Graphics Reference
In-Depth Information
Chapter 6
Multivectors
6.1 Introduction
This is a brief introduction to
multivectors
of
geometric algebra
and we only ex-
plore those elements associated with rotations. Those readers who wish to pursue
the subject further may wish to consult the author's topics:
Geometric Algebra for
Computer Graphics
[5]or
Geometric Algebra: An Algebraic System for Computer
Games and Animation
[6].
We regard vectors as
directed
lines or
oriented
lines, but if
they
exist, why
shouldn't oriented planes and oriented volumes exist? Well they do, which is what
geometric algebra is about. Unfortunately when vectors were invented, the work of
the German mathematician, Hermann Grassmann (1809-1877), was not understood
and consequently ignored. In retrospect this was unfortunate, as Grassmann had
invented an exceedingly powerful algebra for geometry, and it has taken a further
century for it to emerge through the work of William Kingdon Clifford (1845-1879)
and David Hestenes. So let's explore an exciting algebra that offers new ways of
handling rotations.
6.2 Symmetric and Antisymmetric Functions
Symmetric
(
even
) and
antisymmetric
(
odd
) functions play an important role in un-
derstanding multivectors. For example,
f(β)
is a symmetric function if
f(
−
β)
=
f(β)
an example being cos
β
where cos
(
−
β)
=
cos
β
. Whereas,
f(β)
is an antisymmetric
function if
f(
−
β)
=−
f(β)
an example being sin
β
where sin
(
−
β)
=−
sin
β
.
