Abstract
We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working in a constant curvature background space, rather than the usual Euclidean space. A possible benefit, which is explored here, is that it is possible to define cost functions for geometric object matching in computer vision that are fully covariant, in particular invariant under both rotations and translations, unlike the cost functions which have been used in CGA so far. An algorithm is given for application of this method to the problem of matching sets of lines, which replaces the standard matrix singular value decomposition, by computations wholly in Geometric Algebra terms, and which may itself be of interest in more general settings. Secondly, we consider a further perhaps surprising application of the 1d up approach, which is to the context of a recent paper by Joy Christian published by the Royal Society, which has made strong claims about Bell’s Theorem in quantum mechanics, and its relation to the sphere S^7 and the exceptional group E_8, and proposed a new associative version of the division algebra normally thought to require the octonians. We show that what is being discussed by Christian is mathematically the same as our 1d up approach to 3d geometry, but that after the removal of some incorrect mathematical assertions, the results he proves in the first part of the paper, and bases the application to Bell’s Theorem on, amount to no more than the statement that the combination of two rotors from the Clifford Algebra Cl(4, 0) is also a rotor.
Highlights
For the usual conformal geometric algebra (CGA) approach to 3d space (e.g. [6]) it is well known that we add two extra vectors
We look for transformations that keep einvariant, we get spherical geometry
If we look for transformations that keep e invariant, we get hyperbolic geometry, as pictured in Fig. 1 in a 2d example
Summary
For the usual CGA approach to 3d space (e.g. [6]) it is well known that we add two extra vectors. In the spherical case e used as origin means eis never used In both these cases, one can make do with only having one vector extra! Here is the proposal: let us do the geometry we want (even in engineering applications) in either spherical or hyperbolic space, and recover Euclidean results (if needed) by taking the limit as the length scale λ → ∞ at the end (see below for more details of this process). In many cases this will mean 1d less for computations, and can save time in implementations. Further information can be found in [7]
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