Discrete Conics as Distinguished Projective Images of Regular Polygons

摘要

In this paper, we introduce discrete conics, polygonal analogs of conics, as distinguished projective images of regular polygons. We show that discrete conics satisfy a number of nice properties. These include optical (billiard) and isogonal properties analogous to those of conics, the property of being Poncelet polygons with the two associated conics sharing a focus, and a nice property similar to Pascal's Hexagrammum Mysticum Theorem. We conclude by showing that discrete conics arise naturally from two constructions: a discrete negative pedal construction and via a group associated to focus-sharing pencils of conics.