Page 336 - Matrix Analysis & Applied Linear Algebra

332 Chapter 5 Norms, Inner Products, and Orthogonality

Problem: In addition to rotation, how can a polytope (or its image on a com-
puter monitor) be translated?
Solution: Translation of a polytope to a different point in space is accom-
plished by adding a constant to each of its coordinates. For example, to trans-
late the polytope shown in Figure 5.6.7 to the location where vertex 1 is at
T
p =(x 0 ,y 0 ,z 0 ) instead of at the origin, just add p to every point. In partic-
ular, if e is the column of 1’s, the translated vertex matrix is
 
x 0 x 0 ··· x 0
V trans = V orig +  y 0 y 0 ··· y 0  = V orig + pe T (a rank-1 update).
z 0 z 0 ··· z 0
Of course, the edge matrix is not affected by translation.
Problem: How can a polytope (or its image on a computer monitor) be scaled?

Solution: Simply multiply every coordinate by the desired scaling factor. For
example, to scale an image by a factor α, form the scaled vertex matrix
V scaled = αV orig ,

and then connect the scaled vertices with appropriate edges as dictated by the
edge matrix E.
Problem: How can the faces of a polytope that are hidden from the viewer’s
perspective be detected so that they can be omitted from the drawing on the
screen?
Solution: A complete discussion of this tricky problem would carry us too far
3
astray, but one clever solution relying on the cross product of vectors in is
presented in Exercise 5.6.21 for the case of convex polytopes.

Rotations in higher dimensions are straightforward generalizations of rota-
3 3
tions in . Recall from p. 328 that rotation around any particular axis in
amounts to rotation in the complementary plane, and the associated 3 × 3 ro-
tator is constructed by embedding a 2 × 2 rotator in the appropriate position
in a 3 × 3 identity matrix. For example, rotation around the y-axis is rotation
in the xz-plane, and the corresponding rotator is produced by embedding

cos θ sin θ
− sin θ cos θ
in the “ xz-position” of I 3×3 to form
 
cos θ 0 sin θ
P y =  0 1 0  .
− sin θ 0 cos θ
These observations directly extend to higher dimensions.