# Stiefel manifolds and polygons

## TL;DR

This paper explores the use of Stiefel manifolds as a parameter space for understanding, generating, and transforming random polygons by leveraging their geometric properties.

## Contribution

It introduces a novel interpretation of polygons within Stiefel manifolds, enabling new methods for random polygon generation and morphing.

## Key findings

- Stiefel manifolds effectively model polygon parameter spaces.
- The geometry of Stiefel manifolds facilitates polygon generation.
- Methods for morphing polygons using manifold geometry are demonstrated.

## Abstract

Polygons are compound geometric objects, but when trying to understand the expected behavior of a large collection of random polygons -- or even to formalize what a random polygon is -- it is convenient to interpret each polygon as a point in some parameter space, essentially trading the complexity of the object for the complexity of the space. In this paper I describe such an interpretation where the parameter space is a Stiefel manifold and show how to exploit the geometry of the Stiefel manifold both to generate random polygons and to morph one polygon into another.