By Medhat H. Rahim
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Title: 3-D special effects: A Mathematical creation with OpenGL.(Book review)
Author: Medhat H. Rahim
Publication: university technological know-how and arithmetic (Magazine/Journal)
Date: March 1, 2009
Publisher: university technological know-how and arithmetic organization, Inc.
Volume: 109 factor: three web page: 183(2)
Article sort: booklet review
Distributed by way of Gale, part of Cengage studying
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Extra info for 3-D Computer graphics. Mathematical introduction with OpenGL
OpenGL provides a ﬂexible set of tools for positioning vertices, including methods for rotating, scaling, and reshaping objects. 1. The four stages of the rendering pipeline in OpenGL. These tools are called “afﬁne transformations” and are discussed in detail in the next sections. OpenGL uses a 4 × 4 matrix called the “model view matrix” to describe afﬁne transformations. View Selection. This stage is typically used to control the view of the 3-D model. In this stage, a camera or viewpoint position and direction are set.
OpenGL really uses 3-space, however, and so not everything we discuss is exactly correct for OpenGL. We denote the ModelView matrix by M for the rest of this subsection. The purpose of M is to hold a homogeneous matrix representing an afﬁne transformation. We therefore think of M as being a 3 × 3 matrix acting on homogeneous representations of points in 2-space. ) The OpenGL programmer speciﬁes points in 2-space by calling a routine glVertex2f(x,y). As described in Chapter I, this point, or “vertex,” may be drawn as an isolated point or may be the endpoint of a line or a vertex of a polygon.
One notational complication is that a linear transformation A operates on points x = x1 , x2 , whereas a matrix M acts on column vectors. It would be convenient, however, to use both of the notations A(x) and Mx. To make both notations be correct, we adopt the following rather special conventions about the meaning of angle brackets and the representation of points as column vectors: Notation The point or vector x1 , x2 is identical to the column vector xx12 . So “point,” “vector,” and “column vector” all mean the same thing.
3-D Computer graphics. Mathematical introduction with OpenGL by Medhat H. Rahim