John Stillwell - Naive Lie Theory.pdf - Index of

1.5 Quaternion representation of space rotations 17
S ′
P
α
α
S
L
Figure 1.3: Reflection of S and its angle.
1.5.1 If L passes through P,andifS lies on one side of L at angle α (Figure 1.3),
show that S ′ lies on the other side of L at angle α,andthat|PS| = |PS ′ |.
1.5.2 Deduce, from Exercise 1.5.1 or otherwise, that the rotation about P through
angle θ is the result of reflections in any two lines through P that meet at
angle θ/2.
1.5.3 Deduce, from Exercise 1.5.2 or otherwise, that if L , M ,andN are lines
situated as shown in Figure 1.4, then the result of rotation about P through
angle θ, followed by rotation about Q through angle ϕ, is rotation about R
through angle χ (with rotations in the senses indicated by the arrows).
P
R
χ/2
L
N
θ/2
ϕ/2
M
Q
Figure 1.4: Three lines and three rotations.
1.5.4 If L and N are parallel, so R does not exist, what isometry is the result of
the rotations about P and Q?
Now we extend these ideas to R 3 .Arotation about a line through O (called
the axis of rotation) is the product of reflections in planes through O that meet
along the axis. To make the reflections easier to visualize, we do not draw the
planes, but only their intersections with the unit sphere (see Figure 1.5).
These intersections are curves called great circles, andreflection in a great
circle is the restriction to the sphere of reflection in a plane through O.