Geometry

Accurate Circle Configurations and Numerical Conformal Mapping in Polynomial Time

Authors: Warren D. Smith
Comments: Sitting on my web pages since 1991; uploading to vixra for archival purposes.

According to a remarkable re-interpretation of a theorem of E.M. Andreev (1970) by W.P. Thurston (≈1982), there is a unique (up to inversive transformations) packing of interior-disjoint circles in the plane, whose contact graph is any given polyhedral graph G, and such that an analogous "dual" circle packing simultaneously exists, whose contact graph is the planar dual graph G*, and such that the primal and dual circle packings have the same set of tangency points and the primal circles are orthogonal to the dual ones at these tangency points.

This note shows that relatively and absolutely accurate coordinates for the primal and dual circles may be obtained in time polynomial in N, the number of vertices of the polyhedral graph, and D, the number of decimals of accuracy desired.

Consequently one may also accurately "midscribe"' a polyhedron – and simultaneously its dual – in polynomial time.

Also consequently, one may implement Riemann's conformal mapping theorem numerically, in polynomial time with provable accuracy.

Our result is obtained by generalizing and reformulating ideas found in the doctoral thesis of Walter Brägger [Math. Institut, Rheinsprung 21, CH-4051 Basel, Feb. 1991] to reduce our problem to maximizing a smooth convex function. This maximization problem is then solved by using Khachian's "ellipsoid method" or Vaidya's algorithm.
Category: Geometry

A New Home for Bivectors Geometric Algebra on a Sphere

Authors: Norm Cimon
Comments: 20 Pages.

The impetus for the work is this quote:"...as shown by Gel’fand’s approach, we can only abstract a unique manifold if our algebra is commutative."[1] Geometric algebra is non-commutative. Components of different grades can be staged on different manifolds. As operations on those elements proceed, they can effect the promotion and/or demotion of components to higher and/or lower grades, and thus to different manifolds. This paper includes imagery that visually displays bivector addition and rotation on a sphere. David Hestenes interpreted the vector product or rotor in two-dimensions: "as a directed arc of fixed length that can be rotated at will on the unit circle, just as we interpret a vectoras a directed line segment that can be translated at will without changing its length or directionu2026"[2]Rotors can be used to develop addition and multiplication of bivectors on a sphere. For those rotational dynamics, rotors of lengthare the basis elements. The geometric algebra of bivectors — Hamilton’s "pure quaternions" — is thus shown to transparently operate on a spherical manifold.This paper also explores the possible generalizations that emerge from the placement of the graded elements which make up a geometric algebra onto separate manifolds.
Category: Geometry