is 0°, and the latitude of origin is 90° S.DESCRIPTIONAlso known as Vertical Near-Side Perspective orVertical Perspective.This projection is similar to the Orthographicprojection in that its perspective is from space. Inthis projection, the perspective point is not aninfinite distance away; instead, you can specify thedistance. The overall effect of this projection is that itlooks like a photograph taken vertically from asatellite or space vehicle.DistanceRadial scale decreases from true scale at the center tozero on the projection edge. The scale perpendicularto the radii decreases, but not as rapidly.LIMITATIONSThe actual range depends on the distance from theglobe. In all cases, the range is less than 90 degreesfrom the center.USES AND APPLICATIONSUsed as an aesthetic presentation rather than fortechnical applications.PROJECTION METHODPlanar perspective projection. The distance abovethe earth is variable and must be specified before theprojection can be calculated. The greater thedistance, the more closely this projection resemblesthe Orthographic projection. All aspects are circularprojections of an area less than a hemisphere.POINT OF CONTACTA single point anywhere on the globe.LINEAR GRATICULESAll aspects—The central meridian of the projection.Supported map projections• 75

PLATE CARRÉEDirectionNorth, south, east, and west directions are accurate.General directions are distorted, except locally alongthe standard parallels.DistanceThe scale is correct along the meridians and thestandard parallels.The central meridian is 149° W.DESCRIPTIONAlso known as Equirectangular, EquidistantCylindrical, Simple Cylindrical, or Rectangular.This projection is very simple to construct because itforms a grid of equal rectangles. Because of itssimple calculations, its usage was more common inthe past. In this projection, the polar regions are lessdistorted in scale and area than they are in theMercator projection.LIMITATIONSNoticeable distortion of all properties away fromstandard parallels.USES AND APPLICATIONSBest used for city maps or other small areas withmap scales large enough to reduce the obviousdistortion.Used for simple portrayals of the world or regionswith minimal geographic data. This makes theprojection useful for index maps.PROJECTION METHODThis simple cylindrical projection converts the globeinto a Cartesian grid. Each rectangular grid cell hasthe same size, shape, and area. All the graticularintersections are 90 degrees. The traditional PlateCarrée projection uses the equator as the standardparallel. The grid cells are perfect squares. In thisprojection, the poles are represented as straight linesacross the top and bottom of the grid.LINE OF CONTACTTangent at the equator.LINEAR GRATICULESAll meridians and all parallels.PROPERTIESShapeDistortion increases as the distance from thestandard parallels increases.AreaDistortion increases as the distance from thestandard parallels increases.76 • Understanding Map Projections

POLAR STEREOGRAPHICAreaThe farther from the pole, the greater the areal scale.DirectionTrue direction from the pole. Local angles are trueeverywhere.DistanceThe scale increases with distance from the center. Ifa standard parallel is chosen rather than one of thepoles, this latitude represents the true scale, and thescale nearer the pole is reduced.The central meridian is 0°, and the latitude of origin is 90° S.DESCRIPTIONThe projection is equivalent to the polar aspect ofthe Stereographic projection on a spheroid. Thecentral point is either the North Pole or the SouthPole. This is the only polar aspect planar projectionthat is conformal. The Polar Stereographic projectionis used for all regions not included in the UTMcoordinate system, regions north of 84° N and southof 80° S. Use UPS for these regions.LIMITATIONSNormally not extended more than 90 degrees fromthe central pole because of increased scale and areadistortion.USES AND APPLICATIONSPolar regions (conformal).In the UPS system, the scale factor at the pole is0.994, which corresponds to a latitude of true scale(standard parallel) at 81°06'52.3" N or S.PROJECTION METHODPlanar perspective projection, where one pole isviewed from the other pole. Lines of latitude areconcentric circles. The distance between circlesincreases with distance from the central pole.POINT OF TANGENCYA single point, either the North Pole or the SouthPole. If the plane is secant instead of tangent, thepoint of global contact is a line of latitude.LINEAR GRATICULESAll meridians.PROPERTIESShapeConformal; accurate representation of local shapes.Supported map projections• 77

POLYCONICDirectionLocal angles are accurate along the central meridian;otherwise, they are distorted.DistanceThe scale along each parallel and along the centralmeridian of the projection is accurate. Distortionincreases along the meridians as the distance fromthe central meridian increases.The central meridian is 90° W.DESCRIPTIONThe name of this projection translates into ‘manycones’. This refers to the projection methodology.This affects the shape of the meridians. Unlike otherconic projections, the meridians are curved ratherthan linear.PROJECTION METHODMore complex than the regular conic projections, butstill a simple construction. This projection is createdby lining up an infinite number of cones along thecentral meridian. This projection yields parallels thatare not concentric. Each line of latitude representsthe base of its tangential cone.LIMITATIONSDistortion is minimized on large-scale maps, such astopographic quadrangles, where meridians andparallels can be drawn in practice as straight-linesegments. Producing a map library with this kind ofmap sheet is not advisable because errorsaccumulate and become visible when joining sheetsin multiple directions.USES AND APPLICATIONSUsed for 7½- and 15-minute topographic USGS quadsheets, from 1886 until approximately 1957. Note:Some new quad sheets after this date have beenfalsely documented as Polyconic. The presentprojection for east–west State Plane CoordinateSystem zones is Lambert Conformal Conic, andTransverse Mercator for north–south state zones.LINES OF CONTACTMany lines; all parallels of latitude in the projection.LINEAR GRATICULESCentral meridian of the projection and the equator.PROPERTIESShapeNo local shape distortion along the central meridian.Distortion increases with distance from the centralmeridian; thus, east–west distortion is greater thannorth–south distortion.AreaDistortion in area increases with distance from thecentral meridian.78 • Understanding Map Projections

QUARTIC A UTHALICUSES AND APPLICATIONSThematic world maps. The McBryde–Thomas Flat-Polar Quartic projection is based on this projection.The central meridian is 0°.DESCRIPTIONThis pseudocylindrical equal-area projection isprimarily used for thematic maps of the world.PROJECTION METHODPseudocylindrical equal-area projection.LINEAR GRATICULESThe central meridian is a straight line 0.45 times thelength of the equator. Meridians are equally spacedcurves. Parallels are unequally spaced, straightparallel lines perpendicular to the central meridian.Their spacing decreases very gradually as they moveaway from the equator.PROPERTIESShapeGenerally distorted.AreaEqual area.DirectionDirection is generally distorted.DistanceScale is true along the equator. Scale is also constantalong any given latitude and is symmetrical aroundthe equator.LIMITATIONSUseful only as a world map.Supported map projections• 79

