Geometry

1608 Submissions

[4] viXra:1608.0328 [pdf] replaced on 2016-08-27 23:19:26

Additional Solutions of the Limiting Case "CLP" of the Problem of Apollonius via Vector Rotations using Geometric Algebra

Authors: James A. Smith
Comments: 38 Pages.

This document adds to the collection of solved problems presented in References [1]-[6]. The solutions presented herein are not as efficient as those in [6], but they give additional insight into ways in which GA can be used to solve this problem. After reviewing, briefly, how reflections and rotations can be expressed and manipulated via GA, it solves the CLP limiting case of the Problem of Apollonius in three ways, some of which identify the the solution circles' points of tangency with the given circle, and others of which identify the solution circles' points of tangency with the given line. For comparison, the solutions that were developed in [1] are presented in an Appendix.
Category: Geometry

[3] viXra:1608.0217 [pdf] submitted on 2016-08-19 21:41:57

Simplified Solutions of the CLP and CCP Limiting Cases of the Problem of Apollonius via Vector Rotations using Geometric Algebra

Authors: James A. Smith
Comments: 10 Pages.

The new solutions presented herein for the CLP and CCP limiting cases of the Problem of Apollonius are much shorter and more easily understood than those provided by the same author in References 1 and 2. These improvements result from (1) a better selection of angle relationships as a starting point for the solution process; and (2) better use of GA identities to avoid forming troublesome combinations of terms within the resulting equations.
Category: Geometry

[2] viXra:1608.0153 [pdf] submitted on 2016-08-15 04:33:03

The Cartan Model for Equivariant Cohomology

Authors: Xu Chen
Comments: 8 Pages.

In this article, we will discuss a new operator $d_{C}$ on $W(\mathfrak{g})\otimes\Omega^{*}(M)$ and to construct a new Cartan model for equivariant cohomology. We use the new Cartan model to construct the corresponding BRST model and Weil model, and discuss the relations between them.
Category: Geometry

[1] viXra:1608.0103 [pdf] submitted on 2016-08-09 15:55:41

On Quadrics and Pseudoquadrics Inversions in Hyperpseudospheres

Authors: Robert B. Easter
Comments: 4 Pages.

This note on quadrics and pseudoquadrics inversions in hyperpseudospheres shows that the inversions produce different results in a three-dimensional spacetime. Using Geometric Algebra, all quadric and pseudoquadric entities and operations are in the G(4,8) Double Conformal Space-Time Algebra (DCSTA). Quadrics at zero velocity are purely spatial entities in x y z-space that are hypercylinders in w x y z-spacetime. Pseudoquadrics represent quadrics in a three-dimensional (3D) x y w, y z w, or z x w spacetime with the pseudospatial w-axis that is associated with time w=c t. The inversion of a quadric in a hyperpseudosphere can produce a Darboux pseudocyclide in a 3D spacetime that is a quartic hyperbolic (infinite) surface, which does not include the point at infinity. The inversion of a pseudoquadric in a hyperpseudosphere can produce a Darboux pseudocyclide in a 3D spacetime that is a quartic finite surface. A quadric and pseudoquadric can represent the same quadric surface in space, and their two different inversions in a hyperpseudosphere represent the two types of reflections of the quadric surface in a hyperboloid.
Category: Geometry