Computing simple circuits from a set of line segments

We address the problem of connecting line segments to form the boundary of a simple polygon-a simple circuit. However, not every set of segments can be so connected. We present an O(n log n)-time algorithm to determine whether a set of segments, constrained so that each segment has at least one endpoint on the boundary of the convex hull of the segments, admits a simple circuit. Furthermore, this technique can also be used to compute a simple circuit of minimum perimeter, or a simple circuit that bounds the minimum area, with no increase in computational complexity.