RECTIFIED SKEWED ORTHOMORPHICDESCRIPTIONAlso called RSO.This projection is provided with two options for thenational coordinate systems of Malaysia and Bruneiand is similar to the Oblique Mercator.PROJECTION METHODOblique cylindrical projection. A line of true scale isdrawn at an angle to the central meridian.LINE OF CONTACTA single, oblique, great-circle line.LINEAR GRATICULESTwo meridians 180 degrees apart.PROPERTIESShapeConformal. Local shapes are true.AreaIncreases with distance from the center line.DirectionLocal angles are correct.DistanceTrue along the chosen central line.LIMITATIONSIts use is limited to those areas of Brunei andMalaysia for which the projection was developed.USES AND APPLICATIONSUsed for the national projections of Malaysia andBrunei.80 • Understanding Map Projections

ROBINSONDistanceGenerally, scale is made true along latitudes 38° Nand S. Scale is constant along any given latitude andfor the latitude of the opposite sign.LIMITATIONSNeither conformal nor equal area. Useful only forworld maps.USES AND APPLICATIONSDeveloped for use in general and thematic worldmaps.The central meridian is 118° W.DESCRIPTIONAlso called Orthophanic.Used by Rand McNally since the 1960s and by theNational Geographic Society since 1988 for generaland thematic world maps.A compromise projection used for world maps.PROJECTION METHODPseudocylindrical. Meridians are equally spaced andresemble elliptical arcs, concave toward the centralmeridian. The central meridian is a straight line0.51 times the length of the equator. Parallels areequally spaced straight lines between 38° N and S;spacing decreases beyond these limits. The poles are0.53 times the length of the equator. The projectionis based on tabular coordinates instead ofmathematical formulas.LINEAR GRATICULESAll parallels and the central meridian.PROPERTIESShapeShape distortion is very low within 45 degrees of theorigin and along the equator.AreaDistortion is very low within 45 degrees of the originand along the equator.DirectionGenerally distorted.Supported map projections• 81

SIMPLE CONICEquirectangular projection must be used.Use Equirectangular if the standard parallel is theequator.LINES OF CONTACTDepends on the number of standard parallels.Tangential projections (Type 1)—One line, indicatedby the standard parallel.Secant projections (Type 2)—Two lines, specified asfirst and second standard parallels.LINEAR GRATICULESAll meridians.PROPERTIESThe central meridian is 60° W. The first and second standardparallels are 5° S and 42° S. The latitude of origin is 32° S.DESCRIPTIONAlso called Equidistant Conic or Conic.This conic projection can be based on one or twostandard parallels. As the name implies, all circularparallels are an equal distance from each other,spaced evenly along the meridians. This is truewhether one or two parallels are used.PROJECTION METHODCone is tangential if only one standard parallel isspecified and secant if two standard parallels arespecified. Graticules are evenly spaced. The spacebetween each meridian is equal, as is the spacebetween each of the concentric arcs that describe thelines of latitude. The poles are represented as arcsrather than points.If the pole is given as the single standard parallel,the cone becomes a plane and the resultingprojection is the same as a polar AzimuthalEquidistant.If two standard parallels are placed symmetricallynorth and south of the equator, the resultingprojection is the same as Equirectangular, and theShapeLocal shapes are true along the standard parallels.Distortion is constant along any given parallel.Distortion increases with distance from the standardparallels.AreaDistortion is constant along any given parallel.Distortion increases with distance from the standardparallels.DirectionLocally true along the standard parallels.DistanceTrue along the meridians and the standard parallels.Scale is constant along any given parallel butchanges from parallel to parallel.LIMITATIONSRange in latitude should be limited to 30 degrees.USES AND APPLICATIONSRegional mapping of midlatitude areas that have apredominantly east–west extent.Common for atlas maps of small countries.Used by the former Soviet Union for mapping theentire country.82 • Understanding Map Projections

SINUSOIDALDistanceThe scale along all parallels and the central meridianof the projection is accurate.LIMITATIONSDistortion is reduced when used for a single landmass rather than the entire globe. This is especiallytrue for regions near the equator.The central meridian is 117° E.DESCRIPTIONAlso known as Sanson–Flamsteed.As a world map, this projection maintains equal areadespite conformal distortion. Alternative formatsreduce the distortion along outer meridians byinterrupting the continuity of the projection over theoceans and by centering the continents around theirown central meridians, or vice versa.USES AND APPLICATIONSUsed for world maps illustrating area characteristics,especially if interrupted.Used for continental maps of South America, Africa,and occasionally other land masses, where each hasits own central meridian.PROJECTION METHODA pseudocylindrical projection where all parallelsand the central meridian are straight. The meridiansare curves based on sine functions with theamplitudes increasing with the distance from thecentral meridian.LINEAR GRATICULESAll lines of latitude and the central meridian.PROPERTIESShapeNo distortion along the central meridian and theequator. Smaller regions using the interrupted formexhibit less distortion than the uninterruptedsinusoidal projection of the world.AreaAreas are represented accurately.DirectionLocal angles are correct along the central meridianand the equator but distorted elsewhere.Supported map projections• 83

SPACE OBLIQUE MERCATORDESCRIPTIONThis projection is nearly conformal and has littlescale distortion within the sensing range of anorbiting mapping satellite such as Landsat. This is thefirst projection to incorporate the earth’s rotationwith respect to the orbiting satellite. For Landsat 1, 2,and 3, the path range is from 1 to 251. For Landsat 4and 5, the path range is from 1 to 233.Used to tie satellite imagery to a ground-basedplanar coordinate system and for continuousmapping of satellite imagery.Standard format used for data from Landsat 4 and 5.PROJECTION METHODModified cylindrical, for which the central line iscurved and defined by the ground track of the orbitof the satellite.LINE OF TANGENCYConceptual.LINEAR GRATICULESNone.PROPERTIESShapeShape is correct within a few parts per million forthe sensing range of the satellite.AreaVaries by less than 0.02 percent for the sensing rangeof the satellite.DirectionMinimal distortion within the sensing range.DistanceScale is true along the ground track and variesapproximately 0.01 percent within the sensing range.LIMITATIONSPlots for adjacent paths do not match withouttransformation.USES AND APPLICATIONSSpecifically designed to minimize distortion withinthe sensing range of a mapping satellite as it orbitsthe rotating earth.84 • Understanding Map Projections

