Index

Page numbers followed by “f” indicate a figure; and those followed by “t” indicate a table.

A

AABB trees, 1093

Absorption, 737

Abstract coordinate system, 39, 42

to specify scene, 42–44

Abstract geometric vs. ready for rendering, 467

Abstraction, defined, 10

Abstraction, in expressive rendering, 947, 959–961

factorization, 947

kinds of, 947

schematization, 947

simplification, 947

Abstraction considerations, 444

Abstraction distance, 1138

A-buffer, 1057

Acceleration data structures, 472

Accretion, 569

ACC surfaces. See Approximate Catmull-Clark subdivision surfaces

Accumulation buffer, 1056

ACM. See Association for Computing Machinery

ACM Transactions on Graphics, 922

Acne, shadow, 416

Active edge table, 1041

Additive color, 760

Adjacency information, on meshes, 338–339

Adjacent vertex, 637

Adjoint transformation, 253

Affine combination, 154, 160

Affine combination of points, 160

Affine transformations, 182, 234, 259

Affordances (user interfaces), 572

Albedo, 547

Algebra, geometric, 284

Aliasing, 331, 544, 557–559, 837, 1055

in line rendering, 544

Aliasing revisited, 527–729

Alpha-to-coverage, 366

Alpha value, 481

Alternative mesh structures, 187ff, 635ff, 338

AM. See Application model (AM)

Ambient light, 8, 122, 124

Ambient occlusion, 742

Ambient reflection, 136

AMIP. See Application-Model-to-IM-Platform Pipeline (AMIP)

Analytic BSDFs, 358

Angles, 686–688

solid, 686–688

subtended, 687

Animation element, in XAML, 55

Animation(s), 94, 963

burden of temporal coherence in, 985–987

considerations for rendering, 975–987

creating a sailing ship firing a cannon (simulation), 969–972, 970f

creating a walking character, 966–969

double buffering, 975–976

implicit curves in, 631–632

implicit shapes in, 631–632

interlacing, 978–980

level-set approach to, 631–632

motion blur, 980–983

motion perception, 976–978

navigating corridors (motion planning), 972–973

notations related to, 973–975

physically based, 963, 989

problem of the first frame in, 984–985

root frame, 972

stop-motion, 987

temporal aliasing, 980–983

temporal coherence, exploiting, 983–984

triple buffering, 976

ways to produce, 964

Animator, 966, 989

Anisotropic materials, 883

Antialiasing, 498ff, 982, 985, 1055

coverage sampling, 1058–1059

multisample, 1057–1058

spatial, 1055–1060

supersampled, 1056–1057

supersampling techniques for, 985–986

API, 15, 25–26, 32

Appearance modeling, 742

Appel’s ray-casting algorithm, 449, 797, 838

Apple Lisa, 568

Application model (AM), 36, 466–468

Application-Model-to-IM-Platform Pipeline (AMIP), 468–474

Application programming interfaces. See API

Approximate Catmull-Clark subdivision surfaces, 613

Approximation (graphics)

common forms of, 825–831

of the series solution of rendering equation, 847–848

Arcball interface, 280, 584

evaluation of, 584

practical implications of, 584

Architectural considerations, 444–445

Arctan, 152

Area-and-angle preserving, as a property of texture mapping, 555

Area lights, 377–379, 550, 698, 740, 784, 847, 866-868, 888, 915–918, 925–926. See also Area luminaire

hemispherical, 378–379

rectangular, 377–378

reflecting illumination from, 896

Area luminaire, 785, 788, 798, 891, 895. See also Area lights

Area-subdivision algorithms, 1041

Area-weighted radiance, 910. See also Biradiance

Aspect ratios

and field of view, 316–317

of triangles, 197

Associated transformations, 294

Attenuated geometric light source, 133

B

Back buffer, 971, 975

Back face, of polygon, 337

Backface culling, 337, 1023, 1028, 1047–1049

Backscattering, 730

Baking (models), 247

Band, of energies, 672

Band-limiting, 514, 522–523, 524, 534, 541

reconstruction and, 524–527

sampling and, 514–515

Band reconstruction, 534

Barycentric coordinates, 172, 183, 202, 203, 216, 218

analogs of, 182

Barycentric coordinates of x, 219

Barycentric interpolation, code for, 203–207

Basic graphics systems, 20–23

and graphics data, 21–23

Basis functions, 208, 596, 597, 600, 608, 609, 612, 625, 848–850. See also Tent-shaped functions

Basis matrix, 597–598, 603

Beckmann distribution function, 732

Beta phenomenon, 977

Bézier curve, 598, 607

Bézier patches, 607, 608–610, 609f

described, 608

Bicubic tensor product patch, 609

Bidirectional path tracing, 853, 870–871

schematic representation of, 853f

Bidirectional reflectance distribution function (BRDF), 646–647, 703, 783, 814, 834, 852, 946

anisotropic, 883, 885

Blinn-Phong, 883

cosine weighted, 820

glass and, 705–706

Lambertian, 814, 883

mirrors and, 705–706

Phong, 883

reciprocity and, 705–706

Bidirectional scattering distribution function (BSDF), 354–362, 704, 712, 820, 852

analytic, 358

isotropic, 883

local representation of, 882–887

measured, 358

Bidirectional surface scattering reflectance distribution function (BSSRDF), 704, 712, 738

Bidirectional transmittance distribution function (BTDF), 704

Bijective, 152. See also Injective

Billboard clouds, 348

Billboards, 648

and impostors, 347–348

Binary space partition (BSP) trees, 1023–1024, 1030, 1084–1089

building, 1089–1092

C++ implementation of, 1086

conservative ball intersection, 1088

conservative box intersection, 1088

first-ray intersection, 1088

kd tree, 1089

oct tree, 1090

pseudocode for visibility testing in, 1032

quad trees, 1090, 1091f

ray-primitive intersection, 1030–1032

2D binary space partition tree (BSP), 1086f

Binary tree (data structure), 1077

1D example, 1079

Binned rendering, 1137–1138

abstraction distance in, 1138

advantages of, 1138

deferred shading in, 1137–1138

drawbacks of, 1138

excess latency, 1138

full-scene anti-aliasing in, 1137

local memory in, 1137

poor multipass operation, 1138

properties of, 1137–1138

unbounded memory requirements, 1138

Biradiance, 910. See also Area-weighted radiance

Birefringence, 682

Bisection method, 830–831

Bitmaps, 38

Black body, 672

Blending, 362–364

and translucency, 361–364

Blindness, motion-induced, 114

Blinn-Phong BRDF, general form of, 721

Blinn-Phong model (reflection model), 138, 395ff, 414, 721–723

Blobby modeling, 343

Blob tree, 624

Bloom and lens flare, 369

Bloom focus, 336

Blue noise distribution, 921

Blue screening, 485

Blurring, 317, 543, 545, 983

Body-centered Euler angles, 272

Body-centered rotation, 272. See also Object-centered rotation

Boilerplate, 83

Boltzmann’s constant, 674

Bottom-up construction, and composition, 140–144

Boundaries, and light transport, 798

Boundary component, 641

Boundary edge, 194, 637

determining, 638

Boundarylike vertex, 194

Boundary of a simplex, 638

Boundary vertex, 194, 638

Bounded color models, 771

Bounding box, 38, 197, 198f, 285, 420, 429, 631, 983

Bounding-box optimization, 420–421

beyond, 429

Bounding geometry, 1068

Bounding Volume Hierarchy (BVH), 916, 1049, 1092–1093, 1092f

Bounding volumes, 1068

BRDF. See Bidirectional reflectance distribution function (BRDF)

Brewster’s angle, 682

Brightness (light), 108, 750, 756

just noticeable difference (JND), 754

perception of, 750–756

Brush (geometric primitive), 38

Brushstroke coherence, 986

BSDF. See Bidirectional scattering distribution function (BSDF)

B-spline basis matrix, 603

B-splines

cubic, 602–603

nonuniform, 604

nonuniform rational, 604

rational, 604

uniform spacing of, 604

BSSRDF. See Bidirectional surface scattering reflectance distribution function (BSSRDF)

BTDF. See Bidirectional transmittance distribution function (BTDF)

Buckets, 1093, 1093f. See also Grid cells

Buffers, 327–330

color, 328

depth, 329

framebuffer, 329

stencil, 329

Buffer swap, 443

Building blocks of ray optics, 330

Building transformations, from view specification, 303–310

Bump mapping, 547, 550–551

C

Cached and precomputed information on meshes, 340–341

Caching, 983, 1129ff

Callback procedure, 23

Camera(s), 336–337

depth of field, 301

design, 406

focal distance of, 301

orthographic, 315–317

perspective camera specification, 301–303

position of, 301

specifications and transformations, 299–317

transformation and rasterizing renderer ripeline, 310–312

Camera coordinates. See Camera-space coordinates

Camera setup, 460–461

Camera-space coordinates, 22, 299, 928

Camera visibility. See Primary visibility

Candelas (measurement unit of luminous intensity), 751

Capsule (3D volume), 1066

Cartoons, hand-animated, 966

Cathode-ray tubes (CRTs), 20, 770

Catmull-Clark subdivision surfaces, 610–613

Catmull-Rom spline, 540, 540f, 598–601

applications of, 602

generalization of, 601–602

nonuniform, 601

uniform, 601

Caustics, and light transport, 798

Cdf. See Cumulative distribution function (cdf)

