Rotors in triangles and tethrahedra

TL;DR

This paper investigates the geometric properties of rotors within triangles and tetrahedra, providing formulas for curvature at contact points and characterizing the normal lines of rotors in tetrahedra as lying on a quadric surface.

Contribution

It introduces a barycentric formula for curvature in triangle rotors and characterizes the normal lines of rotors in tetrahedra as belonging to a ruling of a quadric surface.

Findings

Curvature of the boundary at contact points in triangles is described by a barycentric formula.

Normal lines at contact points in tetrahedral rotors generally belong to one ruling of a quadric surface.

Abstract

A polytope $P$ is circumscribed about a convex body $\Phi\subset \mathbb{R}^n$ if $\Phi\subset P$ and each facet of $P$ is contained in a support hyperplane of $\Phi$. We say that a convex body $\Phi\subset \mathbb{R}^n$ is a rotor of a polytope $P$ if for each rotation $\rho$ of $\mathbb{R}^n$ there exist a translation $\tau$ so that $P$ is circumscribed about $\tau\rho\Phi$. In this paper we shall prove that if $P$ is a triangle, then there is a baricentric formula that describes the curvature of bd$\Phi$ at the contact points, $\{A_1, A_2,A_3\}$. We prove also that if $\Phi\subset \mathbb{R}^3$ is a convex body which is a rotor in a tetrahedron $T$ and if $\Phi$ intersects the faces of $T$ at the points $\{x_1, \dots, x_4\}$, then the normal lines of $\Phi$ at the contact points with $T$, $\{x_1, \dots, x_4\}$ generically belong to one ruling of a quadric surface.