STATE PLANE COORDINATE SYSTEMDESCRIPTIONAlso known as SPCS, SPC, State Plane, and State.The State Plane Coordinate System is not aprojection. It is a coordinate system that divides the50 states of the United States, Puerto Rico, and theU.S. Virgin Islands into more than 120 numberedsections, referred to as zones. Each zone has anassigned code number that defines the projectionparameters for the region.PROJECTION METHODProjection may be cylindrical or conic. See LambertConformal Conic, Transverse Mercator, and HotineOblique Mercator for methodology and properties.WHY USE STATE PLANEGovernmental organizations and groups who workwith them primarily use the State Plane CoordinateSystem. Most often, these are county or municipaldatabases. The advantage of using SPCS is that yourdata is in a common coordinate system with otherdatabases covering the same area.WHAT IS STATE PLANEThe State Plane Coordinate System was designed forlarge-scale mapping in the United States. It wasdeveloped in the 1930s by the U.S. Coast andGeodetic Survey to provide a common referencesystem to surveyors and mappers. The goal was todesign a conformal mapping system for the countrywith a maximum scale distortion of one part in10,000, then considered the limit of surveyingaccuracy.Three conformal projections were chosen: theLambert Conformal Conic for states that are longereast–west, such as Tennessee and Kentucky; theTransverse Mercator projection for states that arelonger north–south, such as Illinois and Vermont;and the Oblique Mercator projection for thepanhandle of Alaska, because it lays at an angle.To maintain an accuracy of one part in 10,000, it wasnecessary to divide many states into zones. Eachzone has its own central meridian or standardparallels to maintain the desired level of accuracy.The boundaries of these zones follow countyboundaries. Smaller states such as Connecticutrequire only one zone, while Alaska is composed of10 zones and uses all three projections.This coordinate system is referred to here as theState Plane Coordinate System of 1927 (SPCS 27). Itis based on a network of geodetic control pointsreferred to as the North American Datum of 1927(NAD 1927 or NAD27).STATE PLANE AND THE NORTH AMERICANDATUMTechnological advancements of the last 50 yearshave led to improvements in the measurement ofdistances, angles, and the earth’s size and shape.This, combined with moving the origin of the datumfrom Meades Ranch in Kansas to the earth’s center ofmass for compatibility with satellite systems, made itnecessary to redefine SPCS 27. The redefined andupdated system is called the State Plane CoordinateSystem of 1983 (SPCS 83). The coordinates for pointsare different for SPCS 27 and SPCS 83. There areseveral reasons for this. For SPCS 83, all State Planecoordinates published by NGS are in metric units,the shape of the spheroid of the earth is slightlydifferent, some states have changed the definition oftheir zones, and values of longitude and latitude areslightly changed.Officially, SPCS zones are identified by their NGScode. When ESRI ® implemented the NGS codes, theywere part of a proposed Federal InformationProcessing Standard (FIPS). For that reason, ESRIidentifies the NGS zones as FIPS zones. Thatproposed standard was withdrawn, but ESRImaintains the FIPS name for continuity.Sometimes people use an older Bureau of LandManagement (BLM) system. The BLM system isoutdated and doesn’t include codes for some of thenew zones. The values also overlap. You shouldalways use the NGS/FIPS codes.The following zone changes were made fromSPCS 27 to SPCS 83. The zone numbers listed beloware FIPS zone numbers. In addition, false easting andnorthing, or origin, of most zones has changed.California—California zone 7, SPCS 27 FIPSzone 0407, was eliminated and included in Californiazone 5, SPCS 83 FIPS zone 0405.Supported map projections• 85

Montana—The three zones for Montana, SPCS 27FIPS zones 2501, 2502, and 2503, were eliminatedand replaced by a single zone, SPCS 83 FIPSzone 2500.Nebraska—The two zones for Nebraska, SPCS 27FIPS zones 2601 and 2602, were eliminated andreplaced by a single zone, SPCS 83 FIPS zone 2600.USES AND APPLICATIONSUsed for standard USGS 7½- and 15-minute quadsheets.Used for most federal, state, and local large-scalemapping projects in the United States.South Carolina—The two zones for South Carolina,SPCS 27 FIPS zones 3901 and 3902, were eliminatedand replaced by a single zone, SPCS 83 FIPSzone 3900.Puerto Rico and Virgin Islands—The two zones forPuerto Rico and the Virgin Islands, St. Thomas,St. John, and St. Croix, SPCS 27 FIPS zones 5201 and5202, were eliminated and replaced by a single zone,SPCS 83 FIPS zone 5200.UNIT OF LENGTHThe standard unit of measure for SPCS 27 is the U.S.Survey foot. For SPCS 83, the most common unit ofmeasure is the meter. Those states that support bothfeet and meters have legislated which feet-to-metersconversion they use. The difference between the twois only two parts in one million, but that can becomenoticeable when datasets are stored in doubleprecision. The U.S. Survey foot equals 1,200/3,937 m,or 0.3048006096 m.EXAMPLES OF ZONE DEFINITIONSHere are two examples of SPCS 83 parameters:State Alabama East TennesseeZONE 3101 5301FIPS Zone 0101 4100Projection Transverse LambertStandard Parallels35°15'36°25'Central Meridian -85°50' -86°00'Scale Factor Reduction at Central Meridian1:25,000 1:15,000Latitude of Origin 30°30' 34°20'Longitude of Origin -85°50' -86°00'False Easting 200,000 600,000False Northing 0 086 •Understanding Map Projections

STEREOGRAPHICPROPERTIESShapeConformal. Local shapes are accurate.AreaTrue scale at center with distortion increasing withdistance.DirectionDirections are accurate from the center. Local anglesare accurate everywhere.DistanceScale increases with distance from the center.The central meridian is 0°, and the latitude of origin is 90° S.DESCRIPTIONThis projection is conformal.PROJECTION METHODPlanar perspective projection, viewed from the pointon the globe opposite the point of tangency.Stereographic projects points on a spheroid directlyto the plane. See Double Stereographic for adifferent implementation.LIMITATIONSNormally limited to one hemisphere. Portions of theouter hemisphere may be shown, but with rapidlyincreasing distortion.USES AND APPLICATIONSThe oblique aspect has been used to map circularregions on the moon, Mars, and Mercury.All meridians and parallels are shown as circular arcsor straight lines. Graticular intersections are90 degrees. In the equatorial aspect, the parallelscurve in opposite directions on either side of theequator. In the oblique case, only the parallel withthe opposite sign to the central latitude is a straightline; other parallels are concave toward the poles oneither side of the straight parallel.POINT OF CONTACTA single point anywhere on the globe.LINEAR GRATICULESPolar aspect—All meridians.Equatorial aspect—The central meridian and theequator.Oblique aspect—Central meridian and parallel oflatitude with the opposite sign of the central latitude.Supported map projections• 87

TIMESLIMITATIONSUseful only for world maps.USES AND APPLICATIONSUsed for world maps by Bartholomew Ltd., a Britishmapmaking company, in The Times Atlas.The central meridian is 0°.DESCRIPTIONThe Times projection was developed by Moir in 1965for Bartholomew. It is a modified Gall’sStereographic, but Times has curved meridians.PROJECTION METHODPseudocylindrical. Meridians are equally spacedcurves. Parallels are straight lines increasing inseparation with distance from the equator.LINES OF CONTACTTwo lines at 45° N and S.LINEAR GRATICULESAll parallels and the central meridian.PROPERTIESShapeModerate distortion.AreaIncreasing distortion with distance from 45° N and S.DirectionGenerally distorted.DistanceScale is correct along parallels at 45° N and S.88 • Understanding Map Projections

