Chapter 4: Geometry

4.10 OTHER TRANSFORMATIONS OF SPACE
4.10.1 SIMILARITIES
A transformation of space that preserves shapes is called a similarity. Every similarity
of space is obtained by composing a proportional scaling transformation (also
known as a homothety) with an isometry. A proportional scaling transformation centered
at the origin has the form
´Ü Ý Þµ ´Ü Ý Þµ (4.10.1)
where ¼ is the scaling factor (a real number). The corresponding matrix in
homogeneous coordinates is
¾ ¿
¼ ¼ ¼
À ¼ ¼ ¼
¼ ¼ ¼ (4.10.2)
¼ ¼ ¼ ½
In cylindrical coordinates, the transformation is ´ÖÞµ ´ÖÞµ In spherical
coordinates,itis´Öµ ´Öµ
4.10.2 AFFINE TRANSFORMATIONS
A transformation that preserves lines and parallelism (maps parallel lines to parallel
lines) is an af ne transformation. There are two important particular cases of such
transformations:
1. A non-proportional scaling transformation centered at the origin has the form
´Ü Ý Þµ ´Ü Ý Þµ where ¼are the scaling factors (real numbers).
The corresponding matrix in homogeneous coordinates is
À
¾
¿
¼ ¼ ¼
¼ ¼ ¼
¼ ¼ ¼ (4.10.3)
¼ ¼ ¼ ½
2. A shear in the Ü-direction and preserving horizontal planes has the form ´Ü Ý Þµ
´Ü · ÖÞ Ý Þµ, where Ö is the shearing factor. The corresponding matrix in
homogeneous coordinates is
Ë Ö
¾
¿
½ ¼ Ö ¼
¼ ½ ¼ ¼
¼ ¼ ½ ¼ (4.10.4)
¼ ¼ ¼ ½
Every affine transformation is obtained by composing a scaling transformation with
an isometry, or one or two shears with a homothety and an isometry.
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