Mathematical elements for Computer Graphics - David F. Rogers

MATHEMATICAL ELEMENTS FOR COMPUTER GRAPHICS Second Edition

David F. Rogers Professor of Aerospace Engineering United States Naval Academy, Annapolis, Md.

J. Alan Adams Professor of Mechanical Engineering United States Naval Academy, Annapolis, Md.

McGraw-Hill, Inc. New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico Milan Montreal New Delhi Paris San Juan Singapore Sydney Tokyo Toronto

CONTENTS

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Preface Preface to the First Edition ;r 1 l-l 1-2 1-3 1-4 1-5 1-6 1-7 1-8 1-9

1-10 1-11 1-12 1-13 1-14 1-15 1-16 1-17 1-18 1-19 1-20 1-21 1-22 1-23

Introduction To Computer Graphics Overview of Computer Graphics Representing Pictures Preparing Pictures For Presentation Presenting Previously Prepared Pictures Interacting with the Picture Description of Some Graphics Devices Storage Tube Graphics Displays Calligraphic Refresh Graphics Displays Raster Refresh Graphics Displays Cathode Ray Tube Basics Color CRT Raster Scan Basics Video Basics Flat Panel Displays Electrostatic Plotters Laser Printers Dot Matrix Plotters Ink Jet Plotters Thermal Plotters Pen and Ink Plotters Color Film Cameras Active and Passive Graphics Devices Computer Graphics Software References

XI

xiii xvii 1 3 3 5 6 9 18 18 19 24 30 31 32 35 42 44 47 49 50 52 56 57 58 59 Vll

Vlll

MATHEMATICAL ELEMENTS FOR COMPUTER GRAPHICS

Chapter 2 2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10 2-11 2-12 2-13 2-14 2-15 2-16 2-17 2-18 2-19 2-20 2-21 2-22

Chapter 3 3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10 3-11 3-12 3-13 3-14 3-15 3-16 3-17 3-18

Two-Dimensional Transformations

61

Introduction Representation of Points Transformations and Matrices Transformation of Points Transformation of Straight Lines Midpoint Transformation Transformation of Parallel Lines Transformation of Intersecting Lines Rotation Reflection Scaling Combined Transformations Transformation of The Unit Square Solid Body Transformations Translations and Homogeneous Coordinates Rotation About an Arbitrary Point , Reflection Through an Arbitrary Line Projection - A Geometric Interpretation of Homogeneous Coordinates Overall Scaling Points At Infinity Transformation Conventions References

61 61 62 62 65 66 68 69 72 76 78 80 83 86 87 88 89 90 94 95 98 100

Three-Dimensional Transformations

101

Introduction Three-Dimensional Scaling Three-Dimensional Shearing Three-Dimensional Rotation Three-Dimensional Reflection Three-Dimensional Translation Multiple Transformations Rotations About an Axis Parallel to a Coordinate Axis Rotation About an Arbitrary Axis in Space Reflection Through an Arbitrary Plane Affine and Perspective Geometry Orthographic Projections Axonometric Projections Oblique Projections Perspective Transformations Techniques For Generating Perspective Views Vanishing Points Photography and The Perspective Transformation

101 102 106 107 113 115 115 117 121 128 132 135 141 151 157 171 179 185

CONTENTS

3-19 3-20 3-21 3-22

Chapter 4 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 4-9

4-10 4-11

Chapter 5 5-1 5-2 5-3 5-4 5-5 5-6 5-7

„

5-8 5-9

5-10 5-11 5-12 5-13 5-14

Chapter 6 6-1 6-2 6-3 6-4 6-5 6-6 6-7 6-8 6-9

IX

187

Stereographic Projection Comparison of Object Fixed and Center of Projection Fixed Projections Reconstruction of Three-Dimensional Images References

195 200 206

Plane Curves

207

Introduction Curve Representation Nonparametric Curves Parametric Curves Parametric Representation of a Circle Parametric Representation of an Ellipse Parametric Representation of a Parabola Parametric Representation of a Hyperbola A Procedure For Using Conic Sections The General Conic Equations References

207 207 209 211 215 218 223 227 231 231 246

Space Curves

247

Introduction Representation of Space Curves Cubic Splines Normalized Cubic Splines / Alternate Cubic Spline End Conditions ParabolicNBlending Generalized Parabolic Blending Bezier Curves B-spline Curves End Conditions For Periodic B-spline Curves B-spline Curve Fit B-spline Curve Subdivision Rational B-spline Curves References

247 248 250 267 271 278 284 289 305 339 346 351 356 375

Surface Description and Generation

379

Introduction Surfaces of Revolution Sweep Surfaces Quadric Surfaces Piecewise Surface Representation Mapping Parametric Surfaces Bilinear Surface Ruled and Developable Surfaces Linear Coons Surface

379 380 394 400 408 411 414 417 422

X

MATHEMATICAL ELEMENTS FOR COMPUTER GRAPHICS

6-10 6-11 6-12 6-13 6-14 6-15 6-16 6-17

Coons Bicubic Surface Bezier Surfaces B-spline Surfaces B-spline Surface Fitting B-spline Surface Subdivision Gaussian Curvature and Surface Fairness Rational B-spline Surfaces References

426 435 445 456 458 461 465 477

Appendices

481

Appendix Appendix Appendix Appendix Appendix Appendix Appendix

Index

A B C D E F G

Computer Graphics Software Matrix Methods Pseudocode B-spline Surface File Format Problems Programming Projects Algorithms

481 503 507 513 517 527 541 599