# Probabilistic properties of the elliptic motion

## TL;DR

This paper analyzes the probabilistic properties of ellipses generated by random points within a disk attached to a moving circle in an elliptic motion, deriving their moments and distributions.

## Contribution

It introduces a novel probabilistic analysis of the geometric properties of ellipses generated in elliptic motion, including distribution and moments of area and perimeter.

## Key findings

- Derived the distribution of ellipse area and perimeter for random points.
- Calculated moments of these geometric variables.
- Provided probabilistic descriptions for the elliptic motion scenario.

## Abstract

In this paper we consider the plane elliptic motion which occurs if the moving centrode is a circle of radius $r$ and the fixed centrode a circle of radius $2r$. Every point of the moving plane generates an ellipse in the fixed plane. Let a disk of radius $R$, $0 \le R < \infty$, concentric to the moving centrode be attached to the moving plane. If a point $P$ is chosen at random from this disk, then the area and the perimeter of the ellipse generated by $P$ are random variables. We determine the moments and the distributions of these random variables for the case that $P$ is uniformly distributed over the area of the disk.