# Geometric Tools for Computer Graphics (The Morgan Kaufmann Series in Computer Graphics)

## David H. Eberly

Language: English

Pages: 1056

ISBN: 1558605940

Format: PDF / Kindle (mobi) / ePub

Do you spend too much time creating the building blocks of your graphics applications or finding and correcting errors? *Geometric Tools for Computer Graphics* is an extensive, conveniently organized collection of proven solutions to fundamental problems that you'd rather not solve over and over again, including building primitives, distance calculation, approximation, containment, decomposition, intersection determination, separation, and more.

If you have a mathematics degree, this book will save you time and trouble. If you don't, it will help you achieve things you may feel are out of your reach. Inside, each problem is clearly stated and diagrammed, and the fully detailed solutions are presented in easy-to-understand pseudocode. You also get the mathematics and geometry background needed to make optimal use of the solutions, as well as an abundance of reference material contained in a series of appendices.

**Features**

- Filled with robust, thoroughly tested solutions that will save you time and help you avoid costly errors.
- Covers problems relevant for both 2D and 3D graphics programming.
- Presents each problem and solution in stand-alone form allowing you the option of reading only those entries that matter to you.
- Provides the math and geometry background you need to understand the solutions and put them to work.
- Clearly diagrams each problem and presents solutions in easy-to-understand pseudocode.
- Resources associated with the book are available at the companion Web site www.mkp.com/gtcg.

* Filled with robust, thoroughly tested solutions that will save you time and help you avoid costly errors.

* Covers problems relevant for both 2D and 3D graphics programming.

* Presents each problem and solution in stand-alone form allowing you the option of reading only those entries that matter to you.

* Provides the math and geometry background you need to understand the solutions and put them to work.

* Clearly diagrams each problem and presents solutions in easy-to-understand pseudocode.

* Resources associated with the book are available at the companion Web site www.mkp.com/gtcg.

Genetic Programming Theory and Practice XI (Genetic and Evolutionary Computation)

Genetic Programming Theory and Practice XI (Genetic and Evolutionary Computation)

The Little Schemer (4th Edition)

3x1 + 2x2 = 6. 3(6) + 2x2 = 6 18 + 2x2 = 6 2x2 = −12 x2 = −6 yielding another solution u = (6, −6). Indeed, there are an infinite number of solutions. At this point, we can introduce some geometric intuition: if we consider the variables x1 and x2 to correspond to the x-axis and y-axis of a 2D Cartesian coordinate system, the individual solutions consist of points in the plane. Let’s list a few solutions and plot them (Figure 2.2). The set of all solutions to a linear equation of two unknowns

........................................................................................ Point to Ray ......................................................................................... Point to Segment ................................................................................. 190 191 192 Point to Polyline ...................................................................... Point to Polygon ...................................................................... 194 196 Point to

Monterey, CA, 1978. Howard Anton, Elementary Linear Algebra, John Wiley and Sons, New York, 2000. Also quite useful and accessible is Seymour Lipschutz, Schaum’s Outline of Theory and Problems of Linear Algebra, McGraw-Hill, New York, 1968. In the area of computer graphics, the following contain much of interest related to linear algebra: M. E. Mortenson, Mathematics for Computer Graphics Applications, Industrial Press, New York, 1999 (Chapters 1–3). Recommended Reading 61 James D. Foley,

nonparallel directed line segments in the plane constitute a basis for that planar subset of three-dimensional space. Suppose we have a set of vectors V = v1, v2, · · · , vn ∈ V , which are linearly independent as described earlier. Any other vector w that is in the space spanned by 72 Chapter 3 Vector Algebra 2v v 3u u w = 3u + 2v Figure 3.14 A vector as the linear combination of basis vectors. V can be described as a linear combination of V: w = x1v1 + x2v2 + · · · + xnvn, xi ∈ R

rewrite the Head-to-Tail Axiom, by invoking the definition of addition, as P = R + (P − Q) + (Q − R). If we then substitute Q = R, we get P = Q + (P − Q) + (Q − Q). Since (Q − Q) = 0, we have the desired result. vi. (Q + v) − (R + w) = (Q − R) + (v − w). Proof (Q + v) − (R + w) = (Q + v) − R + R − (R + w) = (Q + v) − R + (R − R) − w = (Q + v) − R − w = (Q + v) − Q + [Q − R] − w = v + (Q − Q) + [Q − R] − w = (Q − R) + (v − w) by Head-to-Tail Axiom by part (iv) by part (i) by Head-to-Tail Axiom