《数字媒体：计算机图形学问题求解方法入门指南》

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《数字媒体：计算机图形学问题解决方法》是由Rimon Elias撰写的一本深入浅出的计算机图形学教材。该书以其独特的角度，注重理论与实践相结合，旨在为初学者提供一个系统且易于理解的学习路径。作者通过丰富的习题设计，帮助读者逐步掌握计算机图形学的基础概念和核心算法。书中涵盖了广泛的主题，包括但不限于几何建模、图像处理、渲染技术、动画制作以及三维场景的构建。每一章节都围绕一个具体的计算机图形学问题展开，引导读者运用所学知识去解决问题，从而培养他们的实际操作能力和创新思维。《数字媒体》不仅仅是一本教科书，它更像是一本解决问题的手册，适合那些希望在图形学领域起步的学生和专业人士。 Rimon Elias以德国开罗大学作为学术背景，确保了内容的专业性和权威性。本书的国际标准书号(ISBN)为978-3-319-05136-9和978-3-319-05137-6，电子版可通过DOI 10.1007/978-3-319-05137-6获取。出版商Springer International Publishing将版权保护得非常严格，仅允许在特定情况下进行摘录或复制，如学术评论或教学用途。通过阅读《数字媒体：计算机图形学问题解决方法》，读者不仅可以学习到基础知识，还能了解到如何将理论知识应用于实际项目中，提升技能并为未来的职业发展打下坚实基础。无论是希望从事游戏开发、影视特效制作还是虚拟现实领域的专业人士，这本书都是一个理想的起点。这是一本实用且全面的计算机图形学入门读物，对所有寻求在这个领域深入探索的人来说，无疑是一份宝贵的资源。

Example 3.12 2D general reflection—inhomogeneous coordinates . . . . 84

Example 3.13 2D general reflection—homogeneous coordinates. . . . . . 85

Example 3.14 2D general reflection—homogeneous coordinates. . . . . . 86

Example 3.15 2D general shearing—inhomogeneous coordinates . . . . . 89

Example 3.16 2D general shearing—homogeneous coordinates . . . . . . 90

Example 3.17 2D composite transformations . . . . . . . . . . . . . . . . . . . 93

Example 3.18 2D composite transformations . . . . . . . . . . . . . . . . . . . 94

Example 3.19 2D composite transformations . . . . . . . . . . . . . . . . . . . 95

Example 3.20 2D composite transformations . . . . . . . . . . . . . . . . . . . 96

Example 3.21 2D axes translation—inhomogeneous coordinates. . . . . . 102

Example 3.22 2D axes rotation—inhomogeneous coordinates . . . . . . . 103

Example 3.23 2D axes rotation—inhomogeneous coordinates . . . . . . . 104

Example 3.24 2D axes transformations—inhomogeneous

coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Example 3.25 2D axes transformations—inhomogeneous

coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Example 3.26 2D axes scaling—inhomogeneous coordinates . . . . . . . . 106

Example 3.27 2D axes scaling—inhomogeneous coordinates . . . . . . . . 106

Example 3.28 2D reflection about the x-axis—inhomogeneous

coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Example 4.1 Wireframe modeling. . . . . . . . . . . . . . . . . . . . . . . . . . 113

Example 4.2 Baumgart’s winged-edge data structure. . . . . . . . . . . . . 117

Example 4.3 CSG expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Example 4.4 CSG expressions and trees . . . . . . . . . . . . . . . . . . . . . 121

Example 4.5 Spatial enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Example 4.6 Quadtree partitioning . . . . . . . . . . . . . . . . . . . . . . . . . 127

Example 4.7 Solids as octrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Example 4.8 Bintree partitioning. . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Example 4.9 Bintree partitioning. . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Example 4.10 Models as BSP trees. . . . . . . . . . . . . . . . . . . . . . . . . . 132

Example 4.11 Polygonal modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Example 4.12 Polygonal modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Example 5.1 3D translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Example 5.2 3D rotation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Example 5.3 3D general rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Example 5.4 3D general rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Example 5.5 3D general rotation—angles . . . . . . . . . . . . . . . . . . . . 156

Example 5.6 3D general rotation—rotating vectors . . . . . . . . . . . . . . 158

Example 5.7 3D general rotation—rotating vectors . . . . . . . . . . . . . . 161

Example 5.8 3D general rotation—rotating vectors . . . . . . . . . . . . . . 163

Example 5.9 3D general rotation—vector alignment . . . . . . . . . . . . . 164

Example 5.10 3D general rotation—vector alignment . . . . . . . . . . . . . 165

xviii Examples

Example 5.11 3D general rotation/transformation . . . . . . . . . . . . . . . . 166

