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 wor...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.展开更多
文摘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.
