Graphics Reference
In-Depth Information
7
9
4
9
4
9
2
3
x
¢=
x
+
y
+
z
+
4
9
1
9
8
9
4
3
y
¢=
x
+
y
-
z
-
4
9
8
9
1
9
4
3
.
z
¢=
x
-
y
+
z
-
Generalizing the concept of a rotation is a little less obvious. The simplest way to
get a definition is in a roundabout way by defining a rigid motion first and then use
the orientation-preserving nature of these maps.
Definition.
Let M be a motion of
R
n
and suppose the equations for M are as shown
in Theorem 2.5.1. The motion M is said to be a
rigid motion
if the matrix (a
ij
) is a
special orthogonal matrix.
In analogy to the planar case we get
2.5.4. Theorem.
A rigid motion of
R
n
is an orientation-preserving map. Conversely,
every orientation-preserving motion of
R
n
is a rigid motion.
Proof.
Exercise.
Definition.
A rigid motion R of
R
n
that fixes some point
p
is called a
rotation
. In
that case, we say that R is a
rotation about
p
. The point
p
is called a
center
of the
rotation.
Is this definition of a rotation really what we want and does it generalize the intu-
itively simple notion of a rotation in the plane? Theorem 2.2.6.9 certainly shows that
the new definition is compatible with the old one.
2.5.5. Theorem.
(The Principal Axis Theorem) Every rotation R in
R
3
is a “rotation
about some line.” More precisely, with respect to some appropriate coordinate system,
R is just the
rotation about the z-axis through an angle
q, that is, the equations for R
in that coordinate system are just
xx
¢=
cos
q
-
y
sin
q
y
¢=
x
s
in
q
+
y
c
os
q
z
¢=
z.
(2.30)
In general, if R is a rotation in
R
n
, then we can choose a coordinate system
with respect to which the n ¥ n matrix of coefficients in the equation for R has the
form
