Trapezoid central configurations

TL;DR

This paper classifies all planar four-body central configurations with two pairs of bodies on parallel lines, identifying geometric families and conditions under which mass arrangements determine configuration shape.

Contribution

It provides a complete classification of trapezoid central configurations, including symmetric and non-symmetric cases, using Cartesian coordinates and geometric analysis.

Findings

The set of trapezoid configurations forms a two-dimensional surface.

Rhombus and isosceles trapezoid are boundary families.

Existence of non-symmetric configurations with equal masses.

Abstract

We classify all planar four-body central configurations where two pairs of the bodies are on parallel lines. Using Cartesian coordinates, we show that the set of four-body trapezoid central configurations with positive masses forms a two-dimensional surface where two symmetric families, the rhombus and isosceles trapezoid, are on its boundary. We also prove that, for a given position of the bodies, in some cases an specific order of the masses determine the geometry of the configuration, namely acute or obtuse trapezoid central configuration. We also prove the existence on non-symmetric trapezoid central configuration with two pairs of equal masses.