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Non-uniform rational B-spline 83 Manipulating NURBS objects A number of transformations can be applied to a NURBS object. For instance, if some curve is defined using a certain degree and N control points, the same curve can be expressed using the same degree and N+1 control points. In the process a number of control points change position and a knot is inserted in the knot vector. These manipulations are used extensively during interactive design. When adding a control point, the shape of the curve should stay the same, forming the starting point for further adjustments. A number of these operations are discussed below. [6] Knot insertion As the term suggests, knot insertion inserts a knot into the knot vector. If the degree of the curve is , then control points are replaced by new ones. The shape of the curve stays the same. A knot can be inserted multiple times, up to the maximum multiplicity of the knot. This is sometimes referred to as knot refinement and can be achieved by an algorithm that is more efficient than repeated knot insertion. Knot removal Knot removal is the reverse of knot insertion. Its purpose is to remove knots and the associated control points in order to get a more compact representation. Obviously, this is not always possible while retaining the exact shape of the curve. In practice, a tolerance in the accuracy is used to determine whether a knot can be removed. The process is used to clean up after an interactive session in which control points may have been added manually, or after importing a curve from a different representation, where a straightforward conversion process leads to redundant control points. Degree elevation A NURBS curve of a particular degree can always be represented by a NURBS curve of higher degree. This is frequently used when combining separate NURBS curves, e.g. when creating a NURBS surface interpolating between a set of NURBS curves or when unifying adjacent curves. In the process, the different curves should be brought to the same degree, usually the maximum degree of the set of curves. The process is known as degree elevation. Curvature The most important property in differential geometry is the curvature . It describes the local properties (edges, corners, etc.) and relations between the first and second derivative, and thus, the precise curve shape. Having determined the derivatives it is easy to compute or approximated as the arclength from the second derivate . The direct computation of the curvature with these equations is the big advantage of parameterized curves against their polygonal representations.
Non-uniform rational B-spline 84 Example: a circle Non-rational splines or Bézier curves may approximate a circle, but they cannot represent it exactly. Rational splines can represent any conic section, including the circle, exactly. This representation is not unique, but one possibility appears below: x y z weight 1 0 0 1 1 1 0 0 1 0 1 −1 1 0 −1 0 0 1 −1 −1 0 0 −1 0 1 1 −1 0 1 0 0 1 The order is three, since a circle is a quadratic curve and the spline's order is one more than the degree of its piecewise polynomial segments. The knot vector is . The circle is composed of four quarter circles, tied together with double knots. Although double knots in a third order NURBS curve would normally result in loss of continuity in the first derivative, the control points are positioned in such a way that the first derivative is continuous. (In fact, the curve is infinitely differentiable everywhere, as it must be if it exactly represents a circle.) The curve represents a circle exactly, but it is not exactly parametrized in the circle's arc length. This means, for example, that the point at does not lie at (except for the start, middle and end point of each quarter circle, since the representation is symmetrical). This is obvious; the x coordinate of the circle would otherwise provide an exact rational polynomial expression for , which is impossible. The circle does make one full revolution as its parameter goes from 0 to , but this is only because the knot vector was arbitrarily chosen as multiples of . References • Les Piegl & Wayne Tiller: The NURBS Book, Springer-Verlag 1995–1997 (2nd ed.). The main reference for Bézier, B-Spline and NURBS; chapters on mathematical representation and construction of curves and surfaces, interpolation, shape modification, programming concepts. • Dr. Thomas Sederberg, BYU NURBS, http:/ / cagd. cs. byu. edu/ ~557/ text/ ch6. pdf • Dr. Lyle Ramshaw. Blossoming: A connect-the-dots approach to splines, Research Report 19, Compaq Systems Research Center, Palo Alto, CA, June 1987 • David F. Rogers: An Introduction to NURBS with Historical Perspective, Morgan Kaufmann Publishers 2001. Good elementary book for NURBS and related issues.
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3D Rendering PDF generated using th
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Image-based lighting 64 Image plane
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References Article Sources and Cont
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3D rendering 2 Non real-time Animat
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3D rendering 4 The shaded three-dim
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Ambient occlusion 6 ambient occlusi
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Ambient occlusion 8 • ShadeVis (h
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Anisotropic filtering 10 will only
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Beam tracing 12 Beam tracing Beam t
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Bilinear filtering 14 Are all true.
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Binary space partitioning 16 togeth
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Binary space partitioning 18 Other
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Binary space partitioning 20 [1] Bi
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Bounding interval hierarchy 22 Prop
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Bounding volume 24 bounding boxes b
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Bump mapping 26 Bump mapping Bump m
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CatmullClark subdivision surface 28
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CatmullClark subdivision surface 30
Page 37 and 38: Conversion between quaternions and
Page 39 and 40: Cube mapping 34 Advantages Cube map
Page 41 and 42: Cube mapping 36 Related A large set
Page 43 and 44: Diffuse reflection 38 2), or, of co
Page 45 and 46: Displacement mapping 40 Meaning of
Page 47 and 48: DooSabin subdivision surface 42 Ext
Page 49 and 50: False radiosity 44 False radiosity
Page 51 and 52: Geometry pipelines 46 Geometry pipe
Page 53 and 54: Global illumination 48 Rendering wi
Page 55 and 56: Gouraud shading 50 Gouraud shading
Page 57 and 58: Graphics pipeline 52 Graphics pipel
Page 59 and 60: Graphics pipeline 54 References 1.
