Graphics Reference
In-Depth Information
Chapter 1
Introduction
1.1 Rotation Transforms
In computer graphics the position of an object is expressed by two transforms:
translation and rotation. It is relatively easy to visualise a translation and express
it mathematically, however rotations do present problems. Furthermore, it is not just
objects that require rotating and translating - frames of reference have to be posi-
tioned within the world coordinate system in order to secure different views of the
virtual world. In order to do this, it is necessary to combine rotation and translation
transforms.
When rotating and translating objects, the angles and translation offsets are often
relative to a fixed frame of reference. However, when rotating and translating frames
of reference, the angles and offsets are relative to a changing frame of reference,
which requires careful handling. Primarily, this topic is about rotation transforms,
and how they are used for moving objects and frames of reference in the plane and
in 3D space. But in order to do this within a real computer graphics context, it is
necessary to include the translation transform, which introduces some realism to the
final solution.
The world of mathematics offers a wide variety of rotation techniques to choose
from such as direction cosines, Euler angles, quaternions and multivectors. Each has
strengths and weaknesses, advocates and critics, therefore no attempt will be made
to identify a 'best' technique. However, I will attempt to draw your attention to their
qualities in order that you can draw your own conclusions.
1.2 Mathematical Techniques
Six branches of mathematics play an important role in rotations: trigonometry, com-
plex numbers, vectors, matrices, quaternions and multivectors, which are described
in the following chapters and ensure that this topic is self contained. We only require
to consider certain aspects of trigonometry which will become foundations for the
