Homotopy equivalence between Voronoi medusa and Delaunay medusa

TL;DR

This paper proves that the Voronoi medusa and Delaunay medusa, two space-time structures derived from moving points, are homotopically equivalent, linking their topological properties over time.

Contribution

The paper establishes a homotopy equivalence between Voronoi medusa and Delaunay medusa, extending classical spatial duality into a space-time context.

Findings

Voronoi medusa and Delaunay medusa are homotopic.

The structures form continuous four-dimensional entities.

Homotopy equivalence links their topological features.

Abstract

We trace movements of certain points in space-time along their corresponding continuous path. We partition the space at every moment of time using alpha-Complexes, Voronoi medusa is then the collection or union of restricted Voronoi cells at every moment in time. We can imagine them as a four dimensional structure formed when three dimensional restricted Voronoi cells sweeps continuously through the extra dimension of time. Similarly Delaunay medusa is the collection of the corresponding Delaunay triangulations at each moment in time. In this article we prove that these two structures are homotopic.