TRANSVERSE MERCATORPROPERTIESShapeConformal. Small shapes are maintained. Largershapes are increasingly distorted away from thecentral meridian.AreaDistortion increases with distance from the centralmeridian.DirectionLocal angles are accurate everywhere.The central meridian and the latitude of origin are 0°. The scalefactor is 1.0. Approximately 20 degrees of longitude are shown,which is close to the limit for Transverse Mercator.DESCRIPTIONAlso known as Gauss–Krüger (see that projection).Similar to the Mercator except that the cylinder islongitudinal along a meridian instead of the equator.The result is a conformal projection that does notmaintain true directions. The central meridian isplaced in the center of the region of interest. Thiscentering minimizes distortion of all properties inthat region. This projection is best suited fornorth–south areas. The State Plane CoordinateSystem uses this projection for all north–south zones.The UTM and Gauss–Krüger coordinate systems arebased on the Transverse Mercator projection.PROJECTION METHODCylindrical projection with central meridian placed ina particular region.LINES OF CONTACTAny single meridian for the tangent projection. Forthe secant projection, two almost parallel linesequidistant from the central meridian. For UTM, thelines are about 180 km from the central meridian.LINEAR GRATICULESThe equator and the central meridian.DistanceAccurate scale along the central meridian if the scalefactor is 1.0. If it is less than 1.0, there are twostraight lines with accurate scale equidistant fromand on each side of the central meridian.LIMITATIONSData on a spheroid or an ellipsoid cannot beprojected beyond 90 degrees from the centralmeridian. In fact, the extent on a spheroid orellipsoid should be limited to 15–20 degrees on bothsides of the central meridian. Beyond that range,data projected to the Transverse Mercator projectionmay not project back to the same position. Data on asphere does not have these limitations.USES AND APPLICATIONSState Plane Coordinate System, used forpredominantly north–south state zones.USGS 7½-minute quad sheets. Most new USGS mapsafter 1957 use this projection, which replaced thePolyconic projection.North America (USGS, central meridian scale factor is0.926).Topographic Maps of the Ordnance Survey of GreatBritain after 1920.UTM and Gauss–Krüger coordinate systems. Theworld is divided into 60 north and south zones sixdegrees wide. Each zone has a scale factor of 0.9996and a false easting of 500,000 meters. Zones south ofthe equator have a false northing ofSupported map projections• 89

10,000,000 meters to ensure that all y values arepositive. Zone 1 is at 177° W.The Gauss–Krüger coordinate system is very similarto the UTM coordinate system. Europe is dividedinto zones six degrees wide with the centralmeridian of zone 1 equal to 3° E. The parameters arethe same as UTM except for the scale factor, which isequal to 1.000 rather than 0.9996. Some places alsoadd the zone number times one million to the500,000 false easting value. GK zone 5 could havefalse easting values of 500,000 or 5,500,000 meters.90 •Understanding Map Projections

TWO-POINT EQUIDISTANTpoint represents the correct great circle length butnot the correct great circle path.LIMITATIONSDoes not represent great circle paths.USES AND APPLICATIONSUsed by the National Geographic Society for maps ofAsia.The first point is 117°30' W, 34° N, and the second point is83° W, 40° N.Adapted form used by Bell Telephone system fordetermining the distance used to calculate longdistance telephone rates.DESCRIPTIONThis projection shows the true distance from eitherof two chosen points to any other point on a map.PROJECTION METHODModified planar.POINTS OF CONTACTNone.LINEAR GRATICULESNormally none.PROPERTIESShapeMinimal distortion in the region of the two chosenpoints, if they’re within 45 degrees of each other.Increasing distortion beyond this region.AreaMinimal distortion in the region of the two chosenpoints, if they’re within 45 degrees of each other.Increasing distortion beyond this region.DirectionVarying distortion.DistanceCorrect from either of two chosen points to anyother point on the map. Straight line from eitherSupported map projections• 91

UNIVERSAL POLAR STEREOGRAPHICAreaThe farther from the pole, the greater the area scale.DirectionTrue direction from the pole. Local angles are correcteverywhere.DistanceIn general, the scale increases with distance from thepole. Latitude 81°06'52.3" N or S has true scale. Thescale closer to the pole is reduced.The central meridian is 90° S. The latitude of standard parallelis 81°06'52.3" S. The false easting and northing values are2,000,000 meters.DESCRIPTIONAlso known as UPS.This form of the Polar Stereographic projection mapsareas north of 84° N and south of 80° S that aren’tincluded in the UTM Coordinate System. Theprojection is equivalent to the polar aspect of theStereographic projection of the spheroid withspecific parameters. The central point is either theNorth Pole or the South Pole.LIMITATIONSThe UPS is normally limited to 84° N in the northpolar aspect and 80° S in the south polar aspect.USES AND APPLICATIONSConformal mapping of polar regions.Used for mapping polar regions of the UTMcoordinate system.PROJECTION METHODApproximately (for the spheroid) planar perspectiveprojection, where one pole is viewed from the otherpole. Lines of latitude are concentric circles. Thedistance between circles increases away from thecentral pole. The origin at the intersection ofmeridians is assigned a false easting and falsenorthing of 2,000,000 meters.LINES OF CONTACTThe latitude of true scale, 81°06'52.3" N or S,corresponds to a scale factor of 0.994 at the pole.LINEAR GRATICULESAll meridians.PROPERTIESShapeConformal. Accurate representation of local shape.92 • Understanding Map Projections

UNIVERSAL T RANSVERSE MERCATORDESCRIPTIONAlso known as UTM.The Universal Transverse Mercator system is aspecialized application of the Transverse Mercatorprojection. The globe is divided into 60 north andsouth zones, each spanning six degrees of longitude.Each zone has its own central meridian. Zones 1Nand 1S start at -180° W. The limits of each zone are84° N and 80° S, with the division between northand south zones occurring at the equator. The polarregions use the Universal Polar Stereographiccoordinate system.The origin for each zone is its central meridian andthe equator. To eliminate negative coordinates, thecoordinate system alters the coordinate values at theorigin. The value given to the central meridian is thefalse easting, and the value assigned to the equatoris the false northing. A false easting of500,000 meters is applied. A north zone has a falsenorthing of zero, while a south zone has a falsenorthing of 10,000,000 meters.PROJECTION METHODCylindrical projection. See the Transverse Mercatorprojection for the methodology.LINES OF CONTACTTwo lines parallel to and approximately 180 km toeach side of the central meridian of the UTM zone.LINEAR GRATICULESThe central meridian and the equator.DistanceScale is constant along the central meridian but at ascale factor of 0.9996 to reduce lateral distortionwithin each zone. With this scale factor, lines lying180 km east and west of and parallel to the centralmeridian have a scale factor of one.LIMITATIONSDesigned for a scale error not exceeding 0.1 percentwithin each zone. Error and distortion increase forregions that span more than one UTM zone. UTM isnot designed for areas that span more than a fewzones.Data on a spheroid or an ellipsoid cannot beprojected beyond 90 degrees from the centralmeridian. In fact, the extent on a spheroid orellipsoid should be limited to 15–20 degrees on bothsides of the central meridian. Beyond that range,data projected to the Transverse Mercator projectionmay not project back to the same position. Data on asphere does not have these limitations.USES AND APPLICATIONUsed for United States topographic quadrangles,1:100,000 scale.Many countries use local UTM zones based on theofficial geographic coordinate systems in use.Large-scale topographic mapping of the formerSoviet Union.PROPERTIESShapeConformal. Accurate representation of small shapes.Minimal distortion of larger shapes within the zone.AreaMinimal distortion within each UTM zone.DirectionLocal angles are true.Supported map projections• 93

