Paper detail

Rotors in triangles and tethrahedra

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