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

By Samuel R. Buss

This advent to 3D special effects emphasizes basics and the maths underlying special effects, whereas additionally masking programming options utilizing OpenGL, a platform-independent images programming atmosphere. The minimum necessities make it appropriate for self-study or to be used as a sophisticated undergraduate or introductory graduate textual content because the writer leads step by step from the fundamentals of modifications to complex themes reminiscent of animations and kinematics. Accompanying software program, together with resource code for a ray tracing software program package deal, is out there freely from the book's website.

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Additional info for 3D Computer Graphics : A Mathematical Introduction with OpenGL

Example text

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. In addition, the range and the ﬁeld of view are determined. ” OpenGL uses another 4 × 4 matrix called the “projection matrix” to specify these transformations. Perspective Division. The previous two stages use a method of representing points in 3space by means of homogeneous coordinates. Homogeneous coordinates use vectors with four components to represent points in 3-space.

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. 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.

Let i = 1, 0, 0 , j = 0, 1, 0 , and k = 0, 0, 1 , and let u = A(i), v = A(j), and w = A(k). Set M equal to the matrix (u, v, w), that is, the matrix whose columns are u, v, and w, and thus   u 1 v1 w1 M = u 2 v2 w2  . 6 u 3 v3 w3 Then Mx = A(x) for all x ∈ R3 , that is to say, M represents A. In this way, any linear transformation of R3 can be viewed as being a 3 × 3 matrix. ) A rigid transformation is one that preserves the size and shape of an object and changes only its position and orientation.