TY - JOUR

T1 - On Projective Evolutes of Polygons

AU - Arnold, Maxim

AU - Schwartz, Richard Evan

AU - Tabachnikov, Serge

N1 - Publisher Copyright:
© 2022 Taylor & Francis Group, LLC.

PY - 2024

Y1 - 2024

N2 - The evolute of a curve is the envelope of its normals. In this note we consider a projectively natural discrete analog of this construction: we define projective perpendicular bisectors of the sides of a polygon in the projective plane, and study the map that sends a polygon to the new polygon formed by the projective perpendicular bisectors of its sides. We consider this map acting on the moduli space of projective polygons. We analyze the case of pentagons; the moduli space is two-dimensional in this case. The second iteration of the map has one integral whose level curves are cubic curves, and the transformation on these level curves is conjugated to the map (Formula presented.) mod 1. We also present the results of an experimental study in the case of hexagons.

AB - The evolute of a curve is the envelope of its normals. In this note we consider a projectively natural discrete analog of this construction: we define projective perpendicular bisectors of the sides of a polygon in the projective plane, and study the map that sends a polygon to the new polygon formed by the projective perpendicular bisectors of its sides. We consider this map acting on the moduli space of projective polygons. We analyze the case of pentagons; the moduli space is two-dimensional in this case. The second iteration of the map has one integral whose level curves are cubic curves, and the transformation on these level curves is conjugated to the map (Formula presented.) mod 1. We also present the results of an experimental study in the case of hexagons.

UR - http://www.scopus.com/inward/record.url?scp=85135194953&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85135194953&partnerID=8YFLogxK

U2 - 10.1080/10586458.2022.2102095

DO - 10.1080/10586458.2022.2102095

M3 - Article

AN - SCOPUS:85135194953

SN - 1058-6458

VL - 33

SP - 347

EP - 356

JO - Experimental Mathematics

JF - Experimental Mathematics

IS - 3

ER -