User:Jheald/sandbox/GA/Affine geometric algebra

Dorst et al mention that attempts have been made to extend the Conformal Geometric Algebra to allow ellipsoids to be natural objects, and stretches to be natural bivectorised transformations.

They mention attempts by Perwass and by Doran, but say neither is yet satisfactory.

p.498:
[Two features of Geometric Algebra are essential to the way it allows symmetries and symmetrical objects to be represented automatically in a covariant way:]

• The two-sided versor product preserves the geometric product structure

  $V(XY)V^{-1}=(VXV^{-1})(VYV^{-1}$

• All geometric constructions can be expressed in terms of the geometric product

... [seek a representation in which the symmetries become isometries: distance-preserving transformations]...

Examples...

• Projective geometry. Unfortunately there is not yet an operational model for projective geometry. That would have projective transformations as versors (rather than as linear transformations, as in the homogeneous coordinate approach). The metric of the representation space should probably be based on the cross-ratio. Its blades would naturally represent the conic sections. Initial attempts [15, 48] do not quite have this structure, but we hope that an operational projective model will be developed soon.

• 48: Perwass & Förstner, Uncertain Geometry with Circles, Spheres and Conics. Computational Imaging and Vision, 31, 23--41; also relevant section of Perwass book

  -- can represent ellipsoids, but can no longer represent translations in his algebra.

• 15: Doran & Lasenby, Geometric Algebra for Physicists

Complex signature ?

Seems to me that we already know a bivectorised transformation that can represent stretches: the hyperbolic rotations that occur in mixed-signature Clifford algebras.

Could such "imaginary" rotations not be introduced alongside real rotations, to represent a hyperbolic rotation? eg into the Conformal Geometric Algebra model?

(How much would we then need to complexify to close the algebra?)

Once we have stretches, ellipsoids should appear as the eigenfunctions of a transformation S R_c S^-1. (ie inverse stretch (to turn it into a circle), reflection in a circle, stretch (to turn it back into an ellipse).

Hestenes & followers' party line is against complex CAs... but perhaps this might indicate one way towards their interpretation?