Algebra

1612 Submissions

[2] viXra:1612.0259 [pdf] submitted on 2016-12-16 07:05:01

A Two-Dimensional Vector Space Algebra with Identity 2x2 Matrix Basis Matrix Multiplication Homomorphism

Authors: Claude Michael Cassano
Comments: 3 Pages.

A two-dimensional vector space algebra with identity 2x2 matrix basis matrix multiplication homomorphism There exists a homomorphism between any two-dimensional vector space algebra with identity and a 2x2 matrix basis under ordinary matrix multiplication. This is a statement of constructive existence of an algebra. Given that the vector space of the algebra is known to be 2-dimensional, the algebra product determines the constants: A,B,b ; determining the basis of the algebra. And showing that the basis of a two-dimensional vector space unitary algebra is a cyclic group of order 2
Category: Algebra

[1] viXra:1612.0221 [pdf] submitted on 2016-12-12 03:18:52

Conic and Cyclidic Sections in Double Conformal Geometric Algebra G_{8,2}

Authors: Robert Benjamin Easter, Eckhard Hitzer
Comments: 6 Pages. Proceedings of SSI 2016, Session SS11, pp. 866-871, 6-8 Dec. 2016, Ohtsu, Shiga, Japan, 10 color figures.

The G_{8,2} Geometric Algebra, also called the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), has entities that represent conic sections. DCGA also has entities that represent planar sections of Darboux cyclides, which are called cyclidic sections in this paper. This paper presents these entities and many operations on them. Operations include projection, rejection, and intersection with respect to spheres and planes. Other operations include rotation, translation, and dilation. Possible applications are introduced that include orthographic and perspective projections of conic sections onto view planes, which may be of interest in computer graphics or other computational geometry subjects.
Category: Algebra