The Fourth Geometry I: Difference--Angle Geometry Beyond Euclid, Hyperbolic, and Elliptic

TL;DR

This paper introduces a novel geometry based on difference angles, providing a new axiomatic framework that redefines angles independently of circles or rotations, and reveals unique properties and classical configurations within this system.

Contribution

It develops the first axiomatic system for difference-angle geometry, establishing foundational concepts and demonstrating its potential as a new 'Fourth Geometry' beyond Euclidean, hyperbolic, and elliptic geometries.

Findings

Triangle inequality holds with equality

Sum of interior angles of any triangle is zero

Classical Euclidean configurations reappear in this new setting

Abstract

In this work, we introduce a new geometry based on the difference angle, an angle defined as the difference of slopes of two lines, together with an axiomatic system for angles. This framework provides a constructive approach to the fundamental question ``What is an angle?'', and shows that an angle can be defined independently of circles or rotations, as a primary geometric notion. Within this geometry, one can define difference-angle triangles, norms, bisectors, perpendiculars, and inner products. Several characteristic properties not seen in existing geometries emerge, together with behaviors analogous to those in Euclidean geometry: the triangle inequality always holds with equality, the sum of the interior angles of any triangle is $0$, and a Miquel point exists even for parabolas. In particular, the concurrency of the parabolic Miquel configuration was suggested by Weiss and Odehnal, and our main theorem provides the first explicit and rigorous confirmation of this assertion. We also point out that many classical Euclidean configurations (including Brocard-type configurations) naturally reappear in the setting of difference-angle geometry. These results indicate that difference-angle geometry is a promising candidate for a ``Fourth Geometry'' following the Euclidean, hyperbolic, and elliptic geometries.