2026/07/15 by Georgii Khoruzhevskii
#math.GM
We consider corner-cutting schemes for closed oriented polygons in the plane. At each step every vertex is replaced by two points on the adjacent edges; the same cutting coefficient is used at all vertices of the current polygon, but this coefficient may vary from step to step. For such schemes we prove the existence of a continuous limit curve and derive an explicit formula for its oriented area. The main result separates two contributions: the geometry of the initial contour enters through its oriented area and a local sum of determinants built from neighboring edges, while the entire dependence on the refinement rule is contained in a single numerical constant. We also discuss the geometric meaning of this constant as an area arising in the standard angle. As applications we consider stationary de Rham curves, the Chaikin scheme, the connection with Archimedes' quadrature of the parabola, and special sequences of coefficients leading to a circle and to an arc of a hyperbola; these give corresponding series for π and log 2.