Channels, 483

color, 483

object ID, 485

Chateau (user interface), 589–590

Chromatic aberration, 336–337, 680

Chunking rasterizer. See Tiling rasterizer

CIE chromaticity diagram, 765

applications of, 766

defining complementary colors, 766

excitation purity and, 766

indicating gamuts, 766

CIE Luv color coordinates, 767

CIE L*u*v* uniform color space, 767

Circularly polarized wave, 677

Client area, 37, 39, 46

Clipping, 59, 63, 1045–1046

of data, 24

near-plane, 422, 1044, 1046

Sutherland-Hodgman 2D clipping, 1045–1046

whole-frustum clipping, 1044, 1047

Clipping planes, 122

Closed interval, 150

Closed meshes, 190

Closed oriented meshes, 195

Closed surface, 638

Clustering, 665–666

Clusters, 1043

CMTM. See Composite modeling transformation matrix

CMYK color, 774–775

Coded apertures, 493

Codomain, 151. See also Domain

for texture maps, 553–554

Coefficient of extinction, 682, 728, 738

Coefficient of restitution, 1012

Coherence, 950

spatial, 950

temporal, 950, 962

Colatitude, 688

Collision proxy geometry, 337

Color bleeding, 839

Colorblindness, 746

Color buffer, 328

Color constancy, 110, 748

Color description, 756–758, 771–774

Colorimetry, 747

Color interpolation, 777–779

Color matching, 748

Color models

bounded, 771

RGB, 772–774

YIQ, 775

Color naming, 748

Color palettes, 777

Color perception

peripheral, 781–782

physiology of the eye and, 748–750

strengths and weaknesses, 761

Color percepts, 747

Color(s)

choice, 777

CIE description of, 762–766

CMY, 774–775

CMYK, 774–775

coding, 779–780

complementary, 766

conventional color wisdom, 758–761

description, 756–758, 771–774

implications of, 746

intensity-independent, 765

interpolating, 777–779

matching, 748

naming, 748

nonspectral, 766

palettes, 777

perceived distance between, 767

perception of, 750–756

percepts, 747

perceptual spaces, 767–768

primary, 758–759

RGB sliders and, 761

sensations, 747

standard description of, 761–766

use in computer graphics, 779–780

Color selection interfaces

hue-lightness-saturation (HLS) interface, 776–777

hue-saturation-value (HSV) interface, 776–777

Color sensations, 747

Color specification in WPF, 133

Color wisdom, conventional, 758–761

blue and green make cyan, 760

color is RGB, 761

objects have colors, 759–760

primary colors, 758–759

purple isn’t a real color, 759

Comb function, and transform of, 520

Commission Internationale de l’Éclairage (CIE), 755, 762–766

chromaticity diagram, 765

description of color, 762–766

Complementary colors, 766

Complex applications, processing demands of, 14

Complex conjugate, 512

Component hierarchy, top-down design of, 139–140

Components, reuse of, 144–147

Composion of transformations, 235

Composite component, constructing, 142

Composited image, 485

Composite modeling transformation matrix, 314

Composite transformation matrix, 246, 314, 463

Compositing of images

operations, 488–489

physical units and, 489–490

simplifying, 487

Compression, use of splines for, 605

Compressive sensing, 530

Computability, as a property of texture mapping, 555

Computational photography, 493

Computations, stability of, 278

Computer-based animation industry, 932

Computer graphics, 1–33

2D transformation library, 287–298

3D transformation library, 287–298

applications, 24–25

basic graphics systems, 20–23

brief history of, 7–9

current and future application areas of, 4–6

deep understanding vs. common practice, 12

definition of, 2

examples of, 9–10

goals, resources, and appropriate abstractions, 10–12

graphical user interfaces (GUI) and, 567–574

graphics pipeline, 14–15

interaction in graphics systems, 23–24

introduction to, 1–4

kinds of packages, 25–26

learning, 31–33

numbers and orders of magnitude in, 12–15

physical/mathematical/numerical models of, 11

relationship with art, design, and perception, 19–20

using color in, 779–780

world of, 4

Conceptual design (user interface), 570

Conductive materials, 714

Cones (color receptors), 107, 749

generalized, 757

Conformal mapping, 555

Conservative rasterization, 1096

Conservative rasterizer, 430

Conservative visibility, 1023

Conservative visibility algorithm, 1023

Conservative visibility testing, 1023

backface culling, 1023

frustum culling, 1023

methods of, 1023

spatial data structures and, 1023

Conservative voxelization, 1096

Constancy

application of, 111

color, 110

and its influences, 110–111

shape, 110

size, 110

Constant shading, 127

Constructive solid geometry (CSG), 450

Content, preparing viewport for, 120–122

Continuation, 111–112

applications, 112

Continuous probability. See Continuum probability

Continuum, defined, 808

Continuum probability, 808–810, 815–818

Contour curve, 1048

Contour drawing, 551–552

Contour generator, 953. See also Contour(s)

Contour lines, 616

Contour points, 952

Contour(s), 551, 952, 953

of a smooth surface, 644

suggestive, 957–958

Contribution/detail culling, 470

Control data, 599

Control points, 599

Control templates, 49

Convex boundary polygon, 1045, 1045f

Convex cone, 747

Convex hull property, of cubic B-splines, 603

Convex polygons, 175

Convolution, 500–503

defined, 500

like computations, 504–505

properties of, 503–504

Convolution multiplication theorem, 521

Cook-Torrance model, 731–732

Coons, Steven A., 608

Coordinate frame(s), 240–241

defined, 240

rigid, 240

Coordinates, 153

operations on, 153–155

Coordinate system(s), 90–91

abstract coordinate system, 42–44

floating-point coordinates, 38–39

integer coordinate system, 38–39

physical coordinate system, 38

spectrum of, 44–45

transformations and, 229–230

WPF canvas, 45–46

Coordinate vector, 155

Cornell box, 903, 911, 916–917

Corner-cutting, on polyline, 81, 83

Correspondence (meshes), 661

building, 661

Cosine weighted BRDF, 820

Cosine-weighted sampling

on hemisphere, 815

Cotangent rule, 658

Covectors, 163, 184, 520

transforming, 250–253

Coverage

binary, 1027–1028

partial. See Partial coverage

Coverage of a pixel, 213

Coverage sampling antialiasing (CSAA), 1058–1059

Coverage testing, 422

Crease edges, 953

Critical angle, 682

Cross product, 157–158

CRT. See Cathode-ray tubes (CRTs)

CSG. See Constructive solid geometry (CSG)

C++ Standard Template Library (STL), 1074

CTM. See Composite transformation matrix

Cube map, 554

Cube mapping, 340

Cubic B-spline filter, 540, 540f

Cubic B-splines, 602–603

convex hull property of, 603

formula for, 602–603

nonuniform, 602

uniform, 602–603

Culling

contribution/detail, 470

occlusion, 470

portal, 470

sector-based, 470

view-frustum, 470

Cumulative distribution function (cdf), 685

Curvature, 955

line of, 956, 956f

radial, 957

Curvature shadows, 946

Curved-surface

representation and rendering, 128

Curves

implicit, 616–619

Cyan-Magenta-Yellow (CMY) color, 774–775

Cybersickness, 571

Cylinder kernel, 910–911

D

Dangling edge, 637

Darken operation, 488

Data structures

characterizing, 1077–1079

generic use of, 1077

ordered, 1077

selecting, 1077

spatial, 1065–1102

DDA. See Digital Difference Analyzer (DDA)

