3D Computer Graphics : A Mathematical Introduction with by Samuel R. Buss

By Samuel R. Buss

This creation to 3D special effects emphasizes basics and the math underlying special effects, whereas additionally overlaying programming concepts utilizing OpenGL, a platform-independent portraits programming atmosphere. The minimum necessities make it appropriate for self-study or to be used as a complicated undergraduate or introductory graduate textual content because the writer leads step by step from the fundamentals of adjustments to complex themes comparable to animations and kinematics. Accompanying software program, together with resource code for a ray tracing software program package deal, is offered freely from the book's site.

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Extra resources for 3D Computer Graphics : A Mathematical Introduction with OpenGL

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When the next frame is ready to be displayed, the new frame replaces the old frame on the screen instantaneously (or rather, the next time the screen is redrawn, the new image is used). A region of memory where an image is being created or stored is called a buffer. The image being displayed is stored in the front buffer, and the back buffer holds the next frame as it is being created. When the buffers are swapped, the new image replaces the old one on the screen. Note that swapping buffers does not generally require copying from one buffer to the other; instead, one can just update pointers to switch the identities of the front and back buffers.

More generally, for any other homogeneous representation of the same point, αx1 , αx2 , α with α = 0, the effect of multiplying by N is     αx1 α(ax1 + bx2 + e) N αx2  = α(cx1 + d x2 + f ) , α α which is another representation of the point A(x) in homogeneous coordinates. Thus, the 3 × 3 matrix N provides a representation of the affine map A because, when one works with homogeneous coordinates, multiplying by the matrix N provides exactly the same results as applying the transformation A.

Note that any given point in R2 has many representations in homogeneous coordinates. For example, the point 2, 1 can be represented by any of the following sets of homogeneous coordinates: 2, 1, 1 , 4, 2, 2 , 6, 3, 3 , −2, −1, −1 , and so on. More generally, the triples x, y, w and x , y , w represent the same point in homogeneous coordinates if and only if there is a nonzero scalar α such that x = αx, y = αy, and w = αw. So far, we have only specified the meaning of the homogeneous coordinates x, y, w when w = 0 because the definition of the meaning of x, y, w required dividing by w.

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