Graphics Reference
In-Depth Information
Chapter 8
Frames of Reference in the Plane
8.1 Introduction
In the previous chapter we covered the transforms for rotating points in the plane
with respect to a
fixed
frame of reference. In this chapter we investigate the trans-
forms for computing the coordinates of points in rotated frames of reference using
geometry, matrices and multivectors. We will employ many of the concepts previ-
ously described in order to develop inverse transforms and rotors.
8.2 Frames of Reference
You have probably been on a train waiting to depart from a railway station, and
through the window see another stationary train. Then suddenly you notice move-
ment. To begin with, it is difficult to decide which train is moving, and the problem
is often resolved when the entire scenario is seen with reference to some fixed object
such as a tree or a building. This phenomena reminds us that motion is relative, and
plays an important role in understanding transforms and frames of reference.
When a frame of reference moves - such as a train - the relationship between
the seated passengers and the train remains fixed. The only thing that does change
is the relationship between the train and other frames of references such as a tree or
a building.
One can describe the relative motion between the train and a tree by assuming
that the train remains stationary and the tree moves in an equal and opposite direc-
tion. So if the train's translation is described by
T
, the tree's translation relative to
the train is described by the inverse transform
T
−
1
.
Similarly, the rotation of a swivel chair can be described in two ways. The usual
way is to assume that the swivel chair rotates relative to the desk where it's lo-
cated. The relative motion interpretation proposes that the chair is stationary, whilst
the desk rotates in an equal and opposite direction. So if the chair's rotation is de-
scribed by
R
, the desk's rotation relative to the chair is described by the inverse
transform
R
−
1
.