Debugging, 411–412

rendering and, 915–919

Declarative animation

dynamics animation via, 55–58

Declarative specification, 40

vs. procedural code, 40

Deferred lighting, 441–442

Deferred-rendering method, 440

Deferred shading, 446, 1135–1137

difficulties with, 1136

excess storage and bandwidth in, 1136

goal of, 1135

multi-sample anti-aliasing (MSAA), incompatibility with, 1136

shader-specified visibility and, 1136

Defocus, 1060–1061

Deformation (meshes), 660

Deformation transfer, 660–664

Degenerate transformation, 224

Degree

of an edge, 637

of a vertex, 637

Delta function, 519

Density estimation, 912

Depth buffer, 329, 392, 1023, 1028, 1034–1040

common applications in visibility determination, 1035–1036

common encodings, 1037–1040

depth prepass, 1036

encodings, 1037–1040

screen-space visibility determination and, 1037

Depth buffer encodings, 1037–1040

choices for, 1037

hyperbolic in camera-space z, 1037

linear in camera-space z, 1037

Depth complexity, defined, 446

Depth complexity of a ray, 1028

Depth map. See Depth buffer

Depth of field, 107, 301

Depth prepass, 1036

Depth-sort algorithm, 1042–1043

Depth value, 481

Derivative-based definitions of radiometric terms, 655–656, 700–702

The Design of Everyday Things (Norman), 593

Detail objects, 1051

Development tools, 41

Device code, 432

Device coordinates, 39

Diagonal matrices, 230

Differential coordinates, 655–657

Diffraction, 676, 677

defined, 676

Diffuse, defined, 8

Diffuse reflection, 136

physical models for, 726–727

Diffuse scattering, 713, 716

Diffusion (morphogens), 561

Diffusion curves, 961

Digital cameras, 5

characteristics of, 13–14

Digital Difference Analyzer (DDA), 431

Digital signal processing, 545

Digital video cameras, 5

Direct3D, 452

Directed acyclic graph (DAG), 144, 248

Directed edges, 636

Directed-edge structure, 195

Direct illumination, 372–373

interface to, 372–373

Directional curvature in direction u, 956

Directional hemispherical reflectance, 708

Directional light, 125

Directionally diffuse, 102

Direct light, 370. See also Indirect light

Direct lighting, 834. See also Direct illumination

Direct shadows, 946

Dirty bit flags, 983

Dirty rectangles, 983

Discrete attributes, 651

Discrete differential geometry, 644, 667

Discrete probability, 803–804

relationship to programs, 803–804

Discrete probability space, 803

Displacement, 157

Displacement maps, 344, 547, 557ff, 562f, 647

Display-form-factor independence, 45

Display list, 473

Display transformation, 46–47

Distant objects, 346–348

Distribution (random variable), 806

Distribution ray tracing, 317, 838

Division of modeling principle, 210

DockPanel. See Panel

Dollying (camera control technique), 585

Domain, defined, 151. See also Codomains

Domain restriction, 827

Dominant wavelength, 747

Dot product, 158–159

Double-buffered rendering, 971, 975–976

Draw calls, 434

executing, 442–444

Drawing, Dürer’s, 62f, 64f, 68–72, 1035f

Drawing primitives, 461–462

Dual contouring, 653

Dual paraboloid, 554

Dual space, 163

Dual vectors. See Covectors

Dürer, Albrecht, 61–65

Dürer rendering algorithm implementation, 65–68

Dürer’s drawing, 68–72

Dürer woodcut, 61–65, 1035f

Dynamic Canvas algorithm, 986, 986f

Dynamic range, 8

Dynamics, 463, 989, 996–1008

Dynamics animation, via declarative animation, 55–58

E

Early-depth-test

defined, 446

example, 445–447

Early z-cull, 1136

Edge aligns, 427

Edge collapse, 197

Edge-collapse costs, 649–652

Edge detection, 533, 544, 545

Edge(s), 189

boundary, 637, 954

crease, 953

dangling, 637

directed, 636

interior, 637

sharp, 651

smooth, 953

Edges (computer vision), 952

Edge-swap operation, 197–198

Edge vectors, 175

Electric field

linearly polarized, 678

Electromagnetic spectrum, 330–331, 675f

Elements, 41

animation elements, 55–56

Elliptically polarized light, 679

Embedding topology, 637

Emission, 369, 737

Emissive lighting, 138

Emitters, 334–335

Empirical/phenomenological models, of scattering, 713, 717–725

Energy, photons transport, 333

Energy conservation, 714

Energy function (meshes), 650

Environment map, 549, 550

Environment mapping, 340, 549–550, 939–940

Equations

approximate solutions of, 825–826

approximating, 826

domain restriction, 827

methods for solving, 825–831

Newton’s method for solving, 831

statistical estimators, using, 827–830

using bisection for solving, 830–831

Estimation

summing a series by, 828–830

Estimator random variable, 818

bias, 822–823

consistent sequence of, 818, 822–823

unbiased, 818

variance of, 818

Estimators. See Estimator random variable

Euclidean distance, 767

Euler angles, 267–269

body-centered, 272

Euler characteristic of a mesh surface, 641

Euler integration, 278

Even function, 508

Event (interaction), 92

handling, 85, 92–93

Event (probability), 802, 809

Excitation purity, 747

chromaticity diagram and, 766

defined, 747

Expectation. See Expected value

Expected value, 804–806, 810

properties of, 806–808

of a random variable, 810

related terms, 806–808

Explicit equation, 341

Explicit Euler integration, 1016

Explicit Euler method, 1019. See also Forward Euler method

Explicit trapezoidal methods, 1020

Exponents (display process), 769–771

encoding of, 769–771

Exposure time, 980. See also Shutter time

Expressive rendering, 945–962

abstraction in, 947

challenges of, 949–950

examples of, 948

geometric curve extraction in, 952–959

gradient-based, 952

marks, 950–951

perception and salient features, 951–952

perceptual relevance, 947

research in, 947–948

spatial coherence in, 950

strokes, 950–951

temporal coherence in, 950

Extended marching cubes algorithm, 653

Extensibility via shaders, 453

Extensible Application Markup Language (XAML), 35, 41, 928

animation elements, 55–56

structure of, 41–42

Eye, 106–110

gross physiology of, 106–107

luminous efficiency function for, 751

physiology of, 748–750

receptors, 107–110

resolution, 13

Eye coordinates, 314

Eye path, 796

Eye ray

and camera design notes, 406

generating, 404–406

testing eye-ray computation, 406–407

Eye ray visibility. See Primary visibility

F

Factorization (abstraction), 947

FF. See Fixed-function (FF)

Field of view, and aspect ratio, 316–317

Field radiance, 834, 846

Fields (half-resolution frames), 978

FillEllipse, 39

Fill rate, 14, 636

Filter(s)

applying, 500

Catmull-Rom filter, 542t

cubic B-spline filter, 542t

filtering f with, 502

Gaussian filter, 542t, 543, 545

Mitchell-Netravali filter, 542t

separable, 544

sinc filter with spacing one, 542t

and their Fourier transforms, 542t

unit box filter, 542t

Filtering, 500, 502, 557–559

Final gather step in photon mapping, 913

Finite element method, radiosity, 839

Finite element models, 349

Finite series

summing by sampling and estimation, 828–829

Finite-state automaton (FSA), 574, 857

probabilistic, 858f

Finite support, 535

Finite-support approximations, 540–541

First-person-shooter controls, 588

Fitts, Paul, 572

Fitts’ Law, 572, 587

Fixed-function (FF), 452

era, 452–453

to programmable rendering pipeline, 452–454

Fixed-function 3D-graphics pipeline, 119

Fixed point, 325–326

Flat shading, 20. See also Constant shading

Floating point, 325, 326–327

Floating-point coordinates, 38–39

Flow curve, 1015

Fluorescence, 671

Flux responsivity, 792

Focal distance, 301

Focal points, 951

and caustics, 798

Focus dot (camera manipulation), 586

Fog, 351

Fold set, 953. See also Contour(s)

Foreground image, 485

Form factor (radiosity), 840

computation of, 842

Form factor (display), 58–59

Forward Euler integration, 1018

Forward Euler method, 1019. See also Explicit Euler method

Forward-rendering design, 440

Fourier-like synthesis, 559–560

Fourier transform, 497

applications of, 522

of box, 517

definitions, 511

examples of, 516–517

of function on interval, 511–514

inverse, 520–521

properties of, 521

scaling property of, 521

Fourth-order Runge-Kutta method, 1020

Fovea, 107

Fractional linear transformation, 256

Fragment (pixel), 18, 1055, 1056f

Fragment generation, 17

Fragments, 18

Fragment shaders, 466, 930. See also Pixel shaders

Fragment stage, 433

Framebuffers, 329, 971

front buffer, 971

Frame coherence, 983. See also Temporal coherence

Frames (individual images), 963

Frequency-based synthesis, and analysis, 509–511

Frequency domain, 513

Fresnel, Augustin-Jean, 681

Fresnel equations, 681, 727–729

unpolarized form of, 683

Fresnel reflectance, 683, 727–729

Fresnel’s law, 681–683, 682f

and polarization, 681–683

Frobenius norm, 663

Front buffer, 971, 975

Front face, of polygon, 337

Frontface polygon, 1048

Frustum clipping, 1028, 1045–1046

Frustum culling, 1023, 1028, 1044

Functional design (user interface), 570

Function classes, 505–507

Function L

writing in different ways, 706–707

Functions, 151–152

basis, 208

interpolated, 203

piecewise constant, 187

tabulated, 201

G

G3D (open source graphics system), 241, 295, 321, 356, 933

Game application platforms, 478

Game engines, 25. See also Game application platforms

Gamma, defined, 771

Gamma correction, 769–771

defined, 771

encoding, 398, 769–771

Gamuts (color), 331, 766

chromaticity diagram and, 766

matching problem, 766

Gaussian filter, 542t, 543, 545

GDI, 38

Generalized cone, 757

General position assumption, 291–292, 292f

General purpose computing on GPUs (GPGPU), 1142

Generics in programming languages, 1068

Gentle slope interface, 569

Genus of a surface, 196

Geometric algebra, 284

Geometric curve extraction, 952–959

Geometric light, 124, 133

Geometric model, 2, 41ff, 117ff,

Geometric modeling, 595

Geometric objects, 93–94

Geometric optics, 726

Geometric shapes, 470–472

Geometry

collision proxy, 337

instancing, 349–350

large-scale object, 337

projective, 257

Geometry matrix, 597

Geometry processing, 458–460

Geometry shaders, 931

Geomorph, 649, 650f

GIF. See Graphics Interchange Format (GIF)

Gimbal lock, 269, 994

Glass

BRDF and, 705–706

Global illumination, 340

Glossy highlights, 134, 353, 359–361

Glossy scattering, 414, 716

GLUT (OpenGL Utility Toolkit), 456

Gonioreflectometer, 702

Gouraud, Henri, 128

Gouraud shading, 128, 723, 743, 933. See also Phong shading

fragment shader for, 937

vertex shader for, 935

GPGPU (general purpose computing on GPUs), 1142

GPU. See Graphics Processing Units (GPUs)

GPU architectures, 1108–1111

binned rendering and, 1137–1138

deferred shading and, 1135–1137

Larrabee (CPU/GPU hybrid), 1138–1142

organizational alternatives of, 1135–1142

Grabcut (technology-enabled interface), 590–591

Gradient-domain painting, 961

Graftals (scene-graph elements), 986, 986f

Graphical user interfaces (GUI), 4, 23–24

affordances, 572

arcball interface, 584

choosing the best, 587–588

conceptual design, 570

examples of, 588–591

functional design, 570

lexical design, 570

multitouch, 574–580

natural user interfaces (NUIs), 571

sequencing design, 570

suggestive interface, 589

trackball interface, 580–584

Graphics applications, 21

architectures of, 466–478

kinds of, 24–25

Graphics data, 21–23

Graphics Interchange Format (GIF), 484

Graphics packages

kinds of, 25–26, 451ff

Graphics pipeline, 14–15, 36, 119, 310, 452, 458ff, 927, 1109ff, 432–434

defined, 434

forms of, 927–929

parts of, 17–18

stages of, 16–19

Graphics platforms, 21, 22, 25, 26

Graphics Processing Units (GPUs), 18, 20

Graphics processor architecture, 8, 1107ff

Graphics program

with shaders, 932–937

Graphics workstations, 932

Grays (color), 756

Grayscale, 482

Great circle, 273

Grid, as a class of spatial data structures, 1093–1101

construction, 1093–1095

ray intersection, 1095–1099, 1096f, 1098f

selecting grid resolution, 1099–1101

Grid cells, 1093, 1093f. See also Buckets

Grid resolution, selecting, 1099–1101

GUI. See Graphical user interfaces (GUI)

H

Haar wavelets, 531

transform, 531

Half-edges, 338

Half-open intervals, 150

Half-plane bounded by l, 174

Half-planes

and triangles, 174–175

Half-vector, 721

Hand-animated cartoons, 966

Hash grid, 904, 1095

HDR images. See High dynamic range (HDR) images

Heat, defined, 672

Heat equation, 529

Heightfields, 344

Helmholtz reciprocity, 703–704, 714. See also Reciprocity

Hemicube, 842

Hemisphere

cosine-weighted sampling on, 815

producing a cosine-distributed random sample on, 815

producing a uniformly distributed random sample on, 814

Hemisphere area light, 378–379

Hermite basis functions, 596

Hermite curve, 595–598

Hermite functions, 596

Heun’s method, 1019–1020

Hidden surface removal, 1023. See also Visible surface determination

Hierarchical depth buffer, 1024, 1050. See also Hierarchical z-buffer

Hierarchical modeling, 35, 55, 313ff, 463–464

Hierarchical occlusion culling, 1049–1050

Hierarchical rasterization, 430

Hierarchical z-buffer, 1050. See also Hierarchical depth buffer

High-aspect-ratio triangles, 197

High dynamic range (HDR) images, 481

High-level design, 388–393

High-level vision, 105

Hit point (Unicam), 585

Homogeneous clip space, 429, 1047

Homogenization, 236, 254, 259

Homogenizing transformation, 265

Host code, 432

HoverCam (camera manipulator), 591

Hue (color description), 756

Hue-lightness-saturation (HLS) interface, 776–777

Hue-saturation-value (HSV) interface, 776–777

Human-computer interaction (HCI), 568

arcball interface, 584

interaction event handling, 573–574

mouse-based camera manipulation (Unicam), 584–587

mouse-based object manipulation in 3D, 580–584

multitouch interaction for 2D manipulation, 574–580

prescriptions in, 571–573

suggestive interface, 589

two-contact interaction, 578

Human visual perception, 101–115

Human visual system, 29–30

Hybrid pipeline era, 453

Hyperbolic depth encoding, 1038–1040

complementary or reversed, 1039

Hyperbolic interpolation, 423

I

Identity matrix, 225

iid, see Independent dentically distributed random variables

Illuminant C (CIE chromaticity diagram), 765

Illumination, 9, 340, 362, 370ff, 722, 751, 785

Image

choosing format of, 484–485

composited, 485

compositing, 485–490

defined, 482

enlarging, 534–537

file formats, 483

foreground, 485

gradient, 544

information stored in, 482–483

losslessly compressed, 483

meaning of pixel during compositing, 486

Moiré patterns, 544

other operations and efficiency, 541–544

processing, 492–493

representation and manipulation, 481–494

RGB, 482

scaling down, 537–538

types of, 490–491

Image-based texture mapping, 559

Image display, 29

Image gradient, 544, 961

Images and signal processing, 493–532

Image space, 22, 245

Image-space photon mapping, 876

Immediate mode (IM), 452

vs. retained mode (RM), 39–40

Implementation platform, 393–403

and scene representation, 400–402

and selection criteria, 393–395

and test scene, 402–403

utility classes, 395–400

Implicit curves, 616–619

Implicit functions, representing, 621–624

mathematical models and, 623–624

Implicit lines, 164

Implicitly defined shapes, 164, 615–633

advantages of, 615

in animation, 631–632

disadvantages of, 615

Implicit surfaces, 341–343, 619–620

ray tracing, 631

ray-tracing, 342–343

Importance function, 792, 819

Importance-sampled single-sample estimate theorem, 818–819

Importance sampling, 802, 818–820, 822, 854

integration and, 818–820

multiple, 820, 868–870

Impostors, 348

and billboards, 347–348

Impulses, 356, 713, 784, 1010–1012. See also Snell-transmissive scattering

deriving impulse equations, 1010–1011

magnitude of, 740, 793

Impulse scattering, 715–716, 740, 784. See also Impulses; Snell-transmissive scattering

