Finding the Convex Hull of a Simple Polygon

The problem of finding the convex hull of a planar set of points P, that is, finding the smallest convex region enclosing P, arises frequently in computer graphics. For example, to fit P into a square or a circle, it is necessary and sufficient that H(P), the convex hull of P, fits; and since it is usually the case that H(P) has many fewer points than P has, it is a simpler object to manipulate. It is also the case that many fast graphics algorithms on polygons require that the input polygon be convex, thus making it a useful preprocessing step sometimes to first transform a general polygon into its convex hull. A number of algorithms exist for finding the convex hull of a set of points (e.g., [l, 2,6]), with worst-case complexity O(nlog n) for 1 PI = n. It is also known that O( n log n) is a lower bound just for determining H( P)-that is, not necessarily rendering H(P) in, say, clockwise order [7]. This lower bound is proved for a decision tree model with quadratic tests, which accommodates all the known convex hull algorithms. An interesting case of the convex hull problem that occurs frequently in practice is when the points of P form the vertices of a simple polygon (i.e., a polygon without self-intersections). Several authors have tried to find a fast algorithm for this problem. Sklansky [5] proposed an O(n) algorithm, which Bykat [l] later showed does not always work. It has also been noted that the