Graphics Reference
In-Depth Information
Chapter 9
Rotation Transforms in Space
9.1 Introduction
In this chapter we generalise the 2D transforms covered in Chap. 7 to three dimen-
sions. In particular, we examine rotating points about the fixed three Cartesian axes
and off-set, parallel axes, and about an arbitrary axis. We also explore Euler an-
gles, and their
Achilles' heel
- gimbal lock. Matrices will be used to describe these
geometric scenarios.
9.2 3D Transforms
3D transforms include the scale, translate, reflect, shear and rotate transforms, and
in this chapter we need only consider the translate and rotate operations, which we
will explore individually and in combination. Let's start with the translate transform.
9.2.1 Translate a Point
The matrix for a 3D point translation is basically the same as previously described
but with one extra dimension
T
t
x
,t
y
,t
z
. It requires the homogeneous form and is
written
⎡
⎣
⎤
⎦
100
t
x
010
t
y
001
t
z
000 1
T
t
x
,t
y
,t
z
=