Incremental scanline rasterization, 431

Independent identically distributed (iid) random variables, 808

Independent random variables, 807

properties for, 807

Indexed face sets, 77

Indexed triangle meshes, 338

Indexing arrays, 156

Indexing vectors, 156

Index of refraction, 107, 332. See also Refractive index

Indication (expressive rendering), 948

Indirect light, 370. See also Direct light

Indirect lighting, 834

Infinite series

summing by sampling and estimation, 829–830

Infinite support, 535

Information visualization, 4, 37

Inheritance, as key extraction method, 1073–1074

Initialization in OpenGL, 456–458

Injective function, 151. See also Surjective function

Inner product, 158

Inscattering, 738

Inside/outside testing, 175–177

Instance transform, 139–140

Instancing, described, 450

Instantiated templates, 39, 50

Integral, of spectral radiance, 692

Integral equation, 786

Integration

importance sampling and, 818–820

Intel Core 2 Extreme QX9770 CPU, 1138

Intensity (light), 700, 769–771

encoding of, 769–771

high-light perception of, 770

low-light perception of, 770

Intensity-independent colors, 765

Interaction, keyboard, 95

Interface, 434–444

Interior edge, 637

Interiors of nonsimple polygons, 177

Interior vertices, 194, 638

Interlaced television broadcast and storage formats, 978

Interlacing, 978–980

pulldown, 979–980

telecine, 978–980

International Color Consortium (ICC), 772

Interpolated function

properties of, 203

Interpolated shading (Gouraud), 128–129

Interpolating curve, 600

Interpolation

bilinear, 622

hyperbolic, 423

perspective-correct, 256, 312, 422–424

precision for incremental, 427–428

rational linear, 423

between rotations, 276–278, 277f

spherical linear, 275–276

vs. transformations, 259

Interpolation schemes, 621–622

bilinear interpolation, 622

Intersections, 167–171

of lines, 165–167

ray-plane, 168–170

ray-sphere, 170–171

Interval

Fourier transform on, 518

Invariant under affine transformations, 182

Inverse Fourier transform, 520–521

Inverse function, 151

Inverse tangent functions, 152–153

Invertibility, as a property of texture mapping, 555

Inward edge normal, 175

Irradiance, 697–699. See also Radiosity defined, 697

Irradiance due to a single source, 698

Irradiance map, 557

Isocontour, 341

Isocurves, 616

Isosurfaces, 619. See also Level surfaces

J

Jaggies (image), 31

Joint transform, 140, 146–147

Just noticeable difference (JND), 754

K

k-dimensional structures, 1080–1081

kd tree, 1089

Kernel (photon map), 910

cylinder, 910–911

Kernel (photon mapping), 874, 1142

Key (data structure), 1068, 1077

Keyboard interaction, 95

Key frame, 966, 989. See also Key pose

Key pose, 966, 989. See also Key frame

Keys and bounds, extracting, 1073–1077

inheritance, use of, 1073–1074

traits, 1074–1077

Kinect (interface device), 568

Kinetic energy, 1021

Knee joint

adding, 143–144

Knots, 601

Kubelka-Munk coloring model, 760

L

L² difference. See L² distance

L² distance, 104

L² (space of functions), 506ff

² (space of sequences), 506ff

“Lab” color, 767

Lafortune model (light scattering model), 723–724

Lag. See Latency

Lambertian, 28, 358–359

Lambertian bidirectional reflectance distribution function (BRDF), 720, 883

Lambertian emitter, 695

Lambertian luminaire, 785

Lambertian reflectance, 28, 358, 413, 720, 925

Lambertian reflectors, 719–721

Lambertian scattering, 413–414, 716, 725

Lambertian shading, 353

Lambertian wall paint, 708

Lambert’s Law, 358

Laplacian coordinates, 655–657

applications, 657–660

properties of, 657

Large-scale object geometry, 337

Larrabee (CPU/GPU hybrid), 1138–1142, 1139f

cache coherence, 1140

capability of, 1140

correct provisioning, 1141

efficient parallelization, 1141

flexibility in, 1140

vs. GeForce 9800 GTX, 1140

generality in, 1140

Intel’s IA-32 instruction set architecture (ISA), use of, 1140

latency hiding, 1140

multiple processing cores, 1139

sequence optimization, 1141

specialized, fixed-function hardware, 1139

SPMD and, 1140

texture evaluation, 1139

wide vectors, 1139

Latency, 17, 1123–1126

Lateral inhibition, 108

Law of conservation linear momentum, 1011

Layout, defined, 85

LCD. See Liquid-crystal displays (LCDs)

LED-based interior lighting, 752

Legacy models, 324

Lemniscate of Bernoulli, defined, 616f

Lens flare and bloom, 366, 369

Level of detail (LOD), 347

Level set, 164, 341, 616

Level set methods, 631

Levels of detail (geometric representations), 645–649

determining, 645–646

parametric curves and, 649

surfaces and, 649

Level surfaces, 619. See also Isosurfaces

Lexical design (user interface), 570

Light(s), 26, 330–333, 784

ambient, 122, 124

area, 888

bending of, at an interface, 679–680

defined, 669

direct, 835, 865

directional, 125

elliptically polarized, 679

excitation purity and, 747

geometric, 124

hemisphere area light, 378–379

indirect, 835, 865

infrared, 672

interaction with objects, 118–119

interaction with participating media, 737–738

metameric, 768

point, 886–887, 888–889

representation of, 887–889

light capture, 29

measuring, 692–699

modeling as a continuous flow, 683–692

monospectral, 747

omni-light, 379–380

other measurements of, 700

path, 796

physical properties of, 669–670

quantized, 670

rectangular area light, 377–378

scattering, 388–390

spectral distribution of, 746–748

transport, 335–336, 783–787

ultraviolet, 671

unpolarized, 683

wavelength of, 670

wavelike, 670

wave nature of, 674–677

Light energy and photon arrival rates, 12–13

Light geometry, 133

Lighting, 312

direct, 834, 913

indirect, 834

and materials, 458

Phong, 930

programmable, 930

vs. shading in fixed-function rendering, 127–128

Lighting specification, 120–128

Light maps, 341

Lightness, 755, 756

CIE definition of, 755

Light path, 796

Light transport, 783–787

alternative formulations of, 846–847

boundaries and, 798

caustics and, 798

classification of paths, 796–799

Metropolis, 871–872, 915

perceptually significant phenomena and, 797–799

polarization and, 798

shadow and, 797

symbols used in, 784t

transport equation, 786–787

Light-transport paths classification of, 796–799

Linear combination, 157

Linear depth, 1040

Linear depth encoding, 1040

Linear interpolation, 201

Linearly polarized electric field, 678

Linear radiance, 398

Linear transformation, 221, 259, 307

degenerate (or singular) transformation, 224

examples of, 222–224

multiplication by a matrix as, 224

nonuniform scaling, 223

properties of, 224–233

rotation, 222–223

shearing, 223

Linear waves, 675

Linear z, 1040

Line of curvature, 956, 956f

Lines, intersections of, 165–167

Linked list (data structure), 1077

1D example, 1078–1079

Link of a vertex, 208, 641

Liquid-crystal displays (LCDs), 20

List, as a class of spatial data structures, 1081–1083, 1082f

C++ implementation of conservative ball-primitive intersection in, 1083

C++ implementation of ray-primitive intersection in, 1083

unsorted 1D list, 1081

List-priority algorithms, 1040–1043

BSP sort, 1043

clusters, 1043

depth-sort algorithm, 1042–1043

painter’s algorithm, 1041–1042

Live Paint, 1042

Local flatness (surface), 643, 882

Local Layering, 1042

LOD. See Level of detail (LOD)

Look vectors, 304

LookDirection, 122

Losslessly compressed image, 483

Lossy compression, 471, 483

Low-level vision, 105

Lucasfilm, 932

Luma, 771, 775

Lumens, 707

Luminaire models, 369

and direct and indirect light, 370

and nonphysical tools, 371–372

practical and artistic considerations of, 370–377

and radiance function, 370

Luminaires, 369, 784

area lights, 888

computer graphics, 369

representation of, 888–889

Lambertian, 785

point lights, 888–889

representation of, 888–889

Luminance, 707, 747, 755

of light source, 751

signal representative of, 770–771

Luminous efficiency, 707

Luminous efficiency function, 751

Luminous intensity, 751

M

Mach banding, 20, 211

Macintosh, 568

Magnitude, of impulses, 740, 793

Manifold meshes, 190, 191, 193–195

2D mesh as, 193

boundaries, 194

orientation of triangles in, 193–194

Manifold-with-boundary meshes, 195

operations on, 195

Mappings

application examples of, 557t

cylindrical, 555

examples of, 555–556

reflection, 556

spherical, 555

Marching cubes, 625, 628–629

algorithm, 628

extended, 653

generalization of, 652–653

variants, 652–654

Marching squares, 627

Hermite version of, 653

Markov chains, 857–861

estimating matrix entries with, 858–859, 860–861

Metropolis light transport, 871–872

path tracing and, 856–857

Markov property, 857

Marks (expressive rendering), 950–951

creation of, 951

imitation of artistic technique for creating, 951

physical simulation, 951

scanning/photography approach for creating, 951

Mask, 485–486

Masking, 730

Master templates, 39

Material, in scattering, 712

Material models, 353–358

software interface to, 740–741

Materials

lighting and, 458

Mathematical model, 2, 11. See also Geometric model; Numerical model; Physical models

