Graphics Reference
In-Depth Information
where
t
x
,
t
y
,
t
z
are the
x
-,
y
-,
z
-offsets respectively. It goes without further explana-
tion that the inverse transform
T
−
1
t
x
,t
y
,t
z
is
⎡
⎤
100
−
t
x
⎣
⎦
010
−
t
y
T
−
1
t
x
,t
y
,t
z
=
.
001
−
t
z
000
1
9.2.2 Rotate a Point About the Cartesian Axes
Although we talk about rotating points about another point in space, we require
more precise information to describe this mathematically. We could, for example,
associate a plane with the point of rotation and confine the rotated point to this plane,
but it's much easier to visualise an axis perpendicular to this plane, about which the
rotation occurs. Unfortunately, the matrix algebra for such an operation starts to be-
come fussy, and ultimately we have seek the help of quaternions or multivectors.
So let us begin this investigation by rotating a point about the three fixed Carte-
sian axes. Such rotations are called
Euler rotations
after the Swiss mathematician
Leonhard Euler.
Recall that the transform for rotating a point about the origin in the plane is given
by
⎡
⎤
cos
β
−
sin
β
0
⎣
⎦
.
R
β
=
sin
β
cos
β
0
0
0
1
This can be generalised into a 3D rotation
R
β,z
about the
z
-axis by adding a
z
-
coordinate as follows
⎡
⎣
⎤
⎦
cos
β
−
sin
β
00
sin
β
cos
β
00
R
β,z
=
0
010
0
001
which is illustrated in Fig.
9.1
To rotate a point about the
x
-axis, the
x
-coordinate remains constant whilst the
y
- and
z
-coordinates are changed according to the 2D rotation transform. This is
expressed algebraically as
x
=
x
y
=
y
cos
β
−
z
sin
β
z
=
y
sin
β
+
z
cos
β
