Polygon triangulation

Abstract: An algorithm is described for obtaining an optimal approximation, using triangulation, of a three-dimensional surface defined by randomly distributed points along contour lines. The combinatorial problem of finding the best arrangement of triangles is treated by assuming an adequate objective function. The optimal triangulation is found using classical methods of graph theory. An illustrative example gives the procedure for triangulation of contour lines of a human head for use in radiation therapy planning.

Abstract: Given a set of n vertices in the plane together with a set of noncrossing edges, the constrained Delaunay triangulation (CDT) is the triangulation of the vertices with the following properties: (1) the prespecified edges are included in the triangulation, and (2) it is as close as possible to the Delaunay triangulation. We show that the CDT can be built in optimal O(n log n) time using a divide-and-conquer technique. This matches the time required to build an arbitrary (unconstrained) Delaunay triangulation and the time required to build an arbitrary constrained (nonDelaunay) triangulation. CDTs, because of their relationship with Delaunay triangulations, have a number of properties that should make them useful for the finite-element method. Applications also include motion planning in the presence of polygonal obstacles in the plane and constrained Euclidean minimum spanning trees, spanning trees subject to the restriction that some edges are prespecified.

Abstract: In this paper we consider the problem of finding shortest routes from which every point in a given space is visible (watchman routes). We show that the problem is NP-hard when the space is a polygon with holes even if the polygon and the holes are convex or rectilinear. The problem remains NP-hard for simple polyhedra. We present O(n) and O(nlogn) algorithms to find a shortest route in a simple rectilinear monotone polygon and a simple rectilinear polygon respectively, where n is the number of vertices in the polygon. Finding optimum watchman