and sampled implicit function representations, 623–624

Mathematics, and computer graphics, 30–31

Matrix associated to a transformation, 224

Matrix/matrices, 156

diagonal, 230

identity, 225

invertible, 225

orthogonal, 230

properties of, 230–231

rank of, 231, 261

rotation, 270–272

singular value decomposition (SVD) of, 230

special orthogonal, 230

Matrix multiplication, 161–162

Matrix transformations, 222

interpolating, 280

Matter, 336

Matting problem, 367

MaxBounce (photon mapping), 873

McCloud, Scott, 947

Mean. See Expected value

Measured BSDFs, 358

Measured/captured models, of scattering, 713, 725–726

Measurement,

and sampling, 507

value of, 323–324

Measurement equation, 791–792

Measure of a solid angle, 687

Megakernel tracing, 1033

Memoization (component of dynamic programming technique), 983

Memory practice, 435–437

Memory principles, 434–435, 1117ff

Mesh(es), 338–341

adjacency information on, 338–339

alternative mesh structures, 338

applications, 652–667

beautification, 197

cached and precomputed information on, 340–341

closed, 190, 642

connected unoriented, 639

differential coordinates for, 657

embedding topology for, 637

functions on, 201–220

geometry, 643–644

icosahedral, 648

indexed triangle, 338

Laplacian coordinates for, 657

manifold, 191

manifold-with-boundary meshes, 195

meaning of, 644–645

nonmanifold, 195–196

nontriangle, 637

operations, 197

orientation of triangles in, 193–194

oriented, 191, 639–640

other simplification approaches, 652

per-vertex properties and, 339–340

polygonal, 953

progressive, 649–652

quad, 611, 611f

repair, 654–655

simplices, 208

simplification, 188, 197

subdivision of, 211

terminology, 641

terminology for, 208

topology of, 189, 637–643

triangle, 187, 187f, 188f

unoriented, 191

winged edge polyhedral representation and, 338

Mesh beautification, 197

Mesh flattening, 667

Mesh geometry, 643–644

Mesh Laplacians, 656

Mesh operations, 197

edge collapse, 197

edge-swap, 197–198

mesh beautification, 197

mesh simplification, 197

Mesh repair, 654–655

Mesh specification, 120–128

Mesh structures, 211

memory requirements for, 196–197

Mesh topology, 637–643

Metaball modeling, 343

Metadata, 483

Metameric lights, 768

Metamers, 768

Metropolis light transport, 871–872, 915

Microfacets, 729

Microgeometry, 901

Micropolygon rasterization, 431–432

Micropolygons, 340, 431

Microsoft Office Picture Manager, 569

Minecraft, 964

MIP maps, 217, 491–492, 1120

Mirrors

BRDF of, 705–706

and point lights, 886–887

Mirror scattering, 715, 717–719

Mitchell-Netravali filter, 540f

Mixed probabilities, 820–821

Model, defined, 2

Modeling, defined, 2

Modeling space, 21

Modeling stage, 460

Modeling transformation, 51

Modelview, 314

Modelview matrix, 463

Modelview projection matrix, 314

Modified Euler method, 1020

Modular modeling

motivation for, 138–139

Modulus, of complex numbers, 513

Moiré patterns (image), 544

Monet painting, 948

Monospectral distributions, 747

Monte Carlo approaches, 851–854

bidirectional path tracing, 853

classic ray tracing, 851–852

distribution ray tracing, 838

path tracing, 853

photon mapping, 853–854, 872–876

Monte Carlo integration, 783, 796, 854

Monte Carlo rendering, 786, 922

Moore’s Law, 8, 932

Morphogens, 561

Motion

methods for creating, 966–975

perception, 976–978

root, 969

Motion blur, 980–983, 1061–1062

temporal aliasing and, 980–983

Motion-blur rendering, 922

Motion-induced blindness, 114

Motion perception, 976–978

Beta phenomenon, 977

strobing, 977

Motion planning, 972–973

Mouse-based object manipulation in 3D, 580–584

arcball interface, 584

trackball interface, 580–584

MSAA. See Multi-sample anti-aliasing (MSAA)

Multipass rendering, 441

Multiple importance sampling, 820, 868–870

Multiresolution geometry, 471

Multisample antialiasing (MSAA), 433, 1057–1058, 1136

advantages of, 1058

drawbacks of, 1057–1058

Multitouch interaction for 2D manipulation, 574–580

Munsell color-order system, 762

Mutation strategy, 871

Mutually perpendicular vectors, 229, 240

N

Natural user interfaces (NUIs), 571

Nearest neighbor (density estimation), 912

Nearest-neighbor field, 564

Nearest-neighbor strategy (animation), 967

Near-plane clipping, 1044, 1046

Negative nodes (BSP tree), 1031

Neighborhood (subdivision surfaces), 610

Neighbor-list table, 191

Nit (photometric term), 751

Nonconvex spaces, 211–213

Nonmanifold meshes, 195–196

Nonphotorealistic camera, 3

Nonphotorealistic rendering (NPR), 945, 986

Nonphysical tools, 371–372

Nonspectral colors, 766

Nonspectral radiant exitance, 700. See also Radiosity

Nonuniform B-spline, 604

Nonuniform Catmull-Rom spline, 601–602

Nonuniform rational B-spline, 604

advantages of, 604

CAD systems and, 604

Nonuniform scale. See Nonuniform scaling transformation

Nonuniform scaling, as linear transformation in the plane, 223

Nonuniform scaling transformation, 223

Nonuniform spatial distribution, 1100

Nonzero winding number rule, 177

Norm, of a vector, 157

Normal. See Normal vectors

Normalization process, 72

Normalized Blinn-Phong, 359–361

Normalized device coordinates, 22, 72. See also Camera-space coordinates

Normalized fixed point, 325

Normalizing vector, 157

Normal maps, 647

Normal transform. See Covectors, transforming

Normal vectors, 16, 27–28, 164, 193–194

Notation, mathematical, 150

Numbers

and orders of magnitude in graphics, 12–15

Numerical integration, 29, 801–802

deterministic method, 801–802, 804

probabilistic or Monte Carlo method, 802

Numerical methods for solving ODEs, 1017–1020

Numerical model, 11. See also Mathematical model; Physical models

NURB. See Nonuniform rational B-spline

NVIDIA GeForce 9800 GTX GPU, 1138

Nyquist frequency, 515

Nyquist rate, 544

O

Object-centered rotation, 272. See also Body-centered rotation

Object coordinates, 140, 245. See also Object space

Object ID channel, 485

Object-level scattering, 711–712

Object-oriented API, 41

Objects

detail, 1051

interaction with light, 118–119

and materials, 27–28

Object space, 245. See also Modeling space; Object coordinates

Occlusion, 308, 1023ff

in 2.5D systems, 1042

Occlusion culling, 1023, 1049

hierarchical, 1049–1050

Occlusion function, 1025

Occlusion query, 1049

Oct tree, 629, 1090

Odd winding number rule, 177

OLED. See Organic light-emitting diodes (OLEDs)

Omnidirectional point light

Omni-light, 379–380

One-dimensional (1D) meshes, 189

boundary of, 190

data structure for, 191–192

manifoldmesh, 190 1-ring (vertices), 641

OpenGL, 452

compatibility (fixed-function) profile, 454–455

core API, 466

programmable pipeline, 464–466

program structure, 455–456

utility toolkit, 456

OpenGL ES, 479

OpenGLUtilityToolkit. See GLUT(OpenGLUtilityToolkit)

Optic disk, 107

Optimization

early, 446–447

and performance, 444–447

Ordinary differential equation (ODE), 998

general ODE solver, 1016–1017

numerical methods for, 1017–1020

Oren-Nayar model, 732–734

Organic light-emitting diodes (OLED), 20

Orientation-preserving reflection, 264, 284

Oriented 2D meshes, 194–195

boundaries and, 194–195

Oriented meshes, 191

closed, 195

Oriented simplex, 639

Orthogonal matrix, 230

Orthographic cameras, 315–317

Orthographic projections. See Parallel projections

Outer product (matrices), 260

Output merging stage, 433–434

Output-sensitive time cost, 1079

Outscattering, 738

Outside/inside testing, 175–177

Outward edge normal, 175

Over operator, 365

P

Packet tracing, 1061

Painter’s algorithm, 1028, 1041–1042

Panning (camera controlling technique), 585

Pantone^TM color-matching system, 761

Paraboloid, dual, 554

Parallel projections, 315–316

Parameterization

building tangent vectors from, 552–553

of lines, 155

texture, 555

of triangles, 171

Parameterized model, 76

Parametric equation, 341

Parametric form of the line between P and Q, 155

Parametric-implicit line intersection, 167

Parametric lines, transforming, 254

Parametric-parametric line intersection, 166

Partial coverage, 364–367, 365, 1028, 1054–1062

defocus, 1060–1061

as a material property, 1062

motion blur, 1061–1062

spatial antialiasing, 1055–1060

Participating media, 737–738

Particle collision detection, 1008–1009

Particle collisions, 1008–1012

collision detection, 1008–1009

impulses, 1010–1012

normal forces through transient constraints, 1009

penalty forces, 1009–1010

Particle systems, 350–351

Path mesh, 668

Path tracer

basic, 889–904

building, 864–868

code, 893–901

core procedure in, 893

Kajiya-style, 866, 867

preliminaries, 889–893

symbols used in, 890

Path-tracer code, 893–901

Path tracing, 853, 853f, 855

algorithmic drawbacks to, 855–856

bidirectional, 853, 870–871

constructing a photon map via, 873

and Markov chains, 856–857

path tracer, building, 864–868

Pdf. See Probability density function (pdf)

