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<title>Overview of computing with conformal geometric algebras</title>
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<h1>Overview of computation with conformal geometric algebras (CGA)</h1>
This overview omits features of the toolbox which are common with the
original Clifford Algebra functionality. Although the conformal geometric
algebra functionality depends on the Clifford algebra functionality, it
should be possible to make use of the toolbox for conformal algebras without
needing a detailed knowledge of the Clifford algebra functions. However,
basic familiarity with Matlab® will be needed.
<h2>Basics</h2>
<p>
As with Clifford algebras, before a conformal algebra can be used, it
must be initialized by the function <code>conformal_signature</code>.
This function also initializes a Clifford algebra to provide the underlying
representation of multivectors in the conformal algebra. Called without
parameters, it displays the parameters of the current conformal algebra.
</p>
<p>
A conformal algebra has additional vectors compared to a Clifford algebra,
representing the origin, the point at infinity, and the bivector which
is the outer product of these two. These three additional quantities
are provided as parameterless functions <code>eo, ei</code> and
<code>E</code> respectively. They have default Unicode representations
for display, but these can be overridden when the conformal algebra
is initialized.
</p>
<p>
The three additional vectors are constructed from the underlying
Clifford algebra which has signature (p,q) each one greater than the
signature
of the Conformal Geometric Algebra (for example, the conformal
algebra with signature (3,0) has underlying Clifford algebra (4,1).
Denoting the Clifford algebra vector with the highest index among those
vectors with negative square by en, and that with the highest index
among those vectors with positive square by ep, then we have:
eo = (en - ep)/(λ√2) and ei = (en + ep)λ/√2. The parameter λ may be
specified when a conformal algebra is initialised. If not specified
it defaults to unity.
</p>
<p>
Conformal multivectors are displayed somewhat differently to Clifford
multivectors, as some parts of the multivector are displayed as
coefficients of the origin and infinity vectors, and their bivector
outer product. These 'coefficients' are Clifford multivectors, and
are displayed as such.
</p>
<p>
Note that when a conformal multivector is displayed, the two
Clifford vectors en and ep will not appear (see above). Since these
are the two vectors of highest index from those with negative and
positive square respectively, the remaining Clifford vectors will
not have consecutive indices. This may differ from the way that a
conformal multivector is expressed mathematically, using special
notations for the two additional Clifford vectors needed compared
to the conformal vectors. We call this the 'vector hole' - because
there are apparently two indices missing from the Clifford multivectors.
Eliminating this 'feature' would require some complex coding effort.
</p>
<p>
A relatively small number of functions are specific to conformal
geometric algebras, since in most cases the Clifford function does
what is needed (examples of the latter are functions for addition
and multiplication). The distinction is made clear in the documentation
page for each function.
</p>
<h2>Implementation outline</h2>
<p>
A conformal multivector is actually a Clifford multivector with some
additional information (the conformal signature, and the value of
λ). Because conformal
multivectors are defined as a separate class, Matlab® is able to
handle them differently, and display them differently to Clifford
multivectors. There is however a close relationship between the two
types of multivector. To see the Clifford multivector representation
of a conformal multivector, use the Clifford constructor function
<code>clifford</code>. This will convert a conformal multivector into
its Clifford representation.
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