Chapter 4: Geometry

(also known as a homothety) with an isometry. A proportional scaling transformation
centered at the origin has the form
´Ü ݵ ´Ü ݵ (4.3.1)
where ¼ is the scaling factor (a real number). The corresponding matrix in
homogeneous coordinates is
¾
¿
À ¼ ¼
¼ ¼ (4.3.2)
¼ ¼ ½
In polar coordinates, the transformation is ´Öµ ´Öµ
4.3.2 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:
A non-proportional scaling transformation centered at the origin has the form
´Ü ݵ ´Ü ݵ, where ¼ are the scaling factors (real numbers). The
corresponding matrix in homogeneous coordinates is
¾
¿
À ¼ ¼
¼ ¼ (4.3.3)
¼ ¼ ½
A shear preserving horizontal lines has the form ´Ü ݵ ´Ü · ÖÝ Ýµ, where Ö
is the shearing factor (see Figure 4.6). The corresponding matrix in homogeneous
coordinates is
¾ ¿
Ë Ö ½ Ö ¼
¼ ½ ¼ (4.3.4)
¼ ¼ ½
Every af ne transformation is obtained by composing a scaling transformation with
an isometry, or a shear with a homothety and an isometry.
FIGURE 4.6
A shear with factor Ö ½ ¾ .
Ý
ÖÝ
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