Page 61 and 62: Hidden surface determination 56 imp
Page 63 and 64: High dynamic range rendering 58 Hig
Page 65 and 66: High dynamic range rendering 60 Ton
Page 67 and 68: High dynamic range rendering 62 Fro
Page 69 and 70: High dynamic range rendering 64 •
Page 71 and 72: Irregular Z-buffer 66 Applications
Page 73 and 74: Lambert's cosine law 68 than would
Page 75 and 76: Lambertian reflectance 70 Lambertia
Page 77 and 78: Level of detail 72 Well known appro
Page 79 and 80: Level of detail 74 Hierarchical LOD
Page 81 and 82: Newell's algorithm 76 Newell's algo
Page 83 and 84: Non-uniform rational B-spline 78 Us
Page 85 and 86: Non-uniform rational B-spline 80 of
Page 87: Non-uniform rational B-spline 82 ar
Page 91 and 92: Normal mapping 86 How it works To c
Page 93 and 94: OrenNayar reflectance model 88 Oren
Page 95 and 96: OrenNayar reflectance model 90 , ,
Page 97 and 98: Painter's algorithm 92 The algorith
Page 99 and 100: Parallax mapping 94 • Parallax Ma
Page 101 and 102: Particle system 96 A cube emitting
Page 103 and 104: Path tracing 98 History Further inf
Page 105 and 106: Path tracing 100 Scattering distrib
Page 107 and 108: Phong reflection model 102 Visual i
Page 109 and 110: Phong reflection model 104 Because
Page 111 and 112: Phong shading 106 Visual illustrati
Page 113 and 114: Photon mapping 108 Rendering (2nd p
Page 115 and 116: Photon tracing 110 Advantages and d
Page 117 and 118: Potentially visible set 112 • Can
Page 119 and 120: Potentially visible set 114 Externa
Page 121 and 122: Procedural generation 116 increases
Page 123 and 124: Procedural generation 118 • Softi
Page 125 and 126: Procedural generation 120 Reference
Page 127 and 128: Procedural texture 122 Self-organiz
Page 129 and 130: Procedural texture 124 References [
Page 131 and 132: 3D projection 126 The distance of t
Page 133 and 134: Quaternions and spatial rotation 12
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Quaternions and spatial rotation 13
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Radiosity 140 Overview of the radio
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Radiosity 142 This is sometimes kno
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Radiosity 144 References [1] " Mode
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Ray casting 146 the light will reac
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Ray tracing 148 Typically, each ray
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Ray tracing 150 independence of eac
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Ray tracing 152 On June 12, 2008 In
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Reflection 154 Reflection Reflectio
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Reflection 156 Glossy Reflection Fu
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Reflection mapping 158 Cube mapping
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Render Output unit 160 Render Outpu
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Rendering 162 • indirect illumina
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Rendering 164 Ray tracing Ray traci
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Rendering 166 Academic core The imp
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Rendering 168 • 1984 Distributed
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Retained mode 170 Retained mode In
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Scanline rendering 172 Comparison w
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Screen Space Ambient Occlusion 174
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Screen Space Ambient Occlusion 176
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Shadow mapping 178 Algorithm overvi
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Shadow mapping 180 Drawing the scen
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Shadow mapping 182 Further reading
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Shadow volume 184 There is also a p
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Shadow volume 186 The depth fail me
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Silhouette edge 188 Silhouette edge
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Specular highlight 190 Specular hig
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Specular highlight 192 normalized o
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Sphere mapping 194 Sphere mapping I
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Stencil codes 196 Stencil codes Ste
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Stencil codes 198 Stencils The shap
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Stencil codes 200 [7] Wellein, G et
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Subdivision surface 202 used a four
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Subsurface scattering 204 Subsurfac
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Subsurface scattering 206 External
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Surface normal 208 If a (possibly n
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Surface normal 210 Normal in geomet
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Texture filtering 212 Texture filte
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Texture mapping 214 Texture mapping
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Texture mapping 216 constant distan
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Texture synthesis 218 • Structure
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Texture synthesis 220 Pattern-based
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Texture synthesis 222 • Micro-tex
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UV mapping 224 A UV map can either
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Vertex 226 Polytope vertices are re
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Vertex Buffer Object 228 //Make the
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Vertex Buffer Object 230 GLuint sha
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Vertex Buffer Object 232 vertexes *
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Virtual actor 234 Virtual actor A v
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Virtual actor 236 exercises, and ev
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Volume rendering 238 Volume ray cas
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Volume rendering 240 Maximum intens
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Volume rendering 242 Image-based me
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Volumetric lighting 244 References
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Voxel 246 • Outcast, a game made
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Z-buffering 248 Z-buffering In comp
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Z-buffering 250 } } display COLOR a
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Z-fighting 252 Z-fighting Z-fightin
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Appendix 3D computer graphics softw
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3D computer graphics software 256
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3D computer graphics software 258
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3D computer graphics software 260 2
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Article Sources and Contributors 26
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Article Sources and Contributors 26
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Image Sources, Licenses and Contrib
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Image Sources, Licenses and Contrib
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