By Medhat H. Rahim
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Title: 3-D special effects: A Mathematical advent with OpenGL.(Book review)
Author: Medhat H. Rahim
Publication: university technology and arithmetic (Magazine/Journal)
Date: March 1, 2009
Publisher: tuition technological know-how and arithmetic organization, Inc.
Volume: 109 factor: three web page: 183(2)
Article variety: publication review
Distributed through Gale, part of Cengage studying
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Additional info for 3-D Computer graphics. Mathematical introduction with OpenGL
After a 3-D geometric model has been created, it is necessary to render it on a two-dimensional surface called the viewport. Some common examples of viewports are a window on a video screen, a frame of a movie, and a hard-copy image. There are special transformations, called perspective transformations, that are used to map points from a 3-D model to points on a 2-D viewport. To properly appreciate the uses of transformations, it is important to understand the rendering pipeline, that is, the steps by which a 3-D scene is modeled and rendered.
Example: Let M = 11 02 . 3. To ﬁnd the matrix representation of its inverse M −1 , it is enough to determine M −1 i and M −1 j. It is not hard to see that M −1 1 1 = 0 −1/2 Hint: Both facts follow from M Therefore, M −1 is equal to M −1 and 0 = 1/2 1 0 −1/2 1/2 0 1 0 0 = . 1 1/2 and M . Team LRN 1 0 = 1 1 . 3. An F shape transformed by a linear transformation. The example shows a rather intuitive way to ﬁnd the inverse of a matrix, but it depends on being able to ﬁnd preimages of i and j. One can also compute the inverse of a 2 × 2 matrix by the well-known formula −1 a b c d 1 d −b , det(M) −c a = where det(M ) = ad − bc is the determinant of M.
F u = Then the mapping A can be deﬁned by x1 x2 →M x1 + x2 e f = a b c d x1 + x2 e f = ax 1 + bx2 + e . cx1 + d x2 + f Now deﬁne N to be the 3 × 3 matrix a b e N = c d f . 0 0 1 Using the homogeneous representation x1 , x2 , 1 of x1 , x2 , we see that x1 a b e x1 ax1 + bx2 + e N x2 = c d f x2 = cx1 + d x2 + f . 1 0 0 1 1 1 The effect of N ’s acting on x, y, 1 is identical to the effect of the afﬁne transformation A acting on x, y . The only difference is that the third coordinate of “1” is being carried around.
3-D Computer graphics. Mathematical introduction with OpenGL by Medhat H. Rahim
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