Pen (geometric attribute), 38

Penalty force

application of, 1009–1010

computation of, 1009–1010

defined, 1009

Pencil of rays, 1060

Penumbra, 505, 798

Perception, human visual. See Human visual perception

Perceptual color spaces, 767–768

Peripheral color perception, 781–782

Perlin noise, 560–561, 561f

Perspective camera specification, 301–303

Perspective-correct interpolation, 422–424

Per-vertex properties

of meshes, 339–340

Phong exponent, 736

Phong lighting, 930

Phong lighting equation, 938

Phong model (reflection model), 721–723

Phong reflectance (lighting) model, 134

Phong shader, 937–939

Phong shading, 723. See also Gouraud shading

fragment shader for, 938–939, 940

vertex shader for, 938

Phosphorescence, 671

Photography

computational, 493

Photometry, 670, 700

Photon(s), 369, 670

defined, 872

photon-mapping, 872

propagation, 907–908

Photon arrival rates

and light energy, 12–13

Photon emission, 376–377

Photon map, 872, 908, 913

constructing via photon tracing, 873

Photon mapping, 853–854, 872–876, 904–914

bias in, 875

consistency in, 875

density estimation and, 912

final gather step in, 913

image-space, 876

limitations, 875

main parameters of, 873

phases of, 872

schematic representation of, 853f

Photon propagation, 907–908

Photon tracing, 872

Photopic vision, 753–754

Photorealism, 945

Photorealistic rendering, 31

Physical coordinate system, 38

Physically based animation, 963, 989

Physically based models, of scattering, 713, 727–734

Physical models, 11. See also Numerical model; Physical models

Physical optics, 726

Physical units

and compositing, 489–490

Physics scene graphs, 352–353

Pick correlation, 39, 60, 139, 464

Pick path, 464

Piecewise constant function, 187

Piecewise linear extension, 210

animation, use in, 211

defined, 210

limitations of, 210–211

mesh structure and, 211

Piecewise linear reconstruction, 505

Piecewise-smooth curves, 540

Pie menus, 573, 573f

Pitching, 267

Pixar, 932

Pixel coordinates, 22

Pixel program, 433

Pixels, 5

defined, 29

Pixel shader. See Pixel program

Pixel shaders, 930. See also Fragment shaders

Pixel stages

and vertex stage, 433

Pixel values, 482

Pixmaps. See Bitmaps

Planar wave, 676

Planck, Max, 674

Planck’s constant, 12, 670, 674

Plenoptic function, 370, 693

PNG. See Portable Network Graphics (PNG)

Point lights, 124–125, 133, 886–889

computing direct lighting from, 894

mirrors and, 886–887

reflecting illumination from, 894

Points, 234–235, 288

Point sets, 345–346

Poisson disk process, 921

Polarization, 670, 677–679

circular, 678f

Fresnel’s law and, 681–683

linear, 678f

Polarizers, 679

Polling (interaction loop), 574

Polygonal contour extraction, 955

Polygon coordinates, 249. See also Modeling coordinates

Polygon meshes, 635

Polygons, 175–182

back face, 337

drawing as black box, 23

front face, 337

interiors of nonsimple, 177

micropolygon, 340

normal to polygon in space, 178–179

polygon rate, 14

simple, 175

Polygon soup, 654

Polyhedral manifolds, 191

Polyhedral meshes

conversion to, 625–629

conversion to implicits, 629

Polyline, 81

Polymorphic types, 1068

Polymorphism, 1073

“Poor man’s Fourier series,” 560

Poor sample test efficiency, 420

Popping, 985

Portable Network Graphics (PNG), 484

Portal culling, 470

Portals, 1051, 1052–1054

Positive nodes (BSP tree), 1031

Positive winding number rule, 177

Potential energy, 1021

Potentially visible set (PVS), of primitives, 1050

Power vectors, 875

p-polarized light source, 681

p-polarized waves, 681

Practical lights, 372

Prebaking (models), 247

Premultiplied alpha, 366–367, 487

problem with, 489

Presentation vector graphics, 1042

Primary colors, 758–759

Primary ray, 1027

Primary visibility, 1023, 1024, 1027

Primitive components

geometries of, 140–141

instantiating, 141

Primitives, 6, 38, 461–462, 962, 969, 973, 1028, 1030–1031, 1041–1042, 1044, 1059–1060, 1084–1085, 1087–1092

Primitives per second, 6

Principal curvatures, 956

Principal directions, 956

Probability

of an event, 803

continuum, 808–810

mixed, 820–821

Probability density, 684, 686

Probability density functions (pdf), 684, 808, 810–812

Probability masses, 685, 820

Probability mass function (pmf), 803, 805

Probability of an event, 809

Procedural code, 35, 55, 58

vs. declarative specification, 40

dynamics via, 58

Procedural texturing, 549

Programmable graphics card, 8. See also GPU.

Programmable lighting, 930. See also Programmable shading

Programmable pipeline, 433, 453–454

abstract view of, 464–466

OpenGL, 464–466

Programmable rendering pipeline

fixed-function to, 452–454

Programmable shading, 930, 932. See also Programmable lighting

Programmable units, 433

Programmatic interfaces, 1068–1077

Programmer instruction batching, 1033

Programmer’s model, 17, 454–464

Progressive meshes, 649–652

goal of, 649

Progressive refinement (radiosity), 843

Progressive television formats, 978

Projected solid angle, 690

Projection stage, 460

Projection textures, 555, 629

Projective frame, 265

Projective geometry, 77, 257

Projective transformations, 255, 257, 259–260, 263, 291–293, 308

general position, 291–292, 292f

properties of, 265–266

Projective transformation theorems, 265–266

Propagation, 332–333

Proxies (data structure), 1068

Pseudoinverse

defined, 232

least squares problems and, 232

SVD and, 231–233

Pseudoinverse Theorem, 232

Pulldown (interlacing), 979–980

Q

Quad fragments, 1137

Quad meshes, 611, 611f, 635

Quadratic error function, 653

Quad trees, 1090, 1091f

Quake video game, 1052

Quantitative invisibility, 1028

Quaternions, 273, 283, 993

QuickDraw, 38

R

Radial curvature, 957

Radiance, 333, 397, 693, 694–695

area-weighted, 910

emitted, 785, 832

field, 786, 834, 846

impulse-reflected, 794

linear, 398

surface, 786–787, 834, 846

Radiance computations, 683, 695–697

Radiance function, 370

Radiance propagation, 907–908

Radiant exitance, 699

spectral, 699

Radiant flux, 699. See also Radiant power

Radiant power, 699. See also Radiant flux

Radiometry, 669, 694

Radiosity, 333, 700, 797, 838–844. See also Nonspectral radiant exitance

characteristics, 838

color bleeding and, 839

as finite element method, 839

meshing in, 843

Radiosity equation, 840

Randomized algorithms

random variables and, 802–815

Random parametric filtering (RPF), 922

Random point, 812

Random variable(s)

