We study a geometrically defined return map associated with domains in the class $\mathcal O_C$, where a fixed convex core $C\subset\mathbb R^N$ is linked to the outer boundary $\partial\Omega$ through outward normal rays and a thickness function $d:\partial C\to(0,\infty)$. The radial map $\Phi(c)=c+d(c)\nu(c)$ is combined with a reciprocal map that follows the inward normal to $\partial\Omega$ back to its first intersection with the convex core. Their composition $F=\pi\circ\Phi:\partial C\to\partial C$ defines a geometrically generated discrete dynamical system on $\partial C$. Using the exact normal geometry of the radial parametrization, we derive the local quadratic-remainder formula $... More >