Center of Mass Technique and Affine Geometry

TL;DR

This paper extends the concept of center of mass in geometry by introducing a new framework that includes zero total mass sets, using affine geometry and weighty points to enhance geometric analysis.

Contribution

It introduces a novel space of weighty points and mass dipoles in affine space, overcoming the zero total mass limitation and providing new geometric interpretations and applications.

Findings

Defined a (n+1)-dimensional vector space of weighty points and mass dipoles.

Connected the space to the classical center of mass concept.

Provided applications to geometric problems.

Abstract

The notion of center of mass, which is very useful in kinematics, proves to be very handy in geometry (see [1]-[2]). Countless applications of center of mass to geometry go back to Archimedes. Unfortunately, the center of mass cannot be defined for sets whose total mass equals zero. In the paper we improve this disadvantage and assign to an n-dimensional affine space L over any field k the (n+1)-dimensional vector space over the field k of weighty points and mass dipoles in L. In this space, the sum of weighted points with nonzero total mass is equal to the center of mass of these points equipped with their total mass. We present several interpretations of the space of weighty points and mass dipoles in L, and a couple of its applications to geometry. The paper is self-contained and is accessible for undergraduate students.