Determining a Regular n-Gon from Points on Distinct Sides

Abstract

A convex regular n-gon has four Euclidean degrees of freedom. Four generic distinguishable points in the plane lie on four distinct supporting side-lines of exactly (n − 1)(n − 2)(n − 3) geometric regular n-gons, one per injective assignment of the points to sides modulo a common cyclic shift; the case n = 4 recovers the classical six squares. This count uses infinite supporting lines and is established before any compact-side restriction. We then impose the stricter condition that each sample lie in the relative interior of a distinct side. If side labels are given, three points never determine the n-gon, while four points do so uniquely unless the vector of signed side-offsets lies in a fixed three-dimensional space; midpoint and equal-offset samples are degenerate for every sample size, including n. If labels are unknown, four unlabeled interior samples fail to determine the n-gon on a nonempty open set when n ≥ 5, while five determine it outside a lower-dimensional locus. No sample of size at most n forces uniqueness in the worst case: the vertices of a regular n-gon lie one-per-side on a one-parameter family of concentric regular n-gons. MSC 2020: 51M20, 52A10, 51M04, 68U05. The archive regular_ngon_edge_points_v1.zip contains the LaTeX source, both PDFs, figures, an independent reconstruction implementation, and machine-checkable verification data.