Polygonizations of point sets in the plane

Abstract

We examine the different ways a set of n points in the plane can be connected to form a simple polygon. Such a connection is called a polygonization of the points. For some point sets the number of polygonizations is exponential in the number of points. For this reason we restrict our attention to star-shaped polygons whose kernels have nonempty interiors; these are called nondegenerate star-shaped polygons. We develop an algorithm and data structure for determining the nondegenerate star-shaped polygonizations of a set of n points in the plane. We do this by first constructing an arrangement of line segments from the point set. The regions in the arrangement correspond to the kernels of the nondegenerate star-shaped polygons whose vertices are the original n points. To obtain the data structure representing this arrangement, we show how to modify data structures for arrangements of lines in the plane. This data structure can be computed in O(n4) time and space. By visiting the regions in this data structure in a carefully chosen order, we can compute the polygon associated with each region in O(n) time, yielding a total computation time of O(n5) to compute a complete list of O(n4) nondegenerate star-shaped polygonizations of the set of n points. © 1988 Springer-Verlag New York Inc.