in continuum probability, 809

defined, 803

estimator, 818

expected value of, 810

identically distributed, 808

independent, 807

independent identically distributed (iid), 808

and randomized algorithms, 802–815

random point and, 812

uniform, 807

Random variable with mixed probability, 820

Raster devices, 8

Raster graphics, 209

history of, 931–932

Rasterization, 18, 418–432, 1027, 1061

conservative, 1096

defined, 387, 391

hierarchical, 430

and high-level design, 388–393

incremental scanline, 431

micropolygon, 431–432

and ray casting, 387–449

rendering with API of, 432–434

swapping loops, 418–419

triangles first, 391–393

Rasterizer algorithm, 418

Rasterizing shadows, 428–429

Rasterizing stage, 433

Rasters, 391, 978, 979

Rational B-spline, 604

Rational numbers, 325

Ray bumping, 886, 1027

Ray casting, 1023, 1028, 1029–1034

defined, 387, 391

implementation platform and, 393–403

pixels first, 391–393

and rasterization, 387–449

renderer, 403–404

and sampling framework, 407–408

time cost of, 1029

Ray intersection, 1095–1099

Ray intersection query, 1026

Ray optics

building blocks of, 330

Ray packet, 445

Ray packet tracing, 1027, 1033

Ray-plane intersection, 168–170

Ray-sphere intersection, 170–171

Ray tests, parallel evaluation of, 1032–1034

Ray tracer

steps involved in, 929

Ray-tracing

defined, 391

recursive, 851–852

implicit surfaces, 342–343, 631

Ray-triangle intersection, 408–411, 1073

Reaction (morphogens), 561

Ready for rendering

vs. abstract geometric, 467

Realistic lighting

producing, 124–127

Realistic rendering, building blocks for, 26–31

Real numbers, 324–325

Real-time 3D graphics platforms, 351–480

introduction, 351–352

Reciprocity, 714. See also Helmholtz reciprocity

and BRDF, 705–706

Reconstruction, 505

and band limiting, 524–527

piecewise linear, 505

Rectangular area light, 377–378

Recursive approach, 861–864

Reencoding, 470–471

Reference frame, 963

Reference renderer, 388

Reflectance, 133–138, 702–704, 711ff

ambient reflection, 136

diffuse reflection, 136

emissive lighting, 138

phong reflectance (lighting) model, 134

specular reflection, 137–138

WPF reflectance model, 133–138

Reflectance equation, 703, 786

Reflection mapping, 556

Reflection model, 723. See also Scattering model

Reflective scattering, 697, 715

Reflective surface, 27

Refractive index, 679

Refractive scattering, 716

Rejection sampling, 823

Rendering, 6

animation and, 975–987

binned, 1137–1138

debugging and, 915–919

double-buffered, 971, 975–976

expressive, 945–962

intersection queries in graphics that arise from, 1066–1067

Monte Carlo, 922

motion blur, 980–983

motion-blur, 922

nonphotorealistic, 945, 986

pen-and-ink, 950

photorealistic, 31

stroke-based, 955

temporal aliasing, 980–983

Rendering equation, 373–376, 703, 783, 786–787, 831–836

approximating, 826

approximations of the series solution of, 847–848

discretization approach, 838–844

domain restriction, 827

for general scattering, 789–792

Markov chain approach for solving, 857–861

methods for solving, 825–831

Monte Carlo approaches for solving, 851–854

recursive approach for solving, 861–864

series solution of, 844–846

simplifying, 840

spherical harmonics approach, 848–851

Replication

in spectrum, 523–524

Representations

comparing, 278–279

evaluating, 322–323

of implicit functions, 624–625

of light, 887–888

and standard approximations, 321–384

surface, 882–887

triangle fan, 338

Resolution, 8

eye’s resolution, 13

Resolution dependence, 38

Resolved framebuffer, 1056, 1056f

Restricted transformations, 295–297

advantages of, 295

disadvantages of, 295

Retained mode (RM), 452

vs. immediate mode (IM), 39–40

Retina, 107

Retroreflective scattering, 716

Reuse of components, 144–147

Reyes micropolygon rendering algorithm, 982–983

RGB color cube, 772–773

RGB color model, 772–774

RGB format, 481

RGB image, 482

Ridges, 956–957

apparent, 958–959

Right-handed coordinate system, 158

Rigid coordinate frame, 240

Ringing, 510f, 529, 538

Rodrigues’ formula, 293

Rods (color receptors), 107, 749

saturated, 755

Rolling, defined, 267

Root frame, 972

Root frame animation, 972

Root motion, 969

Rotation, as linear transformation in the plane, 222–223

Rotation about z by the an angle, 266

Rotation around z, 239. See also Rotation in the xy-plane

Rotation by an angle in the xy-plane of 3-space, 266

Rotation in the xy-plane, 239. See also Rotation around z

Rotation matrix, 270–272

finding an axis and angle from, 270–272

Rotation(s), 264, 266

3-sphere and, 273–278

axis-angle description of, 269–270

interpolating between, 276–278

vs. rotation specifications, 279–280

Rule of five, 698, 925

Russian roulette, 874

S

S (normalizing vectors), definition, 157

Sahl, Ibn, 679

Sample-and-hold strategy (animation), 967

Sample and hold reconstruction, 505

Samples (implicit functions), 621

Samples (pixel), 1055, 1056f

Sample shaders, 930

Sampling, 29, 507–508, 557–559, 724–725

approximation of, 519

and band limiting in interval, 514–515

cosine-weighted, 815

importance, 818–820, 854

integration and, 31

multiple importance, 868–870

rejection, 823

stratified, 920

summing a series by, 828–830

Sampling framework, 407–408

Sampling strategy, 854

Sampling theorem, 515

Scalability, 469

Scalar attributes, 651

Scalar multiplication, 157, 158

Scale invariance, 911

Scale transformations, 263–264

Scanline interpolation, 208–210

Scanline rendering, 209

Scanners, 5, 5f

Scattering, 711, 792–793

approximating, 848–851

diffuse, 713, 716

diffuselike, 792

due to transmission, 900

equation, 790

glossy, 716

impulse, 715–716, 784

kinds of, 714–717

Lambertian, 716

of light, 388–390

mirror, 715, 717–719

models, 713

nonspecular, 852

object-level, 711–712

physical constraints on, 713–714

reflective, 715

refractive, 716

rendering equation for, 789–792

retroreflective, 716

Snell-transmissive, 783–784

subsurface, 720, 738–739

surface, 712–714

transmissive, 715

volumetric, 737, 793

Scattering equation, 790

Scattering functions, 354–358

Scattering models, 723. See also Reflection model

Blinn-Phong model, 721–723

Cook-Torrance model, 731–732

empirical/phenomenological models, 713, 717–725

Lafortune model, 723–724

measured/captured models, 713

Oren-Nayar model, 732–734

Phong model, 721–723

physically based models, 713, 727–734

Torrance-Sparrow model, 729–731

types of, 713

wave theory models, 734–736

Scatters, light, 333

Scene, 21, 31, 37

abstract coordinate system to specify, 42–44

planning, 120–124

reduction of complexity, 469

Scene generator, 37

Scene graphs, 39, 118, 351–353

coordinate changes in, 248–250

hierarchical modeling using, 138–147

physics, 352–353

Scene modeling, 945

Scene representation, 400–402

Schematization (abstraction), 947

Schlick approximation, 728–729

Scotopic vision, 753–754

Screen door effect, 986

Screen space, 245

Screen tearing, 976

Second-order Runge-Kutta methods, 1019

Sector (polyhedron), 1050

Sector-based conservative visibility, 1050–1054

mirrors, 1052–1054

portals, 1052–1054

stabbing line, 1051

stabbing tree, 1051–1052

Sector-based culling, 470

Segments (of a spline), 599

Self-shadowing, 1027. See also Shadow acne

Semantic element, 352

Semi-Explicit Euler method. See Semi-Implicit Euler method

Semi-Implicit Euler method, 1019

Sensor response, 791

Separable filter, 544

Sequencing design (user interface), 570

Sets, 150

Shaders, 8, 453. See also Programs

creating, 437–442

defined, 928

extensibility via, 453

fragment, 466, 930

geometry, 931

historical development, 929–932

Phong, 937–939

pixel, 930

sample, 930

in scattering model, 723

simple graphics program with, 932–937

subdivision surface, 931

tessellation, 931

vertex, 930, 931

Shader-specified visibility, 1136

Shader wrapper, 933

G3D, 933

Shades (color), 756

Shading, 412–413, 723, 1055

deferred, 1135–1137

interpolated, 128–129

vs. lighting in fixed-function rendering, 127–128

two-tone, 959

Shading language, 927

Shading normals, 339

Shadow, and light transport, 797

Shadow acne, 325, 416, 1027. See also Self-shadowing

Shadow map, 428, 848

Shadow mapping, 557t

Shadows, 112–113, 414–417

acne, 325, 416, 1027

applications of, 113

curvature, 946

direct, 946

Shannon sampling theorem, 515

Shape constancy, 110

Sharp edges, 651

Sharpening, 543, 545

Shearing transformations, 264

as linear transformation in the plane, 223

Shift-invariant, 529

Shutter time, 980. See also Exposure time

SIGGRAPH (Special Interest Group on GRAPHics and Interactive Techniques), 4, 922

Signal processing, 500

and images, 493–532

Signal, 500

Signed area

of a plane polygon, 177–178

Signed distance transform of the a mesh, 629

Signed normalized fixed-point 8-bit representation, 551

Silhouette, 952

Silicon Graphics, Inc., 931–932

SIMD. See Single Instruction Multiple Data (SIMD)

Simple polygons, 175

Simplex, 208

boundary of, 208

categories of, 208

oriented, 639

Simplicial complices, 198

Simplification, 471

abstraction, 947

of triangle meshes, 188, 649

Single Instruction Multiple Data (SIMD), 430, 1033

Singular transformation. See Degenerate transformation

Singular value decomposition (SVD), 230

computing, 231

matrix properties and, 230–231

and pseudoinverses, 231–233

Singular values of matrix M, 230

Size constancy, 110

Skyboxes, 348–349

Sky sphere. See Skyboxes

Slerp, 275. See also Spherical linear interpolation

Slicing, 935

Smith, Alvy Ray, 498

Smooth edges, 953

Smooth manifolds, 190

Snell’s law, 679, 683, 728

Snell-transmissive scattering, 783–784. See also Impulses

Soft particles, 351

Software-platform independence, 44

Software stack, 468

Solid angles, 370, 686–688

computations with, 688–690

measure of, 687

projected, 690

subtended, 687

Source (texture image), 563

Source polygon, 1045

Spatial acceleration data structures. See Spatial data structures

Spatial antialiasing, 1055–1060

A-buffer, 1057

analytic coverage, 1059–1060

coverage sampling antialiasing (CSAA), 1058–1059

multisample antialiasing (MSAA), 1057–1058

supersampled antialiasing (SSAA), 1056–1057

Spatial coherence, 950

Spatial data structures, 353, 1023, 1065–1102

characterizing, 1077–1079

extracting keys and bounds, 1073–1077

generic use of, 1077

grid, 1093–1101

hash grid, 1095

Huffman’s algorithm and, 1089

intersection methods of, 1069–1073

k-dimensional structures, 1080–1081

list, 1081–1083

ordered, 1077

polymorphic types, 1068

programmatic interfaces, 1068–1077

ray intersection, 1095–1099

selecting, 1077

trees, 1083–1093

Spatial frequencies, 103

SPD. See Spectral power distribution (SPD)

Special orthogonal matrix, 230

Specification

camera, 301–303

color, 133

transformations and camera, 299–317

Spectral irradiance, units of, 698

Spectralon, 720

Spectral power distribution (SPD), 747

incandescent lights, 748

monospectral distributions, 747

Spectral radiance, 692

integral of, 692

Spectral radiant exitance, 699

Spectrum (of a signal), 513

replication in, 523–524

Specular, in graphics, 8

Specular exponent, 137

Specular power. See Specular exponent

Specular reflections, 137, 713

physical models for, 726–727

Specular (mirror) reflections, 353

Specular surface, 27

Sphere mapping, 340

Sphere-to-cylinder projection theorem, 688

Sphere trees, 649, 1093

Spherical harmonics, 531, 843, 848–851

Spherical linear interpolation, 275. See also Slerp

Spline patches, 344

and subdivision surfaces, 343–344

Splines, 343, 595ff, 599, 607ff, 623

Splitting plane, 1030, 1030f

s-polarized wave, 681

Spot color, 766

Spotlight, 133, 702

Square integrable, 506

Square summable, 506

sRGB standard, 774

Stabbing line, 1051

Stabbing tree, 1051–1052

Stamping, 985

Standard basis vectors, 227

Standard deviation, 807

Standard implicit form for a line, 165

Standard parallel view volume, 307

Standard perspective view volume, 307

Star, of a simplex, 208, 208f

Star, of a vertex, 208, 641

Star of an edge, 641

State machine, 454

State variable, 454

State vectors, 1015

Static frame, 462–463

Statistical estimators, 827–830

Stefan-Boltzmann law, 672

Stencil buffer, 329

Steradians, 688

Steven Anson Coons Award, 608

Stratified sampling, 920

blue-noise property of, 921

Strobing (motion perception), 977

Strokes (expressive rendering), 950–951

creation of, 951

imitation of artistic technique for creating, 951

oil-paint, 951

pen-and-ink, 951

physical simulation, 951

scanning/photography approach for creating, 951

Styles, 85

artistic, 947

Subcomponents, 138, 141–144

Subdivision, meshes, 211

Subdivision, of triangle meshes, 188

Subdivision curves, 604

Subdivision surfaces, 344, 607

Catmull-Clark, 610–613

modeling with, 613–614

Subdivision surface shaders, 931

Subsurface reflector, 720

Subsurface scattering, 353, 720, 738–739

computing, 739

modeling, 739

physical modeling, 739

practical effects of modeling, 739

Subtractive color, 760

Suggestive contour generator, 958

Suggestive contours, 957–958

characteristics of, 958

Suggestive interface, 589

Summary measures, of light, 670

Sum-squared difference, 104

Superposition, 361

Supersampled antialiasing (SSAA), 1056–1057

advantages of, 1056

drawbacks of, 1056–1057

implementing, 1056

Surface mesh

embedding of, 642

Surface normal, 16, 27–28

Surface radiance, 786–787, 834, 846

computation from field radiance, 786

Surface radiance function, 787

Surface representations, 882–887

Surface, 390, 607

with boundary, 637–638

closed, 638

implicit, 619–620

orientable, 639

oriented, 639

representations, 882–887

triangulated, 637–638

Surface scattering, 712–714

physical constraints on, 713–714

scattering models, 713

Surface Texture, 132, 547ff

texturing via stretching, 132

texturing via tiling, 132

in WPF, 130–132

Surface with boundary, 638

Surjective, 151. See also Bijective

Sutherland-Hodgman 2D clipping, 1045–1046

Swapping loops, 418–419

T

Tabulated functions, 201

Tagged Image File Format (TIFF), 482

Tangent field, 1015–1016

Tangent-space basis, 340

Tangent vectors

building from a parameterization, 552–553

Target (texture image), 563

Taylor polynomial, 1017

Teddy (user interface), 590

Telecine (interlacing), 978–980

Template (pixels), 563

Templated classes in programming languages, 1068

Temporal aliasing, 980–983

motion blur and, 980–983

Temporal coherence, 950, 962, 983

advantage of, 983

burden of, 985–987

exploiting, 983–984

Tent-shaped functions, 208. See also Basis functions

Tent-shaped graphs. See Basis functions; Tent-shaped functions

Tessellation shaders, 652, 931

Test beds, 72, 81

2D Graphics, 81–98

application of, 95–98

details of, 82–87

structure of, 83–88

using 2D, 82–83

Test scene, 402–403

Texels, 365, 742

Texture aliasing, 216

Texture coordinates, 339, 548

assigning, 555–557

assignment of, 215–216

Texture mapping, 131, 214–215, 547ff

application examples of, 557t

defined, 548

details of, 216

image-based, 559

problems, 216–217

properties of, 555

Texture maps, 15–16, 215, 547ff

codomains for, 553–554

Texture parameterization, 555

Textures

modeling, 630

projection, 555

Texture-space diffusion, 341

Texture synthesis, 559–562

Fourier-like synthesis, 559–560

Perlin noise, 560–561, 561f

reaction-diffusion textures, 561–562

Texturing

bump mapping, 550–551

contour drawing, 551–552

environment mapping, 549–550

variations of, 549–552

via stretching, 132

via tiling, 132

3ds Max transformation widget, 588–589, 588f

3D transformations

building, 237

3D view manipulation widget, 588–589, 588f

3-space

essential mathematics and geometry of, 149–182

3:2 pulldown algorithm (interlacing), 979

TIFF. See Tagged Image File Format (TIFF)

