Chapter 4: Geometry

FIGURE 4.34
The coordinate rays ÇÜ, ÇÝ, ÇÞ, together with their images Ç, Ç, Ç under a rotation, x
the Euler angles associated with that rotation, as follows: ÞÇ, ÜÇÖ ÝÇ×, and
×Ç. (Here the ray ÇÖ is the projection of Ç to the ÜÝ-plane. The ray Ç× is determined
by the intersection of the ÜÝ- and -planes.)
Þ
Ü
Ö
Ý
×
(An alternative decomposition, more natural if we think of the coordinate
system as a rigid trihedron that rotates in space, is the following: a rotation
by about the Þ-axis, followed by a rotation by about the rotated Ý-axis,
followed by a rotation by about the rotated Þ-axis. Note that the order is
reversed.)
Provided that is not a multiple of ½¼ Æ , the decomposition of a rotation in
this form is unique (apart from the ambiguity arising from the possibility of
adding a multiple of ¿¼ Æ to any angle). Figure 4.34 shows how the Euler
angles can be read off geometrically.
Warning: Some references define Euler angles differently; the most common
variation is that the second rotation is taken about the Ü-axis instead of about
the Ý-axis.
5. Screw motion with angle « and displacement 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 (4.9.5)
where Å is given by (4.9.2).
6. Re ec tion
in the ÜÝ-plane: ´Ü Ý Þµ ´Ü Ý Þµ
in the ÜÞ-plane: ´Ü Ý Þµ ´Ü Ý Þµ (4.9.6)
in the ÝÞ-plane: ´Ü Ý Þµ ´ Ü Ý Þµ
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