Transitional geometry

TL;DR

This paper introduces a continuous family of geometries that smoothly connect hyperbolic, Euclidean, and spherical geometries, allowing geometric entities and formulas to transition seamlessly across these classical geometries.

Contribution

It develops a unified framework for transitional geometry, enabling continuous variation of geometric properties between different constant curvature geometries.

Findings

Geometric entities transition continuously across geometries.

Trigonometric formulas adapt smoothly in the transitional setting.

Unified framework bridges hyperbolic, Euclidean, and spherical geometries.

Abstract

We develop a transitional geometry, that is, a family of geometries of constant curvatures which makes a continuous connec-tion between the hyperbolic, Euclidean and spherical geometries. In this transitional setting, several geometric entities like points, lines, dis-tances, triangles, angles, area, curvature, etc. as well as trigonometric formulae and other properties transit in a continuous manner from one geometry to another. AMS classification: 01-99 ; 53-02 ; 53-03 ; 53A35.