Rotors in triangles and tethrahedra

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