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

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.

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main geometric queries are distance measurements, discussed in Chapter 6, and intersection queries, discussed in Chapter 7. Miscellaneous queries of interest are provided in Chapter 8. Chapter 9 provides the definitions for the various three-dimensional objects to which the geometric queries apply. These include lines, rays, line segments, planes and planar objects (two-dimensional objects embedded in a plane in three dimensions), polyhedra and polygon meshes, quadric surfaces (surfaces defined

the previous section that a mapping may be one-to-one or onto. A linear function T : A −→ B is said to be an isomorphism if it is one-to-one and maps A onto B. 2.7 Linear Mappings a b c d e f T a b c d e f g h i j k l T –1 49 g h i j k l Figure 2.8 An invertible mapping. An important aspect of linear mappings is that they are completely determined by how they transform the basis vectors; this can be understood by recalling that any vector v ∈ V can be represented as a linear

just showed that every vector in a particular space is a unique linear combination of a particular set of basis vectors. However, this doesn’t mean that every vector has a unique linear combination representation. For any given vector space V of dimension n, there are an infinite number of linearly independent n-ary subsets of V . That is, any vector w ∈ V can be represented as a linear combination of any arbitrarily chosen set of basis vectors. The vector w = 3u + 2v in Figure 3.14 can also be

algorithms, and so it makes sense to adopt this convenient notation. In terms of vectors, the operation is intuitive and rather obvious. But what about the matrix representation? The vector 122 Chapter 4 Matrices, Vector Algebra, and Transformations v v Figure 4.3 The “perp” operator. v in Figure 4.3 is approximately [ 1 0.2 ]. Intuitively, the “trick” is to exchange the vector’s two components and then negate the first. In matrix terms, we have v ⊥ = [ 1 0.2 ] 0 1 −1 0 = [ −0.2 1 ] In 3D,

....................................................................................... Polynomial Curves .................................................................. 355 356 Bezier Curves ...................................................................................... B-Spline Curves ................................................................................... NURBS Curves .................................................................................... 357 357 358

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