Chapter 4: Geometry

5. Re ec tion in a line with equation Ü · Ý · ¼:
´Ü ݵ
½
¾
¾ ¾
´Ü
¾ · ¾ ¾ ¾ ¾ ݵ ´¾ ¾µ
(4.2.5)
6. Re ec tion in a line going through ´Ü ¼ Ý ¼ µ and making an angle « with the
Ü-axis:
´Ü ݵ ´Ü ¼ Ý ¼ µ·
Ó× ¾« ×Ò ¾«
´Ü Ü
×Ò ¾« Ó× ¾«
¼ Ý Ý ¼ µ (4.2.6)
7. Glide-re ection in a line Ä with displacement : Apply rst a re ection in Ä,
then a translation by a vector of length in the direction of Ä, that is, by the
vector
½ ´¦
¾ §µ (4.2.7)
· ¾ if Ä has equation Ü · Ý · ¼.
4.2.2 FORMULAE FOR SYMMETRIES: HOMOGENEOUS
COORDINATES
All isometries of the plane can be expressed in homogeneous coordinates in terms
of multiplication by a matrix. This fact is useful in implementing these transformations
on a computer. It also means that the successive application of transformations
reduces to matrix multiplication. The corresponding matrices are as follows:
1. Translation by ´Ü ¼ Ý ¼ µ:
Ì´Ü¼Ý ¼µ
¾
¿
½ ¼ Ü ¼
¼ ½ Ý ¼
(4.2.8)
¼ ¼ ½
2. Rotation through « around the origin:
Ê «
¾
¿
Ó× « ×Ò « ¼
×Ò « Ó× « ¼ (4.2.9)
¼ ¼ ½
3. Re ec tion in a line going through the origin and making an angle « with the
Ü-axis:
¾
¿
Ó× ¾« ×Ò ¾« ¼
Å « ×Ò ¾« Ó× ¾« ¼ (4.2.10)
¼ ¼ ½
From this one can deduce all other transformations.
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