Tile fragments, 1137

Tiling rasterizer, 430–431

Tilting principle, 180–181

Time domain, 513

Time-state space, 1013–1015

TIN. See Triangulated Irregular Network (TIN)

Tints (color), 756

T-junction, 642

Tone mapping, 919

Tones (color), 756

Tool trays, 569

Toon-shading, 940–942

fragment shader for improved, 942

pixel shader for, 941

two versions of, 940–942

vertex shader for, 940–941

Torrance-Sparrow model, 729–731

Total internal reflection, 682

Trait data structure, 1074

Traits, as key extraction method, 1074–1077

advantages of, 1076

disadvantages of, 1076

Transformation

linear, 307

modeling hierarchy and camera, 313–315

perspective-to-parallel, 313

projective, 308

rasterizing renderer ripeline and camera, 310–312

unhinging, 307

windowing, 300

Transformation associated to the matrix M, 224

Transformation pipeline, 460

Transformations, 221–286, 288–290

adjoint, 253

affine, 234, 259

AffineTransformation2, 288

associated, 294

change-of-coordinate, 231

classes of, 288–289

composed, 235

and coordinate systems, 229–230

covector, 253

efficiency of, 289–290

finding the matrix for, 226–228

fractional linear, 256

homogenizing, 265

implementation, 290–293

vs. interpolation, 259

linear, 221, 259

LinearTransformation2, 288

matrix, 222

MatrixTransformation2, 288

modeling, 630

parametric lines, 254

projective, 255, 257, 259–260, 263, 291–293

ProjectiveTransformation2, 289

restricted, 295–297

scale, 263–264

shearing, 264

specification of, 290

in three dimensions, 263–285

in two dimensions, 221–262

of vectors, 250–253

windowing, 236–237, 236f

world-centered rotation, 272

Translation, 222, 233–234, 263

Translation equivariant reconstruction, of signal, 546

Translucency, and blending, 361–364

Transmission/rendering, 353, 367–368

reduction, 470–472

Transmissive scattering, 715

Transparent surface, 364

Transport, light, 335–336

Transport equation, 786–787

Transport paths, separation of, 844

Transposition, 156

Trees, as a class of spatial data structures, 1083–1093

binary space partition (BSP) trees, 1084–1089

Bounding Volume Hierarchy (BVH), 1092–1093, 1092f

building BSP trees, 1089–1092

kd tree, 1089

oct tree, 1090

quad trees, 1090, 1091f

Triangle fan, 338

Triangle list, 338

Triangle meshes, 187, 187f, 188f, 635

icosahedral, 187

1D mesh, 189

shape approximation, 187–188

simplification of, 188, 649

subdivision of, 188

uniformity of, 188

Triangle processing, 17

Triangle reordering for hardware efficiency, 664–667

Triangles, 171–175

half-planes and, 174–175

parameterization of, 171

signed area, 177

in space, 173–174

Triangle soup, 338

Triangle strip, 338

Triangulated Irregular Network (TIN), 345

Triangulated surfaces, 637–638

Triple buffering, 976

Trotter, Hale, 149

2D barycentric weights, 424–427

2D coverage sampling, 422

2D graphics

dynamics in 2D graphics using WPF, 55–58

evolution of, 37–41

overview of, 36–37

test beds, 81–98

and Windows Presentation Foundation (WPF), 35–60

2D manipulation

multitouch interaction for, 574–580

2D raster graphics platform, 38

2D scene

WPF to specify, 41–55

2D scissoring, 1044 2D transformations, 238–239

building, 238–239

2-ring (vertices), 641

2-space

essential mathematics and geometry of, 149–182

Two-and-a-half dimensional, 43

Two-tone shading, 959

U

Übershader, 441

UI controls, 39, 41

UI generator, 37

Ulam, Stanislaw, 945

Umbilic points, 956

Umbra, 505, 798

Uncanny valley, 19

Undragging, 581

Unhinging transformation, 307

Unicam, 584–587

Uniform color space, 767

Uniform density

on the sphere, 813

defined, 809

Uniform spline, 601

Uniformity, of triangle meshes, 188

Uniform random variable, 807

Uniform scaling transformation, 223

Units, 333

Unit vector, 157, 229

Unoccluded two-point transport intensity, 847

Unoriented meshes, 191

Unpolarized light, 683

Unsigned normalized, 325

Up direction, 122, 302

User interface (UI), 6–7, 21

User interface examples, 588–591

Chateau, 589–590

first-person-shooter controls, 588

Grabcut, 590–591

Photoshop’s free-transform mode, 589

Teddy, 590

3ds Max transformation widget, 588–589, 588f

Utility classes, 395–400

uv-coordinates, 216

V

Valence, 637

Valleys, 956–957

Value (data structure), 1077

Value of measurement, 323–324

Vanishing point, 77

Variables

change of, 690–692

Variance, 807, 818

Variance reduction, 921

Vectorization (programmer instruction batching), 1033

Vectors, 155–161, 234–235, 288

coordinate, 155

edge, 175

indexing, 156

kinds of, 162–163

length of, 157–161

normal, 164

normalizing, 157

operations, 157–161

transforming, 250–253

Vertex/Vertices, 50, 65, 189

boundary, 194, 638

degree of, 637

interior, 194, 638

link of, 208

locally flat, 643

manifold, 194f

star of, 208

Vertex geometry processing, 17

Vertex geometry transformation, 17

Vertex normal, 129

Vertex shaders, 465, 930, 931

Vertex stage and pixel stages, 433

Vertical synchronization, 976

Video standards, 775–776

View center (camera manipulation), 586

ViewCube (3D view manipulation widget), 588–589, 588f

View-frustum culling, 470, 1023

Viewing stage, 460

Viewport, 37, 302, 455

Viewport3D, 119, 121

View region, 63

View specification

building transformations from, 303–310

View volume, 77, 120, 302

standard parallel, 307

standard perspective, 305, 307

Vignetting, 336

Virtual arcball, 280–283, 584

Virtual parallelism, 1113ff

Virtual sphere, 580

Virtual trackball, 280–283, 580

Virtual transitions, 671

Visibility, 65

conservative, 1023

coverage (binary), 1027–1028

goals for, 1023

list-priority algorithms, 1040–1043

primary, 1023, 1024, 1027

sector-based conservative, 1050–1054

Visibility determination

applications of depth buffer in, 1035–1036

backface culling, 1047–1049

current practice and motivation, 1028–1029

depth buffer, 1034–1040

frustum clipping, 1028, 1045–1046

frustum culling, 1023, 1028, 1044

hardware rasterization renderers and, 1028

hierarchical occlusion culling, 1049–1050

list-priority algorithms, 1040–1043

partial coverage, 1028, 1054–1062

ray casting, 1029–1034

ray-tracing renderers and, 1028

Visibility function, 786, 799, 1025–1027

evaluating, 1026

Visibility problem. See Visibility testing

Visibility testing, 422

Visible contour, 953. See also Contour(s)

Visible points, 390–391

Visible spectrum, 330–332

Visible surface determination, 1023. See also Hidden surface removal

Vision

photopic, 753–754

scotopic, 753–754

Visual cortex, 103, 106, 108, 110

Visual perception, human. See Human visual perception

Visual system, 103–105

applications of, 105, 109–110

components of, 103

Volumetric models, 349–351

Volumetric scattering, 737

Voxelization, conservative, 1096

Voxels, 349–350

VRML, 479

vup, 302

W

Walk cycle, 966

Warnock’s Algorithm, 1041

Warped z-buffer, 1038

Watertight model (meshes), 643

Wavelength, 332, 675

Wave theory models, 734–736

Wave velocity, 675

w-buffer, 1040. See also Depth buffer

WebGL, 479

Weiler-Atherton Algorithm, 1041

Wheel of reincarnation, 18–19

Whites (color), 769

CIE definitions, 769

illuminant C, 769

illuminant E, 769

Whole-frustum clipping, 1044, 1047

Widgets. See UI controls

Wien’s displacement law, 710

Wii (interface device), 568

WIMP (windows, icons, menus, pointers) GUI (WIMP GUI), 6, 8, 567

Winding number, 176–177

Window chrome, 36

Windowing transformation, 236–237, 236f, 300

Windows Presentation Foundation (WPF)

2D Graphics using, 35–60

application/developer interface layers, 40–41

canvas coordinate system, 45–46

data dependencies, 91–92

dynamics in 2D graphics using, 55–58

reflectance model, 133–138

to specify 2D scene, 41–55

surface texture in, 130–132

Winged-edge data structure, 196

Wire-frame model, 65

The Wizard of Oz (film), 950

Woodcut, Dürer, 61–65, 1035

World-centered rotation, 272

World coordinate system, 119

World space, 21, 245

WPF. See Windows Presentation Foundation (WPF)

WPF 3D, 117

design of, 118

high-level overview of, 119–120

X

X3D (language), 479

XAML. See Extensible Application Markup Language (XAML)

Xerox PARC, 8

XToon shading, 942–943

2D texture map for, 942f

atmospheric perspective, 942

vertex and fragment shaders for, 942–943

Y

Yaw, 267

YIQ color model, 775

Z

z-buffer, 306, 310, 392. See also Depth buffer

z-buffer value, 1038, 1039f

z-data, 482

Zero set, 164, 616. See also Isosurfaces

Zero-to-one coordinates. See Normalized device coordinates

z-fighting, 1037

z-values

perspective and, 313