Chapter 4: Geometry

mations of the following types:
1. The identity (which leaves every point fixed)
2. A translation by a vector v
3. A rotation through an angle « around a line Ä
4. A screw motion through an angle « around a line Ä, with displacement
5. A re ection in a plane È
6. A glide-re e ction in a plane È with displacement vector v
7. A rotation-re ection (rotation through an angle « around a line Ä composed
with reflection in a plane perpendicular to Ä).
The identity is a particular case of a translation and of a rotation; rotations are
particular cases of screw motions; reflections are particular cases of glide-reflections.
However, as in the plane case, it is more intuitive to consider each case separately.
4.9.1 FORMULAE FOR SYMMETRIES: CARTESIAN
COORDINATES
In the formulae below, multiplication between a matrix and a triple of coordinates
should be carried out regarding the triple as a column vector (or a matrix with three
rows and one column).
1. Translation by ´Ü ¼ Ý ¼ Þ ¼ µ:
´Ü Ý Þµ ´Ü · Ü ¼ Ý· Ý ¼ Þ· Þ ¼ µ (4.9.1)
2. Rotation through « (counterclockwise) around the line through the origin with
direction cosines (see page 353): ´Ü Ý Þµ Å ´Ü Ý Þµ where Å is
¾the matrix
¿
¾´½ Ó× «µ ·Ó׫ ´½ Ó× «µ ×Ò « ´½ Ó× «µ · ×Ò «
´½ Ó× «µ · ×Ò « ¾´½ Ó× «µ · Ó× « ´½ Ó× «µ ×Ò «
´½ Ó× «µ ×Ò « ´½ Ó× «µ · ×Ò « ¾´½ Ó× «µ · Ó× «
(4.9.2)
3. Rotation through « (counterclockwise) around the line with direction cosines
through an arbitrary point ´Ü ¼ Ý ¼ Þ ¼ µ:
´Ü Ý Þµ ´Ü ¼ Ý ¼ Þ ¼ µ·Å ´Ü Ü ¼ Ý Ý ¼ Þ Þ ¼ µ (4.9.3)
where Å is given by Equation (4.9.2).
4. Arbitrary rotations and Euler angles: Any rotation of space fixing the origin
can be decomposed as a rotation by about the Þ-axis, followed by a rotation
by about the Ý-axis, followed by a rotation by about the Þ-axis. The
numbers , , and are called the Euler angles of the composite rotation,
which acts as: ´Ü Ý Þµ Å ´Ü Ý Þµ where Å is the matrix given by
¾
Ó× Ó× Ó× ×Ò ×Ò ×Ò Ó× Ó× Ó× ×Ò ×Ò Ó×
Ó× Ó× ×Ò ·×Ò Ó× ×Ò Ó× ×Ò · Ó× Ó× ×Ò ×Ò
Ó× ×Ò ×Ò ×Ò Ó×
(4.9.4)
¿
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