Algorithm for planar 4-Body Problem central configurations with given masses

TL;DR

This paper presents an algorithm to determine planar four-body central configurations with specified masses by using orthocentric tetrahedra, rotations, and weighted directed areas to compute distances and configurations.

Contribution

It introduces a novel algorithm that computes four-body central configurations from given masses using geometric and rotational techniques.

Findings

Successfully computes distances for given masses in planar 4-body configurations.

Handles cases with two equal masses effectively.

Provides a method to match computed and given masses through iterative rotation.

Abstract

An algorithm to compute the six distances between particles of a planar Four-Body central configuration is presented according to the following schema. An orthocentric tetrahedron is computed as a function of given masses. Each mass is placed at the corresponding vertex of the tetrahedron. The center of mass (and orthocenter) of the tetrahedron is at the origin of coordinates. The tetrahedron is orientated in a particular position function of the masses: with one of the particles placed on axis 3. The tetrahedron is rotated by two angles (to be tuned variables) around the center of mass until a direction orthogonal to the plane of configuration coincides with axis 3. The four coordinates of the vertices of the tetrahedron along this direction are identified with the weighted directed areas of the central configuration. The central configuration corresponding to these weighted directed areas is computed giving rise to four masses and corresponding distances. The given masses are compared with the computed ones. The two angles of the rotation are tuned until the given masses coincide with the computed. The corresponding distances of this last computation determine the central configuration. The case with two equal masses also is considered.