VAN DER GRINTEN IDistanceScale along the equator is correct.LIMITATIONSCan represent the world, but the most accuraterepresentation is between the 75th parallels oflatitude.USES AND APPLICATIONSUsed for world maps.Formerly the standard world map projection of theNational Geographic Society.The central meridian is 56° E.DESCRIPTIONThis projection is similar to the Mercator projectionexcept that it portrays the world with a curvedgraticule. The overall effect is that area is distortedless than on a Mercator projection, and the shape isdistorted less than on equal area projections.PROJECTION METHODThe Van der Grinten I projection is a compromiseprojection and is not in one of the more traditionalclassifications.LINEAR GRATICULESThe equator and the central meridian of theprojection.PROPERTIESShapeDistortion increases from the equator to the poles.AreaMinimal distortion along the equator and extremedistortion in the polar regions.DirectionLocal angles are correct only at the center.94 • Understanding Map Projections

VERTICAL NEAR-SIDE PERSPECTIVEEquatorial aspect—The equator.Polar aspect—All meridians.PROPERTIESShapeMinimal distortion near the center; maximaldistortion near the edge.AreaMinimal distortion near the center; maximaldistortion near the edge.DirectionTrue direction from the central point.The central meridian is 0°, and the latitude of origin is 90° S.DESCRIPTIONUnlike the Orthographic projection, this perspectiveprojection views the globe from a finite distance.This perspective gives the overall effect of the viewfrom a satellite.PROJECTION METHODPlanar perspective projection, viewed from aspecified distance above the surface. All aspects areeither circular or an area less than a full hemisphere.DistanceThe radial scale decreases with distance from thecenter.LIMITATIONSLimited to a view less than 90 degrees from thecentral point.USES AND APPLICATIONSUses of this projection are aesthetic more thantechnical. The most commonly used aspect for thispurpose is the oblique.Polar aspect—Meridians are straight lines radiatingfrom the center, and the lines of latitude areprojected as concentric circles that become closertoward the edge of the globe.Equatorial aspect—The central meridian and theequator are straight lines. The other meridians andparallels are elliptical arcs.POINT OF CONTACTA single point located anywhere on the globe.LINEAR GRATICULESAll aspects—The central meridian of the projection.Supported map projections• 95

WINKEL IDistanceGenerally, scale is made true along latitudes 50°28' Nand S.LIMITATIONSNeither conformal nor equal area. Useful only forworld maps.The central meridian is 0°.USES AND APPLICATIONSDeveloped for use in general world maps. If thestandard parallels are 50°28' N and S, the total areascale is correct, but local area scales vary.DESCRIPTIONOften used for world maps, the Winkel I projectionis a pseudocylindrical projection that averages thecoordinates from the Equirectangular (EquidistantCylindrical) and Sinusoidal projections. Developedby Oswald Winkel in 1914.PROJECTION METHODPseudocylindrical. Coordinates are the average of theSinusoidal and Equirectangular projections.Meridians are equally spaced sinusoidal curvescurving toward the central meridian. The centralmeridian is a straight line. Parallels are equallyspaced straight lines. The length of the poles and thecentral meridian depends on the standard parallels.If the standard parallel is the equator, Eckert Vresults.LINEAR GRATICULESThe parallels and the central meridian.PROPERTIESShapeGenerally distorted.AreaGenerally distorted.DirectionGenerally distorted.96 • Understanding Map Projections

WINKEL IILIMITATIONSNeither conformal nor equal area. Useful only forworld maps.USES AND APPLICATIONSDeveloped for use in general world maps.The central meridian is 0°.DESCRIPTIONA pseudocylindrical projection that averages thecoordinates from the Equirectangular and Mollweideprojections. Developed by Oswald Winkel in 1918.PROJECTION METHODPseudocylindrical. Coordinates are the average of theMollweide and Equirectangular projections.Meridians are equally spaced curves, curving towardthe central meridian. The central meridian is astraight line. Parallels are equally spaced straightlines. The length of the poles and the centralmeridian depends on the standard parallels.LINEAR GRATICULESThe parallels and the central meridian.PROPERTIESShapeGenerally distorted.AreaGenerally distorted.DirectionGenerally distorted.DistanceGenerally, scale is made true along the standardlatitudes.Supported map projections• 97

WINKEL T RIPELDistanceGenerally, scale is made true along latitudes50.467° N and S or 40° N and S. The second case isused by Bartholomew Ltd., a British mapmakingcompany.LIMITATIONSNeither conformal nor equal area. Useful only forworld maps.The central meridian is 0°, and the standard parallels are at50.467° N and S.DESCRIPTIONA compromise projection used for world maps thataverages the coordinates from the Equirectangular(Equidistant Cylindrical) and Aitoff projections.Developed by Oswald Winkel in 1921.USES AND APPLICATIONSDeveloped for use in general and thematic worldmaps.Used by the National Geographic Society since 1998for general and thematic world maps.PROJECTION METHODModified azimuthal. Coordinates are the average ofthe Aitoff and Equirectangular projections. Meridiansare equally spaced and concave toward the centralmeridian. The central meridian is a straight line.Parallels are equally spaced curves, concave towardthe poles. The poles are around 0.4 times the lengthof the equator. The length of the poles depends onthe standard parallel chosen.LINEAR GRATICULESThe equator and the central meridian.PROPERTIESShapeShape distortion is moderate. In the polar regionsalong the outer meridians, the distortion is severe.AreaDistortion is moderate. In the polar regions along theouter meridians, the distortion is severe.DirectionGenerally distorted.98 • Understanding Map Projections

