D Scaling

The diagonal elements of the generalized transform matrix (4×4) define local and general scaling. To illustrate this, consider the transformation for local scaling:

(2.4)

General scaling can be achieved by using the fourth diagonal element, i.e.:

(2.5)

Ordinary or physical coordinates have the form:

(2.6)

Note that (as in the case of 2D) homogeneous coordinate scaling factor is not equal to 1. As the previous discussion, this means the conversion from volume h = 1 in another volume in the 4D space. Transformed physical coordinates obtained by projecting through the center of the 4D coordinate system go back to the physical volume of h = 1. As before if s <1 there is a uniform expansion. If s> 1 there is a homogeneous coordinate vector compression.

The same result can be obtained using the same coefficients of the local scale. In this case the transformation matrix has the form:

(2.7)

Note that there is a homogeneous coordinate h = 1. Thus, all the query translation takes place in physical volume h = 1.

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