Graphics Reference
In-Depth Information
CHAPTER 4
Transformations and the
Graphics Pipeline
Prerequisites:
Chapters 2 and 3 in [AgoM05]. Chapter 20 for Section 4.14.
4.1
Introduction
In this chapter we combine properties of motions, homogeneous coordinates, pro-
jective transformations, and clipping to describe the mathematics behind the three-
dimensional computer graphics transformation pipeline. With this knowledge one will
then know all there is to know about how to display three-dimensional points on a
screen and subsequent chapters will not have to worry about this issue and can con-
centrate on geometry and rendering issues. Figure 4.1 shows the main coordinate
systems that one needs to deal with in graphics and how they fit into the pipeline. The
solid line path includes clipping, the dashed line path does not.
Because the concept of a coordinate system is central to this chapter, it is worth
making sure that there is no confusion here. The term “coordinate system” for
R
n
means nothing but a “frame” in
R
n
, that is, a tuple consisting of an orthonormal basis
of vectors together with a point that corresponds to the “origin” of the coordinate
system. The terms will be used interchangeably. In this context one thinks of
R
n
purely
as a set of “points” with no reference to coordinates. Given one of these abstract points
p
, one can talk about the coordinates of
p
with respect to one coordinate system or
other. If
p
= (x,y,z) Œ
R
3
, then x, y, and z are of course just the coordinates of
p
with
respect to the standard coordinate system or frame (
e
1
,
e
2
,
e
3
,
0
).
Describing the coordinate systems and maps shown in Figure 4.1 and dealing with
that transformation pipeline in general occupies Sections 4.2-4.7. Much of the dis-
cussion is heavily influenced by Blinn's excellent articles [Blin88b,91a-c,92]. They
make highly recommended reading. Section 4.8 describes what is involved in creat-
ing stereo views. Section 4.9 discusses parallel projections and how one can use the
special case of orthographic projections to implement two-dimensional graphics in a
