Page 38 Table of Contents Index Page 40
Chapters
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30
A, B, C, D, E


CHAPTER 5. AFFINE TRANSFORMATIONS


Figure 5.1: The predicates for analyzing the mathematical properties of a transformation.


system, otherwise returns false. Note that this is a very inclusive category|transformations are
considered reflections even if they distort, scale, or skew the coordinate system, as long as they
invert the handedness.

=> rigid-transformation-p transformation [Generic Function]
Returns true if the transformation transformation transforms the coordinate system as a rigid
object, that is, as a combination of translations, rotations, and pure reflections. Otherwise, it
returns false.

Rigid transformations are the most general category of transformations that preserve magnitudes
of all lengths and angles.

=> even-scaling-transformation-p transformation [Generic Function]
Returns true if the transformation transformation multiplies all x lengths and y lengths by the
same magnitude, otherwise returns false. It does include pure reflections through vertical and
horizontal lines.

=> scaling-transformation-p transformation [Generic Function]
Returns true if the transformation transformation multiplies all x lengths by one magnitude and
all y lengths by another magnitude, otherwise returns false. This category includes even scalings
as a subset.

=> rectilinear-transformation-p transformation [Generic Function]
Returns true if the transformation transformation will always transform any axis-aligned rectan-
gle into another axis-aligned rectangle, otherwise returns false. This category includes scalings
as a subset, and also includes 90 degree rotations.

Rectilinear transformations are the most general category of transformations for which the
bounding rectangle of a transformed object can be found by transforming the bounding rectangle
of the original object.

Minor issue: Supply this figure. | SWM

5.3.2 Composition of Transformations

If we transform from one coordinate system to another, then from the second to a third coordi-
nate system, we can regard the resulting transformation as a single transformation resulting from
composing the two component transformations. It is an important and useful property of affine
transformations that they are closed under composition. Note that composition is not commu-
tative; in general, the result of applying transformation A and then applying transformation B

Page 38 Table of Contents Index Page 40
Chapters
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30
A, B, C, D, E