Mathematical Tools in Computer Graphics with C# Implementations Downloaded from

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13 Index Symbols σ(t), 121, neighborhood, ring, neighborhood, 184, ring, 184, 187 A Absorbance, 39 Adini s method, 177 Advection, 438 Affine combination, 78 Affine invariance, 76, 83, 98, 138 Algebraic curves, 75 Algebraic distance, 369 Aliasing, 386 Ambient, 26 Ambient occlusion, 389 Anisotropic surfaces, 32 Anti-aliasing, 386 Arc length, 113, 200 Attenuation, 29 Attractor, 64 B B-splines, 94, 100 affine invariance, 98 convex hull, 98 interpolation with, 100 non-uniform, 99 periodic interpolation with, 102 uniform, 94 Bézier, 83 Bézier curves, 83 affine invariance, 83 convex hull, 84 piecewise continuous, 86 rational, 91 subdivision, 88 Bézier surfaces, 169 Bézier patch, 170 tensor product surface, 170 triangular, 171 Barycentric coordinates, 172 Barycentric mapping, 223 Basic vectors, 200 Beckmann distribution, 33 Beer-Lambert law, 38, 337 Bent normal, 392 Bernstein polynomials, 83, 172 Bessel-Overhauser splines, 93 Binomial theorem, 84 Binormal, 115 Blending functions, 97 BRIDF, 31 C Cantor set, 71 Capacity, 59 Catmull-Rom splines, 92 Chaikin s scheme, 179 Chebyshev polynomial, 142 first kind, 142 second kind,

14 472 Index Chord length, 175 Cone, 372 Conformal mapping, 226 Conic sections, 91 Continuous, 86 C k, 86 G k, 86 Convex hull, 77, 84, 98 Convex polygons, 53 Cook, 31 Coordinate curves, 200 Cox-de Boor formula, 98 Cramer s rule, 343 Cross product, 3 CSG, 376 Curvature, 112 curvature plot, 113 inflection, 114 radius, 115 Curvature vector, 115 D de Boor algorithm, 98 de Casteljau, 87 Differencing, 238 central, 240 forward, 239 Diffuse, 26 Diffusion, 438 Direct matrix product, 69 Dirichlet kernel, 122 Discrete Fourier transform, 106 Distance surfaces, 369 Divergence, 439 E Emissive, 26 Euclidean distance, 2, 369 Euclidean norm, 2 Euler, 8 Euler transform, 8, 9 F Fair, 114, 183 Fairness, 256 Filter bank, 155 First fundamental form, 201 Flat shading, 54 Fluids Navier-Stokes equations, 438 Fourier, 106 Fourier transform, variation, 149 Fourier matrix, 105 Fractal, 59 Frenet frame, 116 Fresnel equation, 38 G Gaussian curvature, 203 Gaussian distribution, 33 Geometric distance, 369 Geometry image, 219 Geometry images, 219 Gimbal lock, 9 Gouraud shading, 23 Gradient, 438 Graph, 222 Grassmann space, 5 H Hadamard matrix, 156 Harmonic interpolation, 118 affine invariance, 138 even case, 122 non-uniform, 144 odd case, 120 stability, 136 Harmonic subdivision, 207 Harmonic surfaces, 237 tensor product surface, 206, 229 Hausdorff dimension, 59 Helicoid, 203 Helmholtz Hodge decomposition, 439 Hermite polynomials