Example 5.12 3D scaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Example 5.13 3D general reflection—plane parallel to zx-plane . . . . . . 175

Example 5.14 3D general reflection—plane passing through

the origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Example 5.15 3D general reflection—a plane and rotating vectors . . . . 179

Example 5.16 3D general shearing . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Example 5.17 3D composite transformations . . . . . . . . . . . . . . . . . . . 185

Example 5.18 Axes translation—point locations . . . . . . . . . . . . . . . . . 188

Example 5.19 Axes rotation—point locations. . . . . . . . . . . . . . . . . . . 189

Example 5.20 Axes scaling—point coordinates . . . . . . . . . . . . . . . . . 192

Example 6.1 Curve drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Example 6.2 Explicit representation—coefficients for quadratic

curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Example 6.3 Explicit representation—points on curves . . . . . . . . . . . 202

Example 6.4 Explicit representation—transformations

of quadratic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Example 6.5 Parametric representation—interpolation matrix

for quadratic curves . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Example 6.6 Parametric representation—general equation

for quadratic curves . . . . . . . . . . . . . . . . . . . . . . . . . . 210

Example 6.7 Parametric representation—coefficients

for quadratic curves . . . . . . . . . . . . . . . . . . . . . . . . . . 210

Example 6.8 Parametric representation—coefficients

for quadratic curves . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Example 6.9 Parametric representation—transformations

of quadratic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Example 6.10 Parametric representation—transformations

of quadratic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Example 6.11 Parametric representation—general equation

of cubic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Example 6.12 Parametric representation—interpolation matrix

for cubic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Example 6.13 Parametric representation—equation of cubic curves . . . 220

Example 6.14 Parametric representation—points on curves . . . . . . . . . 220

Example 6.15 Bézier curves—points on quadratic curves . . . . . . . . . . 227

Example 6.16 Bézier curves—transformations of cubic curves . . . . . . . 231

Example 6.17 Bézier curves—transformations of cubic curves . . . . . . . 233

Example 6.18 Bézier curves—cubic curves in 3D space . . . . . . . . . . . 234

Example 6.19 Bézier curves—points on cubic curves . . . . . . . . . . . . . 234

Example 6.20 Basis function—N

0;2

ðtÞ using uniform knot vector . . . . . 243

Example 6.21 Basis function—N

0;3

ðtÞ using uniform knot vector . . . . . 245

Example 6.22 Basis function—N

i;p

ðtÞ using uniform knot vector . . . . . 249

Examples xix

Example 6.23 Basis function—N

i;p

ðtÞ using open uniform

knot vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

Example 6.24 B-spline curves—points on quadratic curves using

uniform knot vector . . . . . . . . . . . . . . . . . . . . . . . . . . 258

Example 6.25 B-spline curves—points on quadratic curves using

open uniform knot vector . . . . . . . . . . . . . . . . . . . . . . 258

Example 6.26 B-spline curves—knots on quadratic curves using

uniform knot vector . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Example 6.27 B-spline curves—knots on quadratic curves using

open uniform knot vector . . . . . . . . . . . . . . . . . . . . . . 262

Example 6.28 B-spline curves—knots on quadratic curves using

uniform knot vector . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Example 6.29 B-spline curves—effect of moving control points

on quadratic curves . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Example 6.30 B-spline curves—curve continuity . . . . . . . . . . . . . . . . 265