Selected ReferencesDatums, Ellipsoids, Grids and Grid Reference Systems.Washington, D.C.: NIMA, 1990. Technical Manual 8358.1,www.nima.mil/GandG/pubs.html.Department of Defense World Geodetic System 1984.Third Edition. Washington, D.C.: NIMA, 1997. TechnicalReport 8350.2, www.nima.mil/GandG/pubs.html.European Petroleum Survey Group, EPSG GeodesyParameters, v4.5. www.ihsenergy.com/?epsg/epsg.html,2000.European Petroleum Survey Group, POSC LiteraturePertaining to Geographic and Projected Coordinate SystemTransformations. 2000. Guidance Note Number 7.Geodesy for the Layman. Fourth Edition. Washington, D.C.:NIMA, 1984. Technical Report 80-003,www.nima.mil/GandG/pubs.html.Hooijberg, Maarten, Practical Geodesy: Using Computers.Berlin: Springer–Verlag, 1997.Junkins, D.R., and S.A. Farley, NTv2 Developer’s Guide.Geodetic Survey Division, Natural Resources Canada,1995.Junkins, D.R., and S.A. Farley, NTv2 User’s Guide.Geodetic Survey Division, Natural Resources Canada,1995.Maling, D.H., Coordinate Systems and Map Projections. SecondEdition. Oxford: Pergamon Press, 1993.National Geodetic Survey, NADCON Release Notes,README file accompanying NADCON Version 2.1.NOAA/NGS, July 2000.Rapp, Richard H., Geometric Geodesy: Part I. Department ofGeodetic Science and Surveying, Ohio State University,April 1991.Rapp, Richard H., Geometric Geodesy: Part II. Departmentof Geodetic Science and Surveying, Ohio State University,March 1993.Snyder, John P., Map Projections: A Working Manual. USGSProfessional Paper 1395. Washington, D.C.: USGS, 1993.Snyder, John P., and Philip M. Voxland, An Album of MapProjections. USGS Professional Paper 1453. Washington,D.C.: USGS, 1989.Soler, T., and L.D. Hothem (1989), “ImportantParameters Used in Geodetic Transformations.” Journal ofSurveying Engineering 112(4):414–417, November 1989.Torge, Wolfgang, Geodesy. Second Edition. New York: deGruyter, 1991.Vanicek, Petr, and Edward J. Krakiwsky, Geodesy: TheConcepts. Amsterdam: North-Holland Publishing Company,1982.Voser, Stefan A., MapRef: The Collection of Map Projectionsand Reference Systems for Europe. www.geocities.com/CapeCanaveral/1224/mapref.html, 1997.99

Glossaryangular unitsThe unit of measurement on a sphere or a spheroid,usually in degrees. Map projection parameters such as thecentral meridian and standard parallel are defined inangular units.aspectThe conceptual center of a projection system. See alsoequatorial, oblique, and polar aspect.azimuthAn angle measured from north. Used to define an obliqueaspect of a cylindrical projection or the angle of ageodesic between two points.azimuthal projectionA form of projection where the earth is projected onto aconceptual tangent or secant plane. See planar projection.central meridianThe line of longitude that defines the center and often thex origin of a projected coordinate system.circleA geometric shape for which the distance from the centerto any point on the edge is equal.conformal projectionA projection on which all angles at each point arepreserved. Also called an orthomorphic projection (Snyderand Voxland, 1989).conic projectionA projection resulting from the conceptual projection ofthe earth onto a tangent or secant cone. The cone is thencut along a line extending between the apex and base ofthe cone and laid flat.cylindrical projectionA projection resulting from the conceptual projection ofthe earth onto a tangent or secant cylinder, which is thencut from base to base and laid flat (Snyder and Voxland,1989).datum1. A reference frame defined by a spheroid and thespheroid’s position relative to the center of the earth.2. A set of control points and a spheroid that define areference surface.datum transformationSee geographic transformation.eccentricityA measurement of how much an ellipse deviates from atrue circle. Measured as the square root of the quantity1.0 minus the square of the ratio of the semiminor axis tothe semimajor axis. The square of the eccentricity, ‘e 2 ’, iscommonly used with the semimajor axis, ‘a’, to define aspheroid in map projection equations.ellipseA geometric shape equivalent to a circle that is viewedobliquely; a flattened circle.ellipsoidWhen used to represent the earth, the three-dimensionalshape obtained by rotating an ellipse about its minor axis.This is an oblate ellipsoid of revolution, also called aspheroid.ellipticityThe degree to which an ellipse deviates from a true circle.The degree of flattening of an ellipse, measured as 1.0minus the ratio of the semiminor axis to the semimajoraxis. See also flattening.equal-area projectionA projection on which the areas of all regions are shownin the same proportion to their true areas. Shapes may begreatly distorted (Snyder and Voxland, 1989). Also knownas an equivalent projection.101

equatorThe parallel of reference that defines the origin of latitudevalues, 0° north or south.equatorial aspectA planar projection that has its central point at theequator.equidistant projectionA projection that maintains scale along one or more linesor from one or two points to all other points on the map.equivalent projectionA projection on which the areas of all regions are shownin the same proportion to their true areas. Shapes may begreatly distorted (Snyder and Voxland, 1989). Also knownas an equal-area projection.false eastingA linear value added to the x-coordinate values, usually toensure that all map coordinates are positive. See falsenorthing.false northingA linear value added to the y-coordinate values, usually toensure that all map coordinates are positive. See falseeasting.flatteningA measure of how much a spheroid differs from a sphere.The flattening is the ratio of the semimajor axis minus thesemiminor axis to the semimajor axis. Known as ‘f’ andoften expressed as a ratio. Example: 1/298.3. Also knownas the ellipticity.Gauss–KrügerA projected coordinate system used in Europe and Asiathat divides the area into six degreewide zones. Verysimilar to the UTM coordinate system.geocentric latitudeDefined as the angle between the equatorial plane and aline from a point on the surface to the center of thesphere or spheroid.geodesicThe shortest distance between any two points on thesurface of a spheroid. Any two points along a meridianform a geodesic.geodetic latitudeDefined as the angle formed by the perpendicular to thesurface at a point and the equatorial plane. On a spheroid,the perpendicular doesn’t hit the center of the spheroid inthe equatorial plane except at the equator and the poles.geographic coordinate systemA reference system that uses latitude and longitude todefine the locations of points on the surface of a sphereor spheroid.geographic transformationA method that converts data between two geographiccoordinate systems (datums). Also known as a datumtransformation.Global Positioning SystemA set of satellites operated by the U.S. Department ofDefense. Ground receivers can calculate their locationusing information broadcast by the satellites.GPSSee Global Positioning System.graticuleA network of lines representing a selection of the earth’sparallels and meridians (Snyder and Voxland, 1989).great circleAny circle on the surface of a sphere formed by theintersection of the surface with a plane passing throughthe center of the sphere. The shortest path between anytwo points lies on a great circle and is therefore importantto navigation. All meridians and the equator are greatcircles on the earth defined as a sphere (Snyder andVoxland, 1989).Greenwich prime meridianThe prime meridian located in Greenwich, England.gridA network of lines representing a selection of a projectedcoordinate system’s coordinates.HARNSee High Accuracy Reference Network.102 • Understanding Map Projections

