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Applied Geometry for Computer Graphics and CAD : Springer Undergraduate Mathematics Series - Duncan Marsh

Applied Geometry for Computer Graphics and CAD

Springer Undergraduate Mathematics Series


Published: 3rd January 2005
Ships: 15 business days
15 business days

Focussing on the manipulation and representation of geometrical objects, this book explores the application of geometry to computer graphics and computer-aided design (CAD).

New features in this revised and updated edition include: the application of quaternions to computer graphics animation and orientation; discussions of the main geometric CAD surface operations and constructions: extruded, rotated and swept surfaces; offset surfaces; thickening and shelling; and skin and loft surfaces; an introduction to rendering methods in computer graphics and CAD: colour, illumination models, shading algorithms, silhouettes and shadows.

Over 300 exercises are included, many of which encourage the reader to implement the techniques and algorithms discussed through the use of a computer package with graphing and computer algebra capabilities. A dedicated website also offers further resources and links to other useful websites.

From the reviews:

a nice introduction to the foundations of applications of geometry to computer graphics and computer-aided design....A useful textbook.

Zentralblatt MATH

Mathematics students often ask where they can find a nice introduction to computer graphics and computer-aided design; professors have also been known to pose the same question. They will all find an answer here: Marsh's book should guide them effectively and painlessly towards the applications of mathematics and geometry in graphics and CAD.

Computer-Aided Design 32 (2000)

From the reviews of the second edition:

"This is a mathematics textbook on the basics of the geometry involved in computer graphics and computer-aided design, written at an undergraduate level suitable for students of mathematics, computer science, and engineering. ... All of the book's topics are presented in a clean and concise way, with nice illustrations and attention to both geometric ideas and practical issues of computing. ... This book would be useful for instructors who want a specific reference ... ." (Adam Coffman, Mathematical Reviews, 2005h)

"Images generated by a computer are ubiquitous, they are used in science, in engineering, and by the entertainment business. This textbook is an introduction to the mathematics behind these images. ... This exposition is intended for a broad audience with basic mathematical knowledge (vectors, matrices, calculus)." (P. Schmitt, Monatshefte fur Mathematik, Vol. 151 (4), 2007)

"The title says it all. ... covers exactly what you would expect: theory and background for applying geometric techniques for visualization on a computer. ... This book is part of the SUMS (Springer Undergraduate Mathematics Series). The theory and applications are explained well, and moreover the text contains numerous examples, problems, and fully worked solutions. The book was written with students in computer science, engineering as well as of mathematics in mind ... ." (Pieter Audenaert, Bulletin of the Belgian Mathematical Society, 2007)

