By Medhat H. Rahim
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Title: 3D special effects: A Mathematical creation with OpenGL.(Book review)
Author: Medhat H. Rahim
Publication: university technology and arithmetic (Magazine/Journal)
Date: March 1, 2009
Publisher: college technology and arithmetic organization, Inc.
Volume: 109 factor: three web page: 183(2)
Article kind: ebook review
Distributed via Gale, part of Cengage studying
Read Online or Download 3-D Computer graphics. Mathematical introduction with OpenGL PDF
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Extra info for 3-D Computer graphics. Mathematical introduction with OpenGL
The geometric tools used in computer graphics are mathematically very elegant. Even more important, the techniques discussed in this chapter have the advantage of being fairly easy for an artist or programmer to use and lend themselves to efﬁcient software and hardware implementation. In fact, modernday PCs typically include specialized graphics chips that carry out many of the transformations and interpolations discussed in this chapter. 1 Transformations in 2-Space We start by discussing linear and afﬁne transformations on a fairly abstract level and then see examples of how to use transformations in OpenGL.
During this fourth stage, pixels are given color and depth information, and interpolation methods are used to ﬁll in the interior of polygons. This fourth stage is the only stage dependent on the physical characteristics of the output device. The ﬁrst three stages usually work in a device-independent fashion. The discussion in this chapter emphasizes the mathematical aspects of the transformations used by computer graphics but also sketches their use in OpenGL. The geometric tools used in computer graphics are mathematically very elegant.
Let u be u rotated counterclockwise 90◦ . 2 Every rigid, orientation-preserving, linear transformation is a rotation. 2 holds too: every rotation is obviously a rigid, orientationpreserving, linear transformation. Proof Let A be a rigid, orientation-preserving, linear transformation. Let a, b = A(i). By rigidity, A(i) · A(i) = a 2 + b2 = 1. 6. Therefore, the matrix M representing A is equal to ab −b . Because a 2 + b2 = 1, there must a be an angle θ such that cos θ = a and sin θ = b, namely, either θ = cos−1 a or θ = − cos−1 a.