  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 2-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 $x\mapsto -4x$ mod 1. We also
present the results of an experimental study in the case of hexagons.
