1605 Submissions

[4] viXra:1605.0314 [pdf] replaced on 2016-08-20 22:58:10

Solution of the Special Case "CLP" of the Problem of Apollonius via Vector Rotations using Geometric Algebra

Authors: James A. Smith
Comments: 22 Pages.

NOTE: A new Appendix presents alternative solutions. The famous "Problem of Apollonius", in plane geometry, is to construct all of the circles that are tangent, simultaneously, to three given circles. In one variant of that problem, one of the circles has innite radius (i.e., it's a line). The Wikipedia article that's current as of this writing has an extensive description of the problem's history, and of methods that have been used to solve it. As described in that article, one of the methods reduces the "two circles and a line" variant to the so-called "Circle-Line-Point" (CLP) special case: Given a circle C, a line L, and a point P, construct the circles that are tangent to C and L, and pass through P. This document has been prepared for two very different audiences: for my fellow students of GA, and for experts who are preparing materials for us, and need to know which GA concepts we understand and apply readily, and which ones we do not.
Category: Geometry

[3] viXra:1605.0233 [pdf] replaced on 2016-08-20 21:29:55

The Problem of Apollonius as an Opportunity for Teaching Students to Use Reflections and Rotations to Solve Geometry Problems via Geometric (Clifford) Algebra

Authors: James A. Smith
Comments: 18 Pages.

Note: The Appendix to this new version gives an alternate--and much simpler--solution that does not use reflections. The beautiful Problem of Apollonius from classical geometry ("Construct all of the circles that are tangent, simultaneously, to three given coplanar circles") does not appear to have been solved previously by vector methods. It is solved here via Geometric Algebra (GA, also known as Clifford Algebra) to show students how they can make use of GA's capabilities for expressing and manipulating rotations and reflections. As Viète did when deriving his ruler-and-compass solution, we first transform the problem by shrinking one of the given circles to a point. In the course of solving the transformed problem, guidance is provided to help students ``see" geometric content in GA terms. Examples of the guidance that is given include (1) recognizing and formulating useful reflections and rotations that are present in diagrams; (2) using postulates on the equality of multivectors to obtain solvable equations; and (3) recognizing complex algebraic expressions that reduce to simple rotations of multivectors. As an aid to students, the author has prepared a dynamic-geometry construction to accompany this article.
Category: Geometry

[2] viXra:1605.0232 [pdf] submitted on 2016-05-22 20:17:30

Rotations of Vectors Via Geometric Algebra: Explanation, and Usage in Solving Classic Geometric "Construction" Problems (Version of 11 February 2016)

Authors: James A. Smith
Comments: 76 Pages.

Written as somewhat of a "Schaums Outline" on the subject, which is especially useful in robotics and mechatronics. Geometric Algebra (GA) was invented in the 1800s, but was largely ignored until it was revived and expanded beginning in the 1960s. It promises to become a "universal mathematical language" for many scientific and mathematical disciplines. This document begins with a review of the geometry of angles and circles, then treats rotations in plane geometry before showing how to formulate problems in GA terms, then solve the resulting equations. The six problems treated in the document, most of which are solved in more than one way, include the special cases that Viete used to solve the general Problem of Apollonius.
Category: Geometry

[1] viXra:1605.0024 [pdf] submitted on 2016-05-03 01:13:07

Complements to Classic Topics of Circles Geometry

Authors: Ion Patrascu, Florentin Smarandache
Comments: 180 Pages.

We approach several themes of classical geometry of the circle and complete them with some original results, showing that not everything in traditional math is revealed, and that it still has an open character. The topics were chosen according to authors aspiration and attraction, as a poet writes lyrics about spring according to his emotions.
Category: Geometry