Isoptic surfaces of polyhedra

TL;DR

This paper develops an algorithm to compute and visualize isoptic surfaces of 3D polyhedra, extending the concept from 2D curves to 3D hypersurfaces, with applications to Platonic and Archimedean solids.

Contribution

It introduces a method to determine isoptic surfaces of 3D polyhedra, expanding the theory from 2D curves to 3D hypersurfaces and providing visualizations.

Findings

Computed isoptic surfaces for Platonic solids

Extended isoptic theory to 3D polyhedra

Visualizations created with Wolfram Mathematica

Abstract

The theory of the isoptic curves is widely studied in the Euclidean plane $\bE^2$ (see \cite{CMM91} and \cite{Wi} and the references given there). The analogous question was investigated by the authors in the hyperbolic $\bH^2$ and elliptic $\cE^2$ planes (see \cite{CsSz1}, \cite{CsSz2}, \cite{CsSz5}), but in the higher dimensional spaces there are only a few result in this topic. In \cite{CsSz4} we gave a natural extension of the notion of the isoptic curves to the $n$-dimensional Euclidean space $\bE^n$ $(n\ge 3)$ which are called isoptic hypersurfaces. Now we develope an algorithm to determine the isoptic surface $\mathcal{H}_{\cP}$ of a $3$-dimensional polytop $\mathcal{P}$. We will determine the isoptic surfaces for Platonic solids and for some semi-regular Archimedean polytopes and visualize them with Wolfram Mathematica.