Double Conformal Space-Time Algebra

The Double Conformal Space-Time Algebra (DCSTA) is a high-dimensional 12D Geometric Algebra G(4,8) that extends the concepts introduced with the Double Conformal / D arboux Cyclide Geometric Algebra (DCGA) G(8,2) 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 Space-Time Algebra (STA) G(1,3). Two Conformal Space-Time subalgebras (CSTA) G(2,4) provide spacetime entities for points, flats (incl. worldlines), and hyperbolics, 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 in rounds or hyperbolics. The DCSTA G(4,8) is a doubling product of two G(2,4) CSTA subalgebras that inherits doubled CSTA entities and versors from CSTA and adds new bivector entities for (pseudo) quadrics and Darboux (pseudo) cyclides in spacetime that are also transformed by the doubled versors. The "pseudo" surface entities are spacetime hyperbolics or other surface entities using the time axis as a pseudospatial dimension. The (pseudo) cyclides are the inversions of (pseudo) quadrics in rounds or hyperbolics. An operation for the directed non-uniform scaling (anisotropic dilation) of the bivector general quadric entities is defined using the boost operator and a spatial projection. DCSTA allows general quadric surfaces to be transformed in spacetime by the same complete set of doubled CSTA versor (i.e., DCSTA versor) operations that are also valid on the doubled CSTA point entity (i.e., DCSTA point) and the other doubled CSTA entities. The new DCSTA bivector entities are formed by extracting values from the DCSTA point entity using specifically defined inner product extraction operators. Quadric surface entities can be boosted into moving surfaces with constant velocities that display the length contraction effect of special relativity. DCSTA is an algebra for computing with quadrics and their cyclide inversions in spacetime. For applications or testing, DCSTA G(4,8) can be computed using various software packages, such as Gaalop, the Clifford Multivector Toolbox (for MATLAB), or the symbolic computer algebra system SymPy with the GAlgebra module.