Graphics Reference
In-Depth Information
cos
q
-
s
in
q
0
Ê
ˆ
1
1
Á
Á
Á
Á
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
˜
˜
˜
˜
s
in
q
cos
q
1
1
O
cos
q
-
sin
q
k
k
(2.31)
sin
q
cos
q
k
k
±
1
O
Ë
¯
0
±
1
Conversely, every transformation of
R
n
whose equation has such a matrix of coeffi-
cients is a rotation.
Proof.
See [Lips68]. Note that rotations about the origin are linear transformations
so that one can talk about their associated matrices.
Theorem 2.5.5 suggests that the expression “rotation about a point” is perhaps
misleading in higher dimensions. Although it might be better to say “rotation about
a line,” we shall keep it in order to have a uniform terminology since it makes per-
fectly good sense in the plane. Actually, we shall see shortly in the next section that
one should really talk about directed lines here because the expression “rotation about
a line through an angle q” is
ambiguous
.
The main theorems about motions in
R
n
can now be stated. Their proofs are very
similar to the proofs of the corresponding theorems about motions in the plane and
are omitted.
2.5.6. Theorem.
(1) A motion in
R
n
is completely determined by what it does to n + 1 linearly inde-
pendent points.
(2) A rigid motion in
R
n
is completely determined by what it does to n linearly
independent points.
(3) Every motion in
R
n
can be described as a composition of a translation, a rota-
tion about the origin, and/or a reflection.
(4) Every rigid motion in
R
n
is a composition of a translation and/or a rotation
about the origin.
Proof.
Exercise.
Facts about similarities and affine maps in the plane also generalize to
R
n
.
2.5.7. Theorem.
Every similarity transformation can be expressed by equations of
the form (2.29) where (a
ij
) = (db
ij
), d > 0, and (b
ij
) is an orthogonal matrix. Conversely,
every such system of equations defines a similarity.
Proof.
Exercise.
