Chapter 4: Geometry

This formula also covers passing from a Cartesian system to an oblique system and
vice versa, by taking ¼ Æ or ¼ ¼ Æ .
The relation between two oblique coordinate systems that differ by a translation
is the same as for Cartesian systems. See Equation (4.1.7).
4.2 PLANE SYMMETRIES OR ISOMETRIES
A transformation of the plane (invertible map of the plane to itself) that preserves
distances is called an isometry of the plane. Every isometry of the plane is of one of
the following types:
1. The identity (which leaves every point x ed)
2. A translation by a vector v
3. A rotation through an angle « around a point È
4. A re ection in a line Ä
5. A glide-re e ction in a line Ä with displacement
Although the identity is a particular case of a translation and a rotation, and re ections
are particular cases of glide-re ections, it is more intuitive to consider each
case separately.
4.2.1 FORMULAE FOR SYMMETRIES: CARTESIAN
COORDINATES
In the formulae below, a multiplication between a matrix and a pair of coordinates
should be carried out regarding ¢ the pair as a column vector (or a matrix with two
rows and one column). Thus
£
´Ü ݵ ´Ü · Ý Ü · ݵ.
1. Translation by ´Ü ¼ Ý ¼ µ:
´Ü ݵ ´Ü · Ü ¼ Ý· Ý ¼ µ (4.2.1)
2. Rotation through « (counterclockwise) around the origin:
Ó× « ×Ò «
´Ü ݵ
´Ü ݵ (4.2.2)
×Ò « Ó× «
3. Rotation through « (counterclockwise) around an arbitrary point ´Ü ¼ Ý ¼ µ:
Ó× « ×Ò «
´Ü ݵ ´Ü ¼ Ý ¼ µ·
´Ü Ü
×Ò « Ó× «
¼ Ý Ý ¼ µ (4.2.3)
4. Re ec tion:
in the Ü-axis:
in the Ý-axis:
in the diagonal Ü Ý:
´Ü ݵ ´Ü ݵ
´Ü ݵ ´ Ü Ýµ
´Ü ݵ ´Ý ܵ
(4.2.4)
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