On the number of sides necessary for polygonal approximation of black-and-white figures in a plane

Abstract

A bound on the number of extreme points or sides necessary to approximate a convex planar figure by an enclosing polygon is described. This number is found to be proportional to the fourth root of the figure's area divided by the square of a maximum Euclidean distance approximation parameter. An extension of this bound, preserving its fourth root quality, is made to general planar figures. This is done by decomposing the general figure into nearly convex sets defined by inflection points, cusps, and multiple windings. A procedure for performing actual encoding of this type is described. Comparisons of parsimony are made with contemporary figure encoding schemes.