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3D projection 127 References [1] Ingrid Carlbom, Joseph Paciorek (1978). "Planar Geometric Projections and Viewing Transformations" (http:/ / www. cs. uns. edu. ar/ cg/ clasespdf/ p465carlbom. pdf). ACM Computing Surveys 10 (4): 465–502. doi:10.1145/356744.356750. . [2] Riley, K F (2006). Mathematical Methods for Physics and Engineering. Cambridge University Press. pp. 931, 942. doi:10.2277/0521679710. ISBN 0-521-67971-0. [3] Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, Mass.: Addison-Wesley Pub. Co.. pp. 146–148. ISBN 0-201-02918-9. [4] Sonka, M; Hlavac, V; Boyle, R (1995). Image Processing, Analysis & Machine Vision (2nd ed.). Chapman and Hall. pp. 14. ISBN 0-412-45570-6 External links • A case study in camera projection (http:/ / nccasymposium. bmth. ac. uk/ 2007/ muhittin_bilginer/ index. html) • Creating 3D Environments from Digital Photographs (http:/ / nccasymposium. bmth. ac. uk/ 2009/ McLaughlin_Chris/ McLaughlin_C_WebBasedNotes. pdf) Further reading • Kenneth C. Finney (2004). 3D Game Programming All in One (http:/ / books. google. com/ ?id=cknGqaHwPFkC& pg=PA93& dq="3D+ projection"). Thomson Course. pp. 93. ISBN 978-1-59200-136-1. Quaternions and spatial rotation Unit quaternions provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock. Compared to rotation matrices they are more numerically stable and may be more efficient. Quaternions have found their way into applications in computer graphics, computer vision, robotics, navigation, molecular dynamics and orbital mechanics of satellites. [1] When used to represent rotation, unit quaternions are also called versors, or rotation quaternions. When used to represent an orientation (rotation relative to a reference position), they are called orientation quaternions or attitude quaternions.
Quaternions and spatial rotation 128 Quaternion rotation operations A very formal explanation of the properties used in this section is given by Altmann. [2] The hypersphere of rotations Visualizing the space of rotations Unit quaternions represent the mathematical space of rotations in three dimensions in a very straightforward way. The correspondence between rotations and quaternions can be understood by first visualizing the space of rotations itself. In order to visualize the space of rotations, it helps to consider a simpler case. Any rotation in three dimensions can be described by a rotation by some angle about some axis. Consider the special case in which the axis of rotation lies in the xy plane. We can then specify the axis of one of these rotations by a point on a circle, and we can use the radius of the circle to specify the angle of rotation. Similarly, a rotation whose axis of rotation lies in the "xy" plane can be described as a point on a sphere of fixed radius in three dimensions. Beginning at the north pole of a sphere in three dimensional space, we specify the point at the north pole to be the identity rotation (a zero angle rotation). Just as in the case of the identity rotation, no axis of rotation is defined, and the angle of rotation (zero) is irrelevant. A rotation having a very small rotation angle can be specified by a slice through the sphere parallel to the xy plane and very near the north pole. The circle defined by this slice will be very small, Two rotations by different angles and different axes in the space of rotations. The length of the vector is related to the magnitude of the rotation. corresponding to the small angle of the rotation. As the rotation angles become larger, the slice moves in the negative "z" direction, and the circles become larger until the equator of the sphere is reached, which will correspond to a rotation angle of 180 degrees. Continuing southward, the radii of the circles now become smaller (corresponding to the absolute value of the angle of the rotation considered as a negative number). Finally, as the south pole is reached, the circles shrink once more to the identity rotation, which is also specified as the point at the south pole. Notice that a number of characteristics of such rotations and their representations can be seen by this visualization. The space of rotations is continuous, each rotation has a neighborhood of rotations which are nearly the same, and this neighborhood becomes flat as the neighborhood shrinks. Also, each rotation is actually represented by two antipodal points on the sphere, which are at opposite ends of a line through the center of the sphere. This reflects the fact that each rotation can be represented as a rotation about some axis, or, equivalently, as a negative rotation about an axis pointing in the opposite direction (a so-called double cover). The "latitude" of a circle representing a particular rotation angle will be half of the angle represented by that rotation, since as the point is moved from the north to south pole, the latitude ranges from zero to 180 degrees, while the angle of rotation ranges from 0 to 360 degrees. (the "longitude" of a point then represents a particular axis of rotation.) Note however that this set of rotations is not closed under composition. Two successive rotations with axes in the xy plane will not necessarily give a rotation whose axis lies in the xy plane, and thus cannot be represented as a point on the sphere. This will not be the case with a general rotation in 3-space, in which rotations do form a closed set under composition.
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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
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Conversion between quaternions and
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Cube mapping 34 Advantages Cube map
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Cube mapping 36 Related A large set
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Diffuse reflection 38 2), or, of co
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Displacement mapping 40 Meaning of
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DooSabin subdivision surface 42 Ext
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False radiosity 44 False radiosity
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Geometry pipelines 46 Geometry pipe
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Global illumination 48 Rendering wi
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Gouraud shading 50 Gouraud shading
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Graphics pipeline 52 Graphics pipel
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Graphics pipeline 54 References 1.
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Hidden surface determination 56 imp
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High dynamic range rendering 58 Hig
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High dynamic range rendering 60 Ton
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High dynamic range rendering 62 Fro
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High dynamic range rendering 64 •
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Irregular Z-buffer 66 Applications
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Lambert's cosine law 68 than would
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Lambertian reflectance 70 Lambertia
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Level of detail 72 Well known appro
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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 and 88: Non-uniform rational B-spline 82 ar
Page 89 and 90: Non-uniform rational B-spline 84 Ex
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: 3D projection 126 The distance of t
Page 135 and 136: Quaternions and spatial rotation 13
Page 145 and 146: Radiosity 140 Overview of the radio
Page 147 and 148: Radiosity 142 This is sometimes kno
Page 149 and 150: Radiosity 144 References [1] " Mode
Page 151 and 152: Ray casting 146 the light will reac
Page 153 and 154: Ray tracing 148 Typically, each ray
Page 155 and 156: Ray tracing 150 independence of eac
Page 157 and 158: Ray tracing 152 On June 12, 2008 In
Page 159 and 160: Reflection 154 Reflection Reflectio
Page 161 and 162: Reflection 156 Glossy Reflection Fu
Page 163 and 164: Reflection mapping 158 Cube mapping
Page 165 and 166: Render Output unit 160 Render Outpu
Page 167 and 168: Rendering 162 • indirect illumina
Page 169 and 170: Rendering 164 Ray tracing Ray traci
Page 171 and 172: Rendering 166 Academic core The imp
Page 173 and 174: Rendering 168 • 1984 Distributed
Page 175 and 176: Retained mode 170 Retained mode In
Page 177 and 178: Scanline rendering 172 Comparison w
Page 179 and 180: Screen Space Ambient Occlusion 174
Page 181 and 182: 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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