Transformations of the Planep. 1
Introductionp. 1
Translationsp. 5
Scaling about the Originp. 6
Reflectionsp. 8
Rotation about the Originp. 9
Shearsp. 11
Concatenation of Transformationsp. 13
Applicationsp. 15
Instancingp. 15
Roboticsp. 17
Homogeneous Coordinates and Transformations of the Planep. 19
Introductionp. 19
Homogeneous Coordinatesp. 21
Points at Infinityp. 23
Visualization of the Projective Planep. 24
Line Model of the Projective Planep. 24
Spherical Model of the Projective Planep. 26
Transformations in Homogeneous Coordinatesp. 27
Translationsp. 27
Scaling about the Originp. 28
Rotation about the Originp. 29
Concatenation of Transformationsp. 30
Inverse Transformationsp. 31
Rotation about an Arbitrary Pointp. 33
Reflection in an Arbitrary Linep. 34
Applicationsp. 36
Instancingp. 36
Device Coordinate Transformationp. 37
Point and Line Geometry in Homogeneous Coordinatesp. 38
Homogeneous Coordinates and Transformations of Spacep. 41
Homogeneous Coordinatesp. 41
Transformations of Spacep. 42
Translationsp. 42
Scalings and Reflectionsp. 43
Rotations about the Coordinate Axesp. 43
Rotation about an Arbitrary Linep. 45
Reflection in an Arbitrary Planep. 47
Applicationsp. 49
Computer-aided Designp. 49
Orientation of a Rigid Bodyp. 50
Geometric Methods for Lines and Planes in Spacep. 52
Quaternionsp. 56
Projections and the Viewing Pipelinep. 67
Introductionp. 67
Projections of the Planep. 68
Projections of Three-dimensional Spacep. 72
The Viewplane Coordinate Mappingp. 76
The Viewing Pipelinep. 80
Classification of Projectionsp. 85
Classification of Parallel Projectionsp. 85
Classification of Perspective Projectionsp. 90
Curvesp. 95
Introductionp. 95
Curve Renderingp. 98
Parametric Curvesp. 99
Arclength and Reparametrizationp. 102
Application: Numerical Controlled Machining and Offsetsp. 107
Conicsp. 109
Classification of Conicsp. 112
Conics in Standard Formp. 116
Intersections of a Conic with a Linep. 121
Parametrization of an Irreducible Conicp. 124
Converting from Parametric Form to Implicit Formp. 127
Conics in Spacep. 130
Applications of Conicsp. 132
Bézier Curves Ip. 135
Introductionp. 135
Bézier Curves of Low Degreep. 136
Linear Bézier Curvesp. 136
Quadratic Bézier Curvesp. 136
Cubic Bézier Curvesp. 137
The Effect of Adjusting a Control Pointp. 140
The General Bézier Curvep. 141
Properties of the Bernstein Polynomialsp. 144
Convex Hullsp. 146
Properties of Bézier Curvesp. 147
The de Casteljau Algorithmp. 151
Subdivision of a Bézier Curvep. 154
Applicationsp. 157
Renderingp. 157
Intersection of a Planar Bézier Curve and a Linep. 158
Intersection of Two Bézier Curvesp. 159
Bézier Curves IIp. 161
Spatial Bézier Curvesp. 161
Derivatives of Bézier Curvesp. 162
Conversions Between Representationsp. 166
Piecewise Bézier Curvesp. 168
Rational Bézier Curvesp. 175
Properties of Rational Bézier Curvesp. 177
de Casteljau Algorithm for Rational Curvesp. 180
Projections of Rational Bézier Curvesp. 181
Derivatives of Rational Bézier Curvesp. 185
B-splinesp. 187
Integral B-spline Curvesp. 187
Properties of the B-spline Curvep. 194
B-spline Typesp. 196
Application: Font Designp. 203
Application: Morphing or Soft Object Animationp. 203
The de Boor Algorithmp. 205
Derivatives of a B-splinep. 207
Non-uniform Rational B-Splines (NURBS)p. 212
Projections of NURBS Curvesp. 214
Derivatives of NURBSp. 216
Rational de Boor Algorithmp. 218
Knot Insertionp. 221
Surfacesp. 225
Introductionp. 225
Quadric Surfacesp. 228
Offset Surfacesp. 232
Bézier and B-spline Surfacesp. 234
Properties of Bézier and B-spline Surfacesp. 235
Derivatives of Bézier and B-spline Surfacesp. 238
Surface Constructionsp. 241
Extruded Surfacesp. 241
Ruled Surfacesp. 242
Translationally Swept Surfacesp. 244
Surfaces of Revolutionp. 245
Surface Subdivisionp. 248
Skin and Loft Surfacesp. 251
Geometric Modelling and CADp. 260
Wireframe Modellerp. 261
Surface Modellerp. 261
Constructive Solid Geometry (CSG) Modellersp. 261
Boundary Representations (B-rep)p. 263
Curve and Surface Curvaturesp. 267
Curvature of a Plane Curvep. 267
Curvature and Torsion of a Space Curvep. 275
Curvature of Bézier Curvesp. 283
Surface Curvaturesp. 285
Renderingp. 297
Introductionp. 297
Colourp. 298
An Illumination Model for Reflected Lightp. 299
Diffuse Reflectionp. 300
Specular Reflectionp. 302
Ambient Reflectionp. 304
Attenuationp. 305
Total Intensityp. 305
Shading Algorithmsp. 306
Flat Shadingp. 307
Gouraud Shadingp. 307
Phong Shadingp. 309
Silhouettesp. 309
Shadowsp. 320
Solutionsp. 323
Referencesp. 345
Indexp. 347
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9781852338015
ISBN-10: 1852338016
Series: Springer Undergraduate Mathematics Series
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 350
Published: 3rd January 2005
Publisher: Springer London Ltd
Country of Publication: GB
Dimensions (cm): 23.5 x 17.8  x 2.08
Weight (kg): 0.52
Edition Number: 2
Edition Type: Revised