## Computer Graphics and Geometric ModelingJoseph-Louis Lagrange (1736-1813), one of the greatest mathematicians of the 18th century, made important contributions to the theory of numbers and to analytical and celestial mechanics. His most important work is Mecanique Analytique (1788), the textbook on which all subsequent work in this field is based. A contempo rary reader is surprised to find no diagrams or figures of any kind in this book on mechanics. This reflects one extreme approach to graphics, namely considering it unimportant or even detracting as a teaching tool and not using it. Today, of course, this approach is unthinkable. Graphics, especially computer graphics, is commonly used in texts, advertisements, and movies to illustrate concepts, to emphasize points being discussed, and to entertain. Our approach to graphics has been completely reversed since the days of La grange, and it seems that much of this change is due to the use of computers. Computer graphics today is a mature, successful, and growing field. It is used by many people for many purposes and it is enjoyed by even more people. One criterion for the maturity of a field of study is its size. When a certain discipline becomes so big that no one person can keep all of it in their head, we say that that discipline has matured (or has come of age). This is what happened to computer graphics in the last decade or so. |

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### Contents

1 | |

27 | |

Transformations and Projections | 58 |

Curves | 173 |

Points and Vectors | 174 |

Parametric Blending | 180 |

Curve Representations | 181 |

The Lagrange Polynomial | 198 |

Texturing | 552 |

Bump Mapping | 554 |

Color 1 Color and the Eye | 557 |

The HLS Color Model | 559 |

The RGB Color Model | 561 |

Additive and Subtractive Colors | 563 |

Complementary Colors | 567 |

The CIE Standard | 571 |

The Newton Polynomial | 205 |

Spline Methods for Curves | 206 |

Hermite Interpolation | 207 |

The Cubic Spline Curve | 225 |

The Quadratic Spline | 247 |

Cardinal Splines | 248 |

CatmullRom Curves | 251 |

KochanekBartels Splines | 258 |

Fitting a PC to Experimental Points | 262 |

The Bézier Curve | 266 |

Subdivision Curves | 321 |

The BSpline | 328 |

The Beta Spline | 389 |

Barycentric Sums Revisited | 393 |

Symmetry in Curves | 394 |

Conic Sections | 397 |

Parametric Space of a Curve | 401 |

Curvature and Torsion | 402 |

The Hough Transform | 410 |

Surfaces 415 | 414 |

Input ThreeDimensional Points | 416 |

Basic Concepts | 417 |

Polygonal Surfaces | 419 |

Delaunay Triangulation | 427 |

Bilinear Surfaces | 434 |

Lofted Surfaces | 439 |

Coons Surfaces | 443 |

The Cartesian Product | 456 |

The Biquadratic Surface Patch | 457 |

The Bicubic Surface Patch | 459 |

CatmullRom Surfaces | 468 |

Rectangular Bézier Surfaces | 471 |

Triangular Bézier Surfaces | 483 |

Converting Bézier Patches | 488 |

The Gregory Patch | 495 |

Gordon Surfaces | 498 |

Uniform BSpline Surfaces | 499 |

Surfaces of Revolution | 516 |

Sweep Surfaces | 526 |

Polygonal Surfaces by Subdivision | 530 |

Curves on Surfaces | 533 |

Surface Normals | 535 |

Rendering 1 Introduction | 537 |

A Simple Shading Model | 538 |

Gouraud and Phong Shading | 548 |

Palette Optimization | 549 |

Ray Tracing | 551 |

Computer Animation 1 Background | 575 |

Interpolating Positions | 578 |

I | 583 |

II | 593 |

Nonuniform Interpolation | 600 |

Morphing | 606 |

FreeForm Deformations | 607 |

Image Compression | 609 |

Introduction | 610 |

VariableSize Codes | 611 |

RunLength Encoding | 612 |

Fax Compression | 615 |

Cell Encoding | 622 |

Quadtrees | 624 |

Progressive Image Compression | 630 |

FELICS | 634 |

The Golomb Code | 642 |

Progressive FELICS | 643 |

MLP | 646 |

Differential Lossless Image Compression | 654 |

Wavelets | 656 |

Graphics Standards | 661 |

Boundary Fill | 668 |

Halftoning | 669 |

Dithering | 671 |

Fractals | 680 |

A Fractal Line | 681 |

Branching Rules | 684 |

Iterated Function Systems IFS | 685 |

Image Processing | 688 |

Mathematical Topics 693 | 692 |

Forward Differences | 698 |

Coordinate Systems | 700 |

Vector Algebra | 702 |

Matrices | 709 |

Trigonometric Identities | 711 |

The Greek Alphabet | 715 |

Complex Numbers | 716 |

Quaternions | 717 |

Groups | 719 |

Fields | 720 |

723 | |

Answers to Exercises 733 | 732 |

833 | |

842 | |

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### Common terms and phrases

algorithm angle assume axes axis B-spline barycentric Bernstein polynomials Bézier curve Bézier surface bicubic bitmap blending functions boundary curves calculate camera circle coefficients color compression computer graphics control points coordinate system corner points cube cubic spline decoder defined denote derivatives dimensions direction display distance dither easy encoded endpoints equals Equation example Exercise expression Figure four points given identical integer interpolation iteration key frames Lagrange polynomial length located matrix method midpoint move multiplication nonuniform normal object obtained OOOOOOOOOOO OOOOOOOOOOO OOOOOOOOOOO original parameter parametric curve perpendicular perspective projection pixels points P1 polygon polyline polynomial positive produces projection plane quadratic quaternion reflection result rotation scan screen Section shape shows simple spline segments step straight line straight segment surface patch Table tangent vectors three points three-dimensional transformation triangle two-dimensional unary code unit vector values weights yields zero