Mathematical Physics

   

Double Conformal Space-Time Algebra for General Quadric Surfaces in Space-Time

Authors: Robert B. Easter

The G(4,8) Double Conformal Space-Time Algebra (DCSTA) is a high-dimensional 12D Geometric Algebra that extends the concepts introduced with the G(8,2) Double Conformal / Darboux Cyclide Geometric Algebra (DCGA) with entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) in spacetime with a new boost operator. The base algebra in which spacetime geometry is modeled is the G(1,3) Space-Time Algebra (STA). Two G(2,4) Conformal Space-Time subalgebras (CSTA) provide spacetime entities for points, hypercones, hyperplanes, hyperpseudospheres (and their intersections) and a complete set of versors for their spacetime transformations that includes rotation, translation, isotropic dilation, hyperbolic rotation (boost), planar reflection, and (pseudo)spherical inversion. G(4,8) DCSTA is a doubling product of two orthogonal G(2,4) CSTA subalgebras that inherits doubled CSTA entities and versors from CSTA and adds new 2-vector entities for general (pseudo)quadrics and Darboux (pseudo)cyclides in spacetime that are also transformed by the doubled versors. The "pseudo" surface entities are spacetime surface entities that use the time axis as a pseudospatial dimension. The (pseudo)cyclides are the inversions of (pseudo)quadrics in hyperpseudospheres. An operation for the directed non-uniform scaling (anisotropic dilation) of the 2-vector general quadric entities is defined using the boost operator and a spatial projection. Quadric surface entities can be boosted into moving surfaces with constant velocities that display the Thomas-Wigner rotation and length contraction of special relativity. DCSTA is an algebra for computing with general quadrics and their inversive geometry in spacetime. For applications or testing, G(4,8) DCSTA can be computed using various software packages, such as the symbolic computer algebra system SymPy with the GAlgebra module.

Comments: 24 pages. Written as an invited extended paper, extending the 10-page conference paper Double Conformal Geometric Algebra (ICNPAA 2016), for possible later submission to the journal Mathematics in Engineering, Science and Aerospace (MESA).

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[v1] 2017-08-21 21:54:53

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