High Accuracy Reference NetworkA resurvey of NAD 1983 control points using GPStechniques. The resurvey date is often included as part ofthe datum name—NAD 1983 (1991) or NAD91.High Precision Geodetic (or GPS) NetworkA resurvey of NAD 1983 control points using GPStechniques. The resurvey date is often included as part ofthe datum name—NAD 1983 (1991) or NAD91.HPGNSee High Precision Geodetic (or GPS) Network.Interrupted projectionDiscontinuities and gaps are added to a map to decreasethe overall distortion. The world is divided, usually alongcertain meridians, into sections, or gores. Each section hasits own projection origin.latitudeThe angular distance (usually measured in degrees) northor south of the equator. Lines of latitude are also called asparallels. See geodetic latitude and geocentric latitude.latitude of centerThe latitude value that defines the center (and sometimesorigin) of a projection.latitude of originThe latitude value that defines the origin of they-coordinate values for a projection.linear unitsThe unit of measurement, often meters or feet, on a planeor a projected coordinate system. Map projectionparameters such as the false easting and false northing aredefined in linear units.longitudeThe angular distance (usually measured in degrees) east orwest of a prime meridian.longitude of centerThe longitude value that defines the center (andsometimes origin) of a projection.longitude of originThe longitude value that defines the origin of thex-coordinate values for a projection.major axisThe longer axis of an ellipse or spheroid.map projectionA systematic conversion of locations from angular toplanar coordinates.map scaleThe ratio of a length on a map to its length on the ground.meridianThe reference line on the earth’s surface formed by theintersection of the surface with a plane passing throughboth poles. This line is identified by its longitude.Meridians run north–south between the poles.minor axisThe shorter axis of an ellipse or spheroid.NAD 1927North American Datum of 1927. A local datum andgeographic coordinate system used in North America.Replaced by NAD 1983. Also known as NAD27.NAD 1983North American Datum of 1983. A geocentric datum andgeographic coordinate system used in North America. Alsoknown as NAD83.oblate ellipsoidAn ellipsoid created by rotating an ellipse around its minoraxis.oblique aspectA planar or cylindrical projection with its central pointlocated at some point not on the equator or at the poles.parallelA reference line on the earth’s surface that runs east–westaround a sphere or spheroid and is parallel to the equator.Latitude lines are parallel circles.parametersValues that define a specific instance of a map projection.Parameters differ for each projection and can includecentral meridian, standard parallel, scale factor, or latitudeof origin.Glossary • 103

planar projectionA form of projection where the earth is projected onto aconceptual tangent or secant plane. Usually, a planarprojection is the same as an azimuthal projection (Snyderand Voxland, 1989).polar aspectA planar projection with its central point located at eitherthe North or South Pole.prime meridianA meridian of reference that defines the origin of thelongitude values, 0° east or west.projected coordinate systemA reference system that defines the locations of points ona planar surface.radiusThe distance from the center to the outer edge of a circle.reference ellipsoidSee ellipsoid.rhumb lineA complex curve on the earth’s surface that crosses everymeridian at the same oblique angle; a straight line on theMercator projection. Also called a loxodrome (Snyder andVoxland, 1989).scale factorA value (usually less than one) that converts a tangentprojection to a secant projection. Represented by ‘k 0’ or‘k’. If a projected coordinate system doesn’t have a scalefactor, the standard point or lines of the projection have ascale of 1.0. Other points on the map have scales greateror lesser than 1.0. If a projected coordinate system has ascale factor, the standard point or lines no longer have ascale of 1.0.secant projectionA form of map projection where the conceptual surfaceof the projection (cone, cylinder, or plane) cuts throughthe earth’s surface.semimajor axisThe equatorial radius of a spheroid. Often known as ‘a’.semiminor axisThe polar radius of a spheroid. Often known as ‘b’.sphereA three-dimensional shape obtained by revolving a circlearound its diameter.spherical coordinate systemA system using positions of longitude and latitude todefine the locations of points on the surface of a sphereor spheroid.spheroidWhen representing the earth, the three-dimensional shapeobtained by rotating an ellipse about its minor axis. This isan oblate ellipsoid of revolution, also called an ellipsoid.standard lineA line on a sphere or spheroid that has no lengthcompression or expansion after being projected.Commonly, a standard parallel or central meridian.standard parallelThe line of latitude where the projection surface touchesthe surface. A tangent conic or cylindrical projection hasone standard parallel, while a secant conic or cylindricalprojection has two. A standard parallel has no distortion.State Plane Coordinate SystemA projected coordinate system used in the United Statesthat divides each state into one or more zones to minimizedistortion caused by the map projection. Also known asSPCS and SPC.tangent projectionA form of map projection where the conceptual surfaceof the projection (cone, cylinder, or plane) just touchesthe earth’s surface.true-direction projectionA form of projection that shows lines with correctazimuths from one or two points.unit of measureSee angular units or linear units.Universal Transverse MercatorA projected coordinate system that divides the world into60 north and south zones, six degrees wide.104 • Understanding Map Projections

UTMSee Universal Transverse Mercator.WGS 1984World Geodetic System of 1984. A geocentric datum andgeographic coordinate system created by the United Statesmilitary. Also known as WGS84.Glossary • 105

IndexAAbridged Molodensky method26Aitoff 34Alaska Grid 35Alaska Series E 36Albers 37Angular parametersazimuth 20central meridian 20central parallel 20defined 20latitude of center 20latitude of first point 20latitude of origin 20latitude of second point 20longitude of center 20longitude of first point 20longitude of origin 20longitude of second point 20standard parallel 1 20standard parallel 2 20Angular units 101Azimuth 20, 101Azimuthal Equidistant 38Azimuthal projectionsdefined 12, 101Double Stereographic 46equidistant 38Lambert Azimuthal Equal Area65Orthographic 74Perspective 75Polar Stereographic 77Stereographic 95Universal Polar Stereographic92BBehrmann 39Bipolar Oblique 40Bonne 41Bursa–Wolf method 25CCassini–Soldner 42Central meridian 14, 20, 101Central parallel 20Chamberlin Trimetric 43Circle 101Clarke 1866 spheroid 5Conformal projectionsBipolar Oblique 40defined 12, 101Double Stereographic 46Gauss–Krüger 57, 102Hotine 63Krovak 64Lambert Conformal Conic 66Mercator 70Stereographic 87Transverse Mercator 89UTM 105Conic projectionsAlbers 37Bipolar Oblique 40defined 14, 101Equidistant Conic 53Krovak 64Lambert Conformal Conic 66Polyconic 78Simple Conic 82Coordinate Frame method 25Coordinate systemsgeographic 102projected 104spherical 104Coordinatesin different datums 6Craster Parabolic 44Cylindrical Equal Area 45Cylindrical projectionsBehrmann 39Cassini–Soldner 42Cylindrical Equal Area 45defined 16, 101Equidistant Cylindrical 54Equirectangular 55Gall’s Stereographic 56Gauss–Krüger 57, 102Hotine 63Cylindrical projections(continued)Mercator 70Miller 71Plate Carrée 76Transverse Mercator 89UTM 105DDatum transformations. SeeGeographictransformationsdefined 101Datumscoordinatees in different 6defined 101definition 6earth-centered 6local 6NAD 1927 103NAD 1983 103origin point 6transforming 24versus geographic coordinatesystems 2WGS 1984 105Degrees 2Degrees-minutes-seconds 2Developable surfaces 13Distortionsand map projections 11DMS. See Degrees-minutessecondsDouble Stereographic 46EEarthas a sphere 4as a spheroid 4Earth-centered datums 6Eccentricity 101Eckert I 47Eckert II 48Eckert III 49Eckert IV 50Eckert V 51

