Optimal Scaling in Double-Contact Regular Polygon Containment

Double-Contact Polygon Scaling

Özet

This study presents a detailed derivation of a trigonometric identity governing the optimal scaling of a regular m-gon inscribed within a regular n-gon underdouble-contact constraints. Building on prior work that established containment inequalities for nested polygons in the complex plane, we focus on the symmetric configuration where rotational and vertical translation components vanish (b = 0, d = 0). In this setting, we derive a closed-form expression for the scaling factor c by equating two distinct contact conditions involving edge-vertex interactions. The resulting identity incorporates cosine and cotangent terms and reveals how geometric symmetry leads to algebraic simplification. We also provide a long-form factorization and numerical examples to illustrate the identity’s behavior across different polygon pairs. This work contributes to the broader theory of polygonal optimization and symbolic encoding in geometric configurations.