The Algebra of Projective Spheres on Plane, Sphere and Hemisphere

• 作者: István Lénárt
• 来源期刊: 应用数学与应用物理(英文)
• 学科: 数学
• 关键词: Projective Spheres Binary Operation Incidence Relation Pappos Desargues Hesse-Chasles Configurations Napier Pentagram and Hyperbolic Napier Pentagon Polarity in Cartesian Coordinates Differentiable Functions
• 年，卷（期）: 2020,（10）
• 页码范围: 2286-2333
• 页数: 48页
• 分类号: O17

摘要

Numerous authors studied polarities in incidence structures or algebrization of projective geometry <a href="#1">[1]</a> <a href="#2">[2]</a>. The purpose of the present work is to establish an algebraic system based on elementary concepts of spherical geometry, extended to hyperbolic and plane geometry. The guiding principle is: “<em>The point and the straight line are one and the same</em>”. Points and straight lines are not treated as dual elements in two separate sets, but identical elements within a single set endowed with a binary operation and appropriate axioms. It consists of three sections. In Section 1 I build an algebraic system based on spherical constructions with two axioms: <em>ab</em> = <em>ba</em> and (<em>ab</em>)(<em>ac</em>) = <em>a</em>, providing finite and infinite models and proving classical theorems that are adapted to the new system. In Section Two I arrange hyperbolic points and straight lines into a model of a projective sphere, show the connection between the spherical Napier pentagram and the hyperbolic Napier pentagon, and describe new synthetic and trigonometric findings between spherical and hyperbolic geometry. In Section Three I create another model of a projective sphere in the Cartesian coordinate system of the plane, and give methods and techniques for using the model in the theory of functions.