Eckert VI 52ED 1950 6Ellipse 101Ellipsoidsdefined 101eccentricity 101ellipticity 101flattening 102major axis 103minor axis 103semimajor axis 104semiminor axis 104Ellipticity 101Equal-area projectionsAlbers 37Behrmann 39Bonne 41Craster Parabolic 44Cylindrical Equal Area 45defined 12, 101Eckert II 48Eckert IV 50Eckert VI 52Lambert Azimuthal Equal Area65McBryde–Thomas Flat-PolarQuartic 69Mollweide 72Quartic Authalic 79Sinusoidal 83Equation-based methods 25Equator 102Equatorial aspect 102Equidistant Conic 53Equidistant Cylindrical 54Equidistant projectionsAzimuthal 38defined 12, 102Equidistant Conic 53Equidistant Cylindrical 54Simple Conic 82Two-Point Equidistant 91Equirectangular 55Equivalent projectionsdefined 102European Datum of 1950 6FFalse easting 20, 102False northing 20, 102Flat-Polar Quartic 69Flattening 4, 102GGall’s Stereographic 56Gauss–Krüger 57, 102Geocentric Coordinate System 58Geocentric latitudes 102Geocentric Translation method25Geodesics 102Geodetic latitudes 102Geographic Coordinate System59Geographic coordinate systems102definition 2parts of 2size and shape 4sphere-based 4spheroid-based 4transforming 24Geographic transformationsAbridged Molodensky 26Bursa–Wolf 25Coordinate Frame 25defined 102Geocentric Translation 25HARN 27HARN method 102HPGN 27Molodensky 26NADCON 27NTv1 28NTv2 28Position Vector 25seven-parameter 25three-parameter 25GK. See Gauss–KrügerGnomonic 60Graticules 2Great Britain National Grid 61Great circles 3, 102Greenwich prime meridian 2,102Grid-based methods 27Grids 102GRS 1980 5GRS80 5HHammer–Aitoff 62HARN 102. See High AccuracyReference NetworkHigh Accuracy ReferenceNetwork 7, 27High Precision Geodetic Network7, 27Hotine Oblique Mercator 63HPGN. See High PrecisionGeodetic NetworkKKrovak 64LLambert Azimuthal Equal Area65Lambert Conformal Conic 66Latitude 2Latitude of center 20, 103Latitude of first point 20Latitude of origin 20, 103Latitude of second point 20Latitude range 2Latitudesdefined 103geocentric 102geodetic 102Linear parametersdefined 20false easting 20false northing 20scale factor 20Linear units 103Local Cartesian Projection 67Local datums 6108 • Understanding Map Projections

Longitude 2Longitude length 3Longitude of center 20, 103Longitude of first point 20Longitude of origin 20, 103Longitude of second point 20Longitude range 2Longitudes 103Loximuthal 68MMajor axes 103Map projections 103definition 11Map scales 103McBryde–Thomas Flat-PolarQuartic 69Mercator 16, 70Meridians 2, 103Meridional convergence 3Miller Cylindrical 71Minor axes 103Mollweide 72Molodensky method 26NNAD 1927 6, 7, 27, 28, 103NAD 1983 27, 28, 103NAD 1983 and WGS 1984 7NAD27 7NAD83 7NADCON method 27National Transformation version 128National Transformation version 228New Zealand National Grid 73North American Datum of 1927 7North American Datum of 1983 7NTv2. See NationalTransformation version 2OOblique aspect 103Oblique Mercator 63Oblique projectionsAzimuthal Equidistant 38Bipolar 40Double Stereographic 46Hotine 63Krovak 64Lambert Azimuthal Equal Area65Orthographic 74Polar Stereographic 77Stereographic 87Vertical Near-Side Perspective95Old Hawaiian datum 27Origin point 6Orthographic 18, 74PParallels 2, 103Parametersangular 20azimuth 101central meridian 101defined 103false easting 102false northing 102latitude of center 103latitude of origin 103linear 20longitude of center 103longitude of origin 103scale factor 104standard lines 104standard parallel 104Perspective 75Planar projectionsAzimuthal Equidistant 38defined 17, 104Double Stereographic 46Gnomonic 60Orthographic 74Perspective 75Polar Stereographic 77Planar projections (continued)Stereographic 87Vertical Near-Side Perspective75, 95Plate Carrée 76Polar aspect 104Polar Stereographic 77Polyconic 78Position Vector method 25Prime meridians 2defined 104Greenwich 102Projected coordinate systems 104and geographic coordinatesystems 10definition 10units 10x-axis 10y-axis 10Projection parametersazimuth 20central meridian 20central parallel 20defined 20false easting 20false northing 20latitude of center 20latitude of first point 20latitude of origin 20latitude of second point 20longitude of center 20longitude of first point 20longitude of origin 20longitude of second point 20scale factor 20standard parallel 1 20standard parallel 2 20Projection surfaces 11Projection types 13Projectionsazimuthal 17, 101conformal 12, 101conic 14, 101cylindrical 16, 101distortions 11equal-area 12, 101equidistant 12, 102equivalent 102Index • 109

Projections (continued)planar 17, 104secant 104tangent 104true-direction 12, 104QQuartic Authalic 79RRadius 104Rectified Skewed Orthomorphic 80Reference ellipsoids. See EllipsoidsRhumb lines 16, 104Robinson 81RSO. See Rectified Skewed OrthomorphicSSanson–Flamsteed 83Scale factor 20, 104Secant projections 13, 14, 104Semimajor axes 4, 104Semiminor axes 4, 104Seven-parameter method 25Simple Conic 82Sinusoidal 83Space Oblique Mercator 84SPCS. See State Plane Coordinate SystemSpheres 104Spherical coordinate systems 104Spheroids 104discussion 4eccentricity 101ellipticity 101flattening 4, 102major axis 103minor axis 103semimajor axis 4, 104semiminor axis 4, 104using different 4Standard lines 13, 104Standard parallel 1 20Standard parallel 2 20Standard parallels 14, 104State Plane Coordinate System 85, 104Stereographic 87Polar case 77TTangent projections 13, 104Three-parameter method 25Times 88Transformationsdatum 101Transverse Mercator 16, 89Transverse projections 16True-direction projections 104defined 12Two-Point Equidistant 91UUnits of measure 104angular 101linear 103Universal Polar Stereographic 92Universal Transverse Mercator 93, 105UPS 92UTM 93, 105VVan Der Grinten I 94Vertical Near-Side Perspective 75, 95WWGS 1984 105WGS 1984 and NAD 1983 7Winkel I 96Winkel II 97Winkel Tripel 98XX-axis 10XYZ coordinate systems 24YY-axis 10110 • Understanding Map Projections