Example 6.31 B-spline curves—points on cubic curves using

uniform knot vector . . . . . . . . . . . . . . . . . . . . . . . . . . 268

Example 6.32 B-spline curves—points on cubic curves using

open uniform knot vector . . . . . . . . . . . . . . . . . . . . . . 269

Example 6.33 B-spline curves—knots on cubic curves using

uniform knot vector . . . . . . . . . . . . . . . . . . . . . . . . . . 271

Example 6.34 B-spline curves—effect of moving control points

on cubic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

Example 6.35 NURBS curves—points on quadratic curves using

different knot vectors . . . . . . . . . . . . . . . . . . . . . . . . . 276

Example 6.36 NURBS curves—changing homogeneous weights

for quadratic curves using different knot vectors . . . . . . 278

Example 6.37 NURBS curves—a circular arc as a quadratic

curve segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

Example 6.38 NURBS curves—a full circle as quadratic

curve segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

Example 6.39 NURBS curves—changing homogeneous weights

for cubic curves using different knot vectors . . . . . . . . . 286

Example 6.40 NURBS curves—points on 3D curve . . . . . . . . . . . . . . 287

Example 7.1 Bézier surfaces—points on biquadratic surfaces . . . . . . . 299

Example 7.2 Bézier surfaces—points on isoparametric curve . . . . . . . 299

Example 7.3 Bézier surfaces—points on biquadratic surfaces . . . . . . . 301

Example 7.4 Bézier surfaces—points on bicubic surfaces . . . . . . . . . 303

Example 7.5 NURBS surfaces—points on biquadratic surfaces

using different knot vectors . . . . . . . . . . . . . . . . . . . . . 307

Example 7.6 NURBS surfaces—changing homogeneous weights

for biquadratic surfaces using open uniform

knot vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

xx Examples

Example 7.7 NURBS surfaces—points on bicubic surfaces

using different knot vectors . . . . . . . . . . . . . . . . . . . . . 310

Example 7.8 NURBS surfaces—changing homogeneous weights

for bicubic surfaces using open uniform knot vector. . . . 315

Example 8.1 Axonometric projection. . . . . . . . . . . . . . . . . . . . . . . . 329

Example 8.2 Isometric projection—points given a view plane

normal vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Example 8.3 Isometric projection—points given a view plane

normal vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

Example 8.4 Isometric projection—points given a view plane

normal vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

Example 8.5 Dimetric projection—points given a view plane

normal vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

Example 8.6 Dimetric projection—projection matrix given a view

plane normal vector . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Example 8.7 Oblique projection—length ratio given angle a ....... 346

Example 8.8 Oblique projection—angle a given length ratio . . . . . . . 346

Example 8.9 Oblique projection—orthographic projection . . . . . . . . . 346

Example 8.10 One-point perspective projection—deriving the

projection matrix when COP is at the origin and view

plane is at x ¼ d ............................ 350

Example 8.11 One-point perspective projection—deriving the

projection matrix when COP is at the origin and view

plane is at y ¼ d ............................ 351

Example 8.12 One-point perspective projection—projection matrix

when COP is at ½0; 0; d

and view plane

is at z ¼ 0................................. 353

Example 8.13 One-point perspective projection—points after

projection when COP is at ½0; 0; d

and view plane is at z ¼ 0...................... 354

Example 8.14 One-point perspective projection—vanishing point

when COP is at ½0; 0; d

and view plane

is at z ¼ 0................................. 356

Example 8.15 One-point perspective projection—points after

projection when COP is at

;

; d



and view plane is at z ¼ 0...................... 356

Example 8.16 One-point perspective projection—vanishing point

when COP is at

;

; d



and view

plane is at z ¼ 0............................. 358

Example 8.17 Two-point perspective projection—points after

projection when COP is at the origin and view plane

is parallel to the y-axis . . . . . . . . . . . . . . . . . . . . . . . . 361

Examples xxi

Example 8.18 Two-point perspective projection—deriving projection

matrix when COP is at an arbitrary point and view

plane is parallel to the y-axis . . . . . . . . . . . . . . . . . . . . 363

Example 8.19 Two-point perspective projection—deriving projection

matrix onto the xy-plane when COP is at an arbitrary

point and view plane is parallel to the y-axis. . . . . . . . . 364

Example 8.20 Two-point perspective projection—points after

projection onto the xy-plane when COP is at an

arbitrary point and view plane is parallel

to the y-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

Example 8.21 Three-point perspective projection—points after

projection given view plane orientation when

COP is at the origin . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Example 8.22 Three-point perspective projection—points after

projection given view plane orientation when COP

is at the origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Example 8.23 Three-point perspective projection—vanishing points

along the x-, y- and z-axes given a general view plane

when COP is at the origin. . . . . . . . . . . . . . . . . . . . . . 372

Example 8.24 Three-point perspective projection—deriving the

projection matrix when COP is at an arbitrary point

and the view plane has a general

equation n

x þ n

y þ n

z ¼ d .................... 375

Example 8.25 Three-point perspective projection—vanishing point

along an arbitrary direction when COP is at an

arbitrary point and the view plane has a general

equation n

x þ n

y þ n

z ¼ d .................... 377

Example 8.26 One-point perspective projection—vanishing point

when COP is at the origin and the view

plane is at z ¼ d............................. 377

Example 8.27 Three-point perspective projection—vanishing point

when COP is at the origin and the view plane

has a general equation n

x þ n

y þ n

z ¼ d .......... 378

Example 8.28 Projection conditions . . . . . . . . . . . . . . . . . . . . . . . . . 379

Example 8.29 Projection conditions . . . . . . . . . . . . . . . . . . . . . . . . . 382

Example 9.1 View reference coordinate system—expressing points

in view reference coordinate system given their

world coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

Example 9.2 Isometric projection—points given a view plane

normal vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

Example 9.3 View reference coordinate system—expressing points

in world coordinates given their view reference

coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

xxii Examples

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