Angle contraction between geodesics

TL;DR

This paper generalizes a discrete dynamical system based on angle bisection from the Euclidean plane to regular surfaces in three-dimensional space, exploring properties of geodesics and their applications in classifying surface points.

Contribution

It extends known Euclidean angle bisection systems to curved surfaces and analyzes their properties using geodesics, providing new insights into surface point classification.

Findings

Properties of angle bisection systems are similar on surfaces and in Euclidean plane.

Geodesic-based systems can classify surface points as elliptic, hyperbolic, or parabolic.

The generalization reveals new geometric behaviors on curved surfaces.

Abstract

We consider here a generalization of a well known discrete dynamical system produced by the bisection of reflection angles that are constructed recursively between two lines in the Euclidean plane. It is shown that similar properties of such systems are observed when the plane is replaced by a regular surface in ${\mathbb R}^3$ and lines are replaced by geodesics. An application of our results to the classification of points on the surface as elliptic, hyperbolic or parabolic is also presented.