On the Homogeneous Model of Euclidean Geometry

Abstract

We attach the degenerate signature (n, 0, 1) to the projectivized dual Grassmann algebra P( R (n+1) * ) to obtain the Clifford algebra P(R * n,0,1 ) and explore its use as a model for euclidean geometry. We avoid problems with the degenerate metric by constructing an algebra isomorphism J between the Grassmann algebra and its dual that yields non-metric meet and join operators. We focus on the cases of n = 2 and n = 3 in detail, enumerating the geometric products between k-and l-blades. We establish that sandwich operators of the form X → gX g provide all euclidean isometries, both direct and indirect. We locate the spin group, a double cover of the direct euclidean group, inside the even subalgebra of the Clifford algebra, and provide a simple algorithm for calculating the logarithm of such elements. We conclude with an elementary account of euclidean rigid body motion within this framework.

Geometric Algebras for Euclidean Geometry

“…If one attaches such a Q to RP n then, for many choices of Q, Cayley and Klein showed it is possible to define a distance function on a subset M ⊂ RP n which makes M into a constant-curvature metric space ([Kle26], Ch. 6, [Gun11b], §3.1, or [Gun11a], Ch. 4).…”

“…2.2. The representation can be extended to include points and planes; details can be found in [Gun11b], §15.6. Also note that PGA replaces the irregular transformation formula for the sandwich operators of the biquaternions and of the motor algebra (Sect.…”

“…The algebra first appeared in the modern literature in [Sel00] and [Sel05], and was then extended and embedded in the metric-neutral toolkit described in [Gun11a]. A compact, self-contained treatment is given in [Gun11b] (extended version [Gun11c]). …”

“…Readers who would like to confirm the claimed covariance for themselves, and do not have access to [Gun11b] or [Gun11a] are referred to Appendix A, which provides a detailed discussion of the versors of P(R * 2,0,1 ) and their associated sandwich operators.…”

Doing Euclidean Plane Geometry Using Projective Geometric Algebra

“…PGA for euclidean geometry first appeared in the modern literature in [Sel00] and [Sel05] and was extended and developed in [Gun11a], [Gun11b], [Gun11c], and [Gun16b]. Readers unfamiliar with duality or projectivization, or just interested in a fuller, more rigorous treatment, should consult the latter references.…”