Page 289 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

For completeness, we note that the standard matrix for a counterclockwise rotation through an angle about an axis in ,

which is determined by an arbitrary unit vector that has its initial point at the origin, is

(17)

The derivation can be found in the book Principles of Interactive Computer Graphics, by W. M. Newman and R. F. Sproull
(New York: McGraw-Hill, 1979). The reader may find it instructive to derive the results in Table 7 as special cases of this more
general result.

Dilation and Contraction Operators

If k is a nonnegative scalar, then the operator on or is called a contraction with factor k if and a

dilation with factor k if . The geometric effect of a contraction is to compress each vector by a factor of k (Figure 4.2.6a),

and the effect of a dilation is to stretch each vector by a factor of k (Figure 4.2.6b). A contraction compresses or uniformly

toward the origin from all directions, and a dilation stretches or uniformly away from the origin in all directions.

Figure 4.2.6

The most extreme contraction occurs when , in which case reduces to the zero operator , which

compresses every vector into a single point (the origin). If , then reduces to the identity operator ,

which leaves each vector unchanged; this can be regarded as either a contraction or a dilation. Tables 8 and 9 list the dilation

and contraction operators on and .

Table 8

Operator Illustration Equations Standard Matrix