Graphics Reference
In-Depth Information
Chapter 12
Bivector Rotors
12.1 Introduction
In Chap. 6 we explored multivectors, and in Chap. 11 we saw how quaternions are
used to rotate points and frames of reference about an arbitrary vector. In this chapter
we will see how these two ideas merge into one to form bivector rotors. In order to
show how such rotors operate, we begin with reflections and show how these can
effect a rotation.
12.2 The Three Reflections Theorem
The three reflections theorem states that '
each isometry of the Euclidean plane is
the composite of one, two, or three reflections.
' To begin with, an
isometry of the
Euclidean plane
is a way of transforming the plane that preserves length. Such
isometries include rotation, translation, reflection and glide reflections. The latter is
a combination of a reflection in a line and a translation along that line. John Stillwell
provides an elegant proof for this theorem in his topic
Numbers and Geometry
[8].
The isometry we are particularly interested in is reflection, where the distance
between two points is preserved in their reflection. Consider, for example, the 2D
scenario shown in Fig.
12.1
where two lines
M
and
N
are imaginary mirrors sepa-
rated by an angle
θ
. The real point
P
subtends an angle
α
to mirror
M
and creates
a virtual image
P
R
which subtends an equal but opposite angle.
Although it is not physically possible, we can imagine that the virtual image
P
R
is reflected in the second mirror
N
. To begin with,
P
R
subtends an angle
θ
α
to
mirror
N
and creates another virtual image
P
which subtends an equal and opposite
angle. What is interesting about this configuration is that although the mirrors are
separated by
θ
, the angle between
P
and
P
is 2
θ
. In order to take advantage of this
effect we need to know how vectors are reflected using multivectors, which is the
subject of the next section.
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