3D Computer Graphics. Mathem. Intro with OpenGL by Buss

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T. β = 0 or γ = 0} . This set of triples is, of course, a plane in R3 with a hole at the origin. Thus, we can identify this two-dimensional subspace of R3 (that is, the plane) with the line in the projective plane. If, on the other hand, L is the line at infinity, then it corresponds in the same way to the two-dimensional subspace { x1 , x2 , 0 : x1 , x2 ∈ R}. These considerations give rise to another way of understanding the two-dimensional projective plane. The “points” of the projective plane are one-dimensional subspaces of R3 .

The code fragment draws wireframe spheres with commands glutWireSphere( radius, slices, stacks ); The value of radius is the radius of the sphere. The integer values slices and stacks control the number of “wedges” and horizontal “stacks” used for the polygonal model of the sphere. The sphere is modeled with the “up” direction along the z-axis, and thus “horizontal” means parallel to the x y-plane. Team LRN 40 Transformations and Viewing The glColor3f(red, green, blue) commands are used to set the current drawing color.

Every linear transformation A in R3 can be represented by a 3 × 3 matrix M as follows. 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.

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