15 Index 473 cubic, 429 Hermitian, 149 Hilbert curve, 66 Homogeneous coordinates, 4 Hooke s law, 434 Hypersurface, 167 I Implicit surfaces, 368 Inclusion monotonic, 381 Infinite cylinder, 371 Interpolation, 100 Intersection ray-plane, 342 ray-sphere, 342 ray-triangle, 343 Interval, 380 centered forms, 384 mean value form, 385 midpoint, 383 Newton iteration, 385 root finding, 381, 382 width, 383 Interval arithmetic, 380 operations, 380 transcendental functions, 381 Interval extension, 381 Isocurve, 168, 169 Iterated function system, 64 J Jacobi identity, 3 Jacobian, 433 Jacobian matrix, 382 Jitter, 388 Julia set, 62 K κ, 113 Klein bottle, 248 Koch snowflake, 67 Kronecker delta, 79 Kronecker product, 69, 71 multiple, 70 L Lagrange, 78 Lagrange interpolation, 78, 146 trigonometric, 107, 122 λ, 32, 37 Lambert, 27 Lambert s Law, 27 Laplacian, 439 Lebesgue covering dimension, 60 Lighting, 23 ambient term, 26 Cook-Torrance model, 31 diffuse term, 26 emissive term, 26 specular, 27 spot lights, 30 Lighting model Phong, 25 Lindenmayer systems, 65 Line segments, 4 Linear combination, 78 M Möbius band, 248 Mandelbrot set, 60 Matrix class, 17 Max norm, 383 Mean curvature, 203 Mesh parameterization, 219 METAPOST, 108 Meusnier s theorem, 202 Mock curvature, 109 Moments, 145 Monte Carlo methods, 396 N Navier-Stokes equations, 438 Newton-Raphson, 379 Normal, 170 infinite plane, 338 sphere, 338

16 474 Index triangle, 339 Normalized, 3 NURBS, 104 Nyquist limit, 387 O Obscurance, 389 Orthogonal, 201 Orthogonal parameterization, 201 Osculating circle, 115 Osculating plane, 115 Overhand knot, 248 P Parameterization, 222 mean value coordinates, 227 shape-preserving, 223 stretch minimization, 225 Parameterized curve, 112 Parametric curves, 76 Parametric surface, 167 Partition of unity, 76, 83, 138 Pauli matrices, 15 Periodic interpolation, 102 Permutation matrix, 118 Perpendicular, 3 Perspective projection matrix, 12 Phong, 25 Phong shading, 23 Planar, 222 Poisson equation, 439 Principal directions, 202 Principal normal, 115 Q Quasi-interpolation, 200 Quaternion class, 20 Quaternions, 13 R Rational Bézier curves, 91 Raytracing, 333 Reflection, 338 Refraction, 339 Rendering, 238 Root finding interval bisection, 381 interval Newton-Raphson, 382 Rotation, 7, 9 generalized, 9 quaternions, 13 S Scalar product, 2 Scaling, 6 Second fundamental form, 202 Shading, 23 Gouraud, 23 Phong, 23 Sierpinski carpet, 71 triangle, 73 Signed distance bound, 369 Snell s law, 37, 339 Solomon s Seal knot, 249 Spectral representation, 118 Specular, 27 Sphere parametric, 168 Sphere tracing, 369 Stochastic supersampling, 387 Stretch, 225 Subdivision 3, 185 Catmull-Clark, 191 curves, 177 Doo-Sabin, 192 Loop, 180 modified butterfly, 183 stationary, 179 Superquadric, 372 Supersampling, 386 stochastic, 387 Surface, 167 Surface normal, 170

17 Index 475 Surface of revolution, 248 T Tension-Continuity-Bias splines, 93 Tensor product, 69 Tensor product surfaces, 168 Toeplitz, 149 Toeplitz matrix, 149 Topological dimension, 60 Torrance, 31 Torsion, 115 Torus, 372 parametric, 168 Transforms affine, 5, 6 concatenation, 10 identity, 6 linear, 6 rotation, 7 scaling, 6 translation, 7 Translation, 7 Transport notation, 394 Transpose, 2 Tridiagonal, 102 Trigonometric interpolation, 104 Twist vector, 176 U Umbilical point, 202 USFFT, 148 V Valence, 179 Vandermonde matrix, 145 Vector class, 15 Vector field, 437 Vector identity, 3 Vector product, 3 Vector space, 1 Viscosity, 438 W Wavelets, 153 father wavelet, 153 Haar, 157 Haar wavelet transform, 154 lifting scheme, 158 mother wavelet, 154 nonstandard construction, 159 scaling functions, 156 Weighted average, 78, 84 World coordinates, 436