Geometric properties of a sine function extendable to arbitrary normed planes

TL;DR

This paper explores a metric generalization of the sine function in normed planes, characterizes Radon planes, and establishes conditions for Euclidean geometry, including a Law of Sines adaptation.

Contribution

It introduces a new sine function generalization, characterizes Radon and Euclidean planes, and studies sine-preserving mappings in normed planes.

Findings

The generalized sine function has specific properties in normed planes.

An angular measure compatible with the sine exists only in Euclidean planes.

A version of the Law of Sines is established for Radon planes.

Abstract

In this paper we study a metric generalization of the sine function which can be extended to arbitrary normed planes. We derive its main properties and give also some characterizations of Radon planes. Furthermore, we prove that the existence of an angular measure which is "well-behaving" with respect to the sine is only possible in the Euclidean plane, and we also define some new constants that estimate how non-Radon or non-Euclidean a normed plane can be. Sine preserving self-mappings are studied, and a complete description of the linear ones is given. In the last section we exhibit a version of the Law of Sines for Radon planes.