Mathematical Tools In Computer Graphics With C# Implementations

By: Alexandre Hardy, Willi-hans Steeb

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Paperback | 7 January 2008

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494 Pages

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Mathematics is vital for an understanding of computer graphics. This volume helps the reader gain such an understanding by presenting all introductory and most advanced topics in the field of computer graphics with mathematical descriptions and derivations. Offering a balance of theory, applications, and code, the underlying numerical methods and algorithms are derived and a large number of examples are given. The book begins with a discussion of basic graphics tools such as vectors, matrices, and quaternions, and then builds up to more advanced topics such as the intersection of three-dimensional objects. Both classical and newer topics, such as parameterization, wavelets, fractals, and geometry images, are covered. In particular, the book contains all of the classes in C# necessary for computer graphics, providing a full explanation of the C# code and C# implementations for almost all algorithms.

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Non-Fiction Computing & I.T.Computer Programming & Software Development Programming & Scripting Languages Software Engineering Graphical & Digital Media Applications Business Applications Mathematical & Statistical Software Booktopia Publisher Services World Scientific Publishing

List of Figures	p. xiii
Notation	p. xvii
Vectors, Matrices and Transforms	p. 1
Vector Spaces	p. 1
Points and Vectors	p. 4
Homogeneous Coordinates	p. 4
Representing Objects by Points	p. 5
Affine Transformations	p. 5
Introduction and Definitions	p. 5
Scaling	p. 6
Translation	p. 7
Rotation	p. 7
Concatenation of Transforms	p. 10
Projection	p. 11
Quaternions	p. 13
C# Implementation	p. 15
Lighting	p. 23
Shading	p. 23
Affine Transforms and Normal Vectors	p. 24
Local Lighting Models	p. 25
The Phong Lighting Model	p. 25
Emissive Properties	p. 26
Ambient Reflection	p. 26
Diffuse Reflection	p. 26
Specular Reflection	p. 27
Multiple Colored Light Sources	p. 28
Attenuation	p. 29
Lights	p. 30
Spot Lights	p. 30
Transparent Objects	p. 31
Cook-Torrance Model	p. 31
Bidirectional Reflectivity	p. 31
Cook-Torrance Model	p. 32
Microfacet Distribution Term	p. 33
Geometric Surface Occlusion Term	p. 33
Fresnel Term	p. 37
Beer-Lambert Law	p. 38
C# Implementation	p. 39
Rasterization	p. 49
Pixels	p. 49
Drawing Lines	p. 49
Bresenham's Algorithm for Lines	p. 50
Drawing Circles	p. 51
Bresenham's Algorithm for Circles	p. 51
Filling	p. 53
Gouraud Shading	p. 54
Rasterization in C#	p. 55
Drawing Pixels	p. 57
Bresenham's Algorithms in C#	p. 57
Fractals	p. 59
Mandelbrot Set	p. 60
Julia Set	p. 62
Iterated Function Systems	p. 64
L-Systems and Fractals	p. 65
Kronecker Product and Fractals	p. 69
Definitions	p. 69
Kronecker Product Fractals	p. 71
Curves	p. 75
Introduction	p. 75
Affine Invariance	p. 76
Convex Hull	p. 77
Lagrange Interpolation	p. 78
C# Implementation	p. 79
Bezier Curves	p. 83
Affine Invariance	p. 83
Convex Hull	p. 84
Derivative at Edges	p. 84
Piecewise Continuous Bezier Curves	p. 86
Rendering	p. 87
Rational Bezier Curves	p. 91
Bezier Curves: Conic Sections	p. 91
C# Implementation	p. 91
Catmull-Rom Splines	p. 92
Bessel-Overhauser Splines	p. 93
Tension-Continuity-Bias Splines	p. 93
Uniform B-Splines	p. 94
Affine Invariance	p. 98
Convex Hull	p. 98
Cox-de Boor Formula	p. 98
C# Implementation	p. 99
Non-Uniform B-Splines	p. 99
Interpolating with B-Splines	p. 100
Periodic Interpolation	p. 102
Non-Uniform Rational B-Splines	p. 104
Trigonometric Interpolation	p. 104
METAPOST and Bezier Curves	p. 108
METAPOST Example	p. 110
Curvature and Torsion	p. 112
Harmonic Interpolation	p. 117
Interpolation	p. 118
Odd Case	p. 120
Even Case	p. 122
Examples	p. 125
Curvature Plots	p. 132
Numerical Stability	p. 136
Affine Invariance	p. 138
Convex Hull Property	p. 141
C# Implementation of Harmonic Interpolation	p. 141
Chebyshev Polynomials	p. 142
Odd Case	p. 143
Even Case	p. 144
Non-Uniform Harmonic Interpolation	p. 144
Wavelets	p. 153
Introduction	p. 153
One-Dimensional Wavelets	p. 154
Two-Dimensional Wavelets	p. 158
Curves	p. 162
C# Implementation	p. 164
Surfaces	p. 167
Parametric Surfaces	p. 167
Tensor Product Surfaces	p. 168
Bezier Surfaces	p. 169
Tensor Product Bezier Surfaces	p. 170
Triangular Bezier Surfaces	p. 171
Rational Bezier Surfaces	p. 173
Bezier Surface Interpolation	p. 173
B-Spline Tensor Product Surfaces	p. 173
B-Spline Surface Interpolation	p. 174
Subdivision Surfaces	p. 177
Loop Subdivision	p. 180
Modified Butterfly Subdivision	p. 183
[radical]3 Subdivision	p. 185
Interpolating [radical]3 Subdivision	p. 187
Catmull-Clark Subdivision	p. 191
Doo-Sabin Subdivision	p. 192
Comparison	p. 195
Interpolation with Subdivision Surfaces	p. 198
Curvature of Surfaces	p. 200
Harmonic Surfaces	p. 206
Tensor Product Surface	p. 206
Harmonic Subdivision	p. 207
Local Harmonic Subdivision	p. 208
Local Harmonic Interpolation for Curves	p. 209
Parametric Distance	p. 210
Subdivision Rules	p. 212
Irregular Vertices	p. 214
Boundaries	p. 217
Geometry Images and Parameterization	p. 219
Cutting a Mesh into a Disk	p. 219
Parameterization	p. 222
Rasterization of the Geometry Image	p. 228
Interpolation of Geometry Images	p. 229
Geometry Images - Approximation	p. 237
Rendering	p. 238
Approximating Basis Functions	p. 240
Combined Results	p. 243
Curvature	p. 245
C# Implementation	p. 257
Raytracing	p. 333
Raytracing Process	p. 333
Representation of a Ray	p. 338
Reflection	p. 338
Refraction	p. 339
Intersections	p. 341
Sphere	p. 342
Infinite Plane	p. 342
Triangles	p. 343
Effect of Transforms	p. 344
C# Implementation of a Raytracer	p. 344
Implicit Surfaces	p. 368
Sphere Tracing	p. 369
Distance Functions	p. 371
C# Implementation	p. 374
CSG Objects	p. 376
C# Implementation	p. 377
Parametric Surfaces	p. 379
Interval Arithmetic	p. 380
Interval Root Finding - Bisection	p. 381
Interval Root Finding - Newton-Raphson	p. 382
Ray Tracing Harmonic Surfaces	p. 385
Lighting Models	p. 386
Supersampling	p. 386
Regular Supersampling	p. 387
Stochastic Supersampling	p. 387
Adaptive Supersampling	p. 388
Ambient Occlusion	p. 389
Ray Marching	p. 392
Photon Mapping	p. 393
Transport Notation	p. 394
Path Tracing	p. 394
Creating the Photon Map	p. 395
Photon Tracing	p. 396
Photon Map Data Structure	p. 397
Radiance Estimate	p. 399
C# Implementation	p. 401
Radiosity	p. 409
Light Transport Notation	p. 409
Radiosity Matrix	p. 410
Solving for Radiosity Values	p. 411
Solving: Jacobi Method	p. 411
Solving: Gauss-Seidel Iteration	p. 411
Solving: Shooting Method	p. 412
Form Factors	p. 412
Numerical Solution	p. 413
Raytracing Method	p. 413
Hemicube Method	p. 413
Rendering	p. 415
C# Implementation	p. 415
Animation	p. 427
Traditional Animation Techniques	p. 427
Keyframing	p. 427
Motion Capture	p. 428
Physics Models	p. 428
Animation of Position	p. 429
Arc length parameterization	p. 430
Orientation	p. 430
Articulated Structures (Kinematics)	p. 431
Forward Kinematics	p. 432
Vertex Blending	p. 433
Inverse Kinematics	p. 433
Mass Spring Systems	p. 434
Particle Systems	p. 435
Free Form Deformations	p. 435
Fluids	p. 436
Navier-Stokes Equations	p. 438
Advection	p. 439
Diffusion	p. 440
Projection	p. 440
Boundary Conditions	p. 441
C# Implementation	p. 441
Free Surface	p. 449
C# Implementation of Free Surfaces	p. 450
Bibliography	p. 459
Index	p. 471
Table of Contents provided by Ingram. All Rights Reserved.

Mathematical Tools In Computer Graphics With C# Implementations

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Paperback

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