Plane Geometry - Bruce E. Shapiro

More documents

Recommendations

Info

$NAME: Math 103L: Exercise on Functions ... - Bruce E. Shapiro$

$Introducing Latex - Bruce E. Shapiro$

$Applied Differential Equations, Math 280 at CSUN - Bruce E. Shapiro$

$Math 103 Section 1.2: Linear Equations and ... - Bruce E. Shapiro$

310 SECTION 52. ARC LENGTHThus the length of any curve γ between the points A = γ(a) and B = γ(b)on the surface of the unit sphere is given byL(γ) =∫ bSince θ ′ (t) sin 2 φ(t) ≥ 0, we conclude thata»θ ′2 (t) sin 2 ϕ(t) + ϕ ′2 (t) dtTaking square roots,Thusθ ′2 (t) sin 2 ϕ(t) + ϕ ′2 (t) ≥ ϕ ′2 (t)»θ ′2 (t) sin 2 ϕ(t) + ϕ ′2 (t) ≥ |ϕ ′ (t)|L(γ) ≥≥∣∫ ba∫ ba|ϕ ′ (t)| dt|ϕ ′ (t)|dt∣= |ϕ(b) − ϕ(a)|Without any loss of generality we can assume that the points A and B havethe same longitude. To see this, observe that since A, B, and O (the center)are three distinct points that we can use them to define a plane that cutsthe sphere in half. Use this plane to define the xz plane of our coordinatesystem. Then we can alwayws find a rigid motion and rotation that takesthis coordinate system to any other coordinate system.Since the two points are on the same line of longitude,and henceµ(∠BOA) = |ϕ(b) − ϕ(a)|L(γ) ≥ µ(∠BOA)If we measure the angle in radians, then the arc-length from A to B isL(ÃB) = rµ(∠BOA) = µ(∠BOA)« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 52. ARC LENGTH 311where we have introduced the symbol ÃB to represent an arc of a greatcircle,and where r = 1 is the radius of the sphere. ThusL(γ) ≥ L(ÃB)We have proven the following theorem.Theorem 52.4 The path of shortest distance, on the surface of a sphere,between two points A and B on the surface of the sphere, lies along a greatcircle connecting the two points. More specifically, if γ(t) : [a, b] ↦→ S is anycurve on on the sphere such that γ(a) = A and γ(b) = B thenL(γ) ≥ L(ÃB)Figure 52.3: The arc of the great circle from A to B is shorter than anyother curve on the surface of the sphere connecting A to B.γ(t)Therefore we will make the following assumption: a line in spherical geometrycorresponds to the arc of a great circle connecting the two points.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
Page 1:
Foundations of GeometryLecture Note
Page 4 and 5:
ivCONTENTS30 Triangles in Neutral G
Page 6 and 7:
2 SECTION 1. PREFACEare enrolled in
Page 8 and 9:
4 SECTION 1. PREFACEre-learning it
Page 10 and 11:
6 SECTION 2. NCTM3. To guide in the
Page 12 and 13:
8 SECTION 2. NCTMTechnology. Techno
Page 14 and 15:
10 SECTION 2. NCTM« CC BY-NC-ND 3.
Page 16 and 17:
12 SECTION 3. CA STANDARDIn 2005 Ca
Page 18 and 19:
14 SECTION 3. CA STANDARDThe Califo
Page 20 and 21:
16 SECTION 3. CA STANDARDCalifornia
Page 22 and 23:
18 SECTION 4. COMMON COREAt all lev
Page 24 and 25:
20 SECTION 5. LOGIC AND PROOFour th
Page 26 and 27:
22 SECTION 5. LOGIC AND PROOFRene D
Page 28 and 29:
24 SECTION 5. LOGIC AND PROOF“I
Page 30 and 31:
26 SECTION 6. REAL NUMBERSThe inter
Page 32 and 33:
28 SECTION 6. REAL NUMBERS“The Su
Page 34 and 35:
30 SECTION 7. EUCLID’S ELEMENTSEu
Page 36 and 37:
32 SECTION 7. EUCLID’S ELEMENTSFi
Page 38 and 39:
34 SECTION 8. HILBERT’S AXIOMSiom
Page 40 and 41:
36 SECTION 8. HILBERT’S AXIOMS«
Page 42 and 43:
38 SECTION 9. BIRKHOFF/MACLANE AXIO
Page 44 and 45:
40 SECTION 9. BIRKHOFF/MACLANE AXIO
Page 46 and 47:
42 SECTION 10. THE SMSG AXIOMSP is
Page 48 and 49:
44 SECTION 10. THE SMSG AXIOMS« CC
Page 50 and 51:
46 SECTION 11. THE UCSMP AXIOMSin s
Page 52 and 53:
48 SECTION 11. THE UCSMP AXIOMS« C
Page 54 and 55:
50 SECTION 12. VENEMA’S AXIOMS3.
Page 56 and 57:
52 SECTION 12. VENEMA’S AXIOMS«
Page 58 and 59:
54 SECTION 13. INCIDENCE GEOMETRYFi
Page 60 and 61:
56 SECTION 13. INCIDENCE GEOMETRYEx
Page 62 and 63:
58 SECTION 13. INCIDENCE GEOMETRYFi
Page 64 and 65:
60 SECTION 13. INCIDENCE GEOMETRY«
Page 66 and 67:
62 SECTION 14. BETWEENNESSFigure 14
Page 68 and 69:
64 SECTION 14. BETWEENNESSTheorem 1
Page 70 and 71:
66 SECTION 14. BETWEENNESS(a) To sh
Page 72 and 73:
68 SECTION 14. BETWEENNESSThe follo
Page 74 and 75:
70 SECTION 14. BETWEENNESSExample 1
Page 76 and 77:
72 SECTION 14. BETWEENNESSorf(A) >
Page 78 and 79:
74 SECTION 14. BETWEENNESS« CC BY-
Page 80 and 81:
76 SECTION 15. THE PLANE SEPARATION
Page 82 and 83:
78 SECTION 15. THE PLANE SEPARATION
Page 84 and 85:
80 SECTION 16. ANGLESFigure 16.1: T
Page 86 and 87:
82 SECTION 16. ANGLESCorollary 16.6
Page 88 and 89:
84 SECTION 16. ANGLESFigure 16.6: I
Page 90 and 91:
86 SECTION 16. ANGLESD cannot be in
Page 92 and 93:
88 SECTION 17. THE CROSSBAR THEOREM
Page 94 and 95:
90 SECTION 17. THE CROSSBAR THEOREM
Page 96 and 97:
92 SECTION 18. LINEAR PAIRSFigure 1
Page 98 and 99:
94 SECTION 18. LINEAR PAIRSγ + ∠
Page 100 and 101:
96 SECTION 19. ANGLE BISECTORSwith
Page 102 and 103:
98 SECTION 19. ANGLE BISECTORSAngle
Page 104 and 105:
100 SECTION 20. THE CONTINUITY AXIO
Page 106 and 107:
102 SECTION 20. THE CONTINUITY AXIO
Page 108 and 109:
104 SECTION 21. SIDE-ANGLE-SIDEBirk
Page 110 and 111:
106 SECTION 21. SIDE-ANGLE-SIDEFigu
Page 112 and 113:
108 SECTION 22. NEUTRAL GEOMETRYEll
Page 114 and 115:
110 SECTION 22. NEUTRAL GEOMETRY«
Page 116 and 117:
112 SECTION 23. ANGLE-SIDE-ANGLEFig
Page 118 and 119:
114 SECTION 23. ANGLE-SIDE-ANGLE«
Page 120 and 121:
116 SECTION 24. EXTERIOR ANGLESTheo
Page 122 and 123:
118 SECTION 24. EXTERIOR ANGLES« C
Page 124 and 125:
120 SECTION 25. ANGLE-ANGLE-SIDEBy
Page 126 and 127:
122 SECTION 26. SIDE-SIDE-SIDECase
Page 128 and 129:
124 SECTION 26. SIDE-SIDE-SIDEBy si
Page 130 and 131:
126 SECTION 27. SCALENE AND TRIANGL
Page 132 and 133:
128 SECTION 27. SCALENE AND TRIANGL
Page 134 and 135:
130 SECTION 28. CHARACTERIZATION OF
Page 136 and 137:
132 SECTION 28. CHARACTERIZATION OF
Page 138 and 139:
134 SECTION 29. TRANSVERSALSFigure
Page 140 and 141:
136 SECTION 29. TRANSVERSALSCorolla
Page 142 and 143:
138 SECTION 30. TRIANGLES IN NEUTRA
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
146 SECTION 31. QUADRILATERALS IN N
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
156 SECTION 32. THE EUCLIDEAN PARAL
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
168 SECTION 33. RECTANGLESLemma 33.
Page 174 and 175:
170 SECTION 33. RECTANGLESFigure 33
Page 176 and 177:
172 SECTION 33. RECTANGLES« CC BY-
Page 178 and 179:
174 SECTION 34. THE PARALLEL PROJEC
Page 180 and 181:
176 SECTION 34. THE PARALLEL PROJEC
Page 182 and 183:
178 SECTION 35. SIMILAR TRIANGLESBy
Page 184 and 185:
180 SECTION 35. SIMILAR TRIANGLESCo
Page 186 and 187:
182 SECTION 36. TRIANGLE CENTERSFig
Page 188 and 189:
184 SECTION 36. TRIANGLE CENTERSThe
Page 190 and 191:
186 SECTION 36. TRIANGLE CENTERSCen
Page 192 and 193:
188 SECTION 37. AREADefinition 37.5
Page 194 and 195:
190 SECTION 37. AREAToday’s Lesso
Page 196 and 197:
192 SECTION 38. THE PYTHAGOREAN THE
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 206 and 207:
202 SECTION 39. CIRCLESFigure 39.1:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
208 SECTION 39. CIRCLESTheorem 39.1
Page 214 and 215:
210 SECTION 39. CIRCLESLet D be the
Page 216 and 217:
212 SECTION 39. CIRCLES« CC BY-NC-
Page 218 and 219:
214 SECTION 40. CIRCLES AND TRIANGL
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
220 SECTION 41. EUCLIDEAN CIRCLESFi
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
228 SECTION 41. EUCLIDEAN CIRCLESBy
Page 234 and 235:
230 SECTION 42. AREA AND CIRCUMFERE
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
Page 242 and 243:
238 SECTION 43. INDIANA BILL 246the
Page 244 and 245:
240 SECTION 44. ESTIMATING πFigure
Page 246 and 247:
242 SECTION 44. ESTIMATING πso tha
Page 248 and 249:
244 SECTION 44. ESTIMATING πn π n
Page 250 and 251:
246 SECTION 44. ESTIMATING πTable
Page 252 and 253:
248 SECTION 44. ESTIMATING πnums =
Page 254 and 255:
250 SECTION 44. ESTIMATING π« CC
Page 256 and 257:
252 SECTION 45. EUCLIDEAN CONSTRUCT
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
Page 264 and 265: 260 SECTION 45. EUCLIDEAN CONSTRUCT
Page 266 and 267: 262 SECTION 45. EUCLIDEAN CONSTRUCT
Page 268 and 269: 264 SECTION 46. HYPERBOLIC GEOMETRY
Page 276 and 277: 272SECTION 47.PERPENDICULAR LINES I
Page 278 and 279: 274SECTION 47.PERPENDICULAR LINES I
Page 280 and 281: 276 SECTION 48. PARALLEL LINES IN H
Page 288 and 289: 284 SECTION 49. TRIANGLES IN HYPERB
Page 294 and 295: 290 SECTION 50. AREA IN HYPERBOLIC
Page 302 and 303: 298 SECTION 51. THE POINCARE DISK M
Page 310 and 311: 306 SECTION 52. ARC LENGTHAs we see
Page 312 and 313: 308 SECTION 52. ARC LENGTHProof. Su
Page 316 and 317: 312 SECTION 52. ARC LENGTHMeasuring
Page 318 and 319: 314 SECTION 53. SPHERICAL GEOMETRYW
Page 320 and 321: 316 SECTION 53. SPHERICAL GEOMETRYF
Page 322 and 323: 318 SECTION 53. SPHERICAL GEOMETRYP
Page 324 and 325: 320 APPENDIX A. SYMBOLS USEDAppendi
Page 326 and 327: 322 APPENDIX A. SYMBOLS USEDTechnol
show all

Plane Geometry - Bruce E. Shapiro

Create successful ePaper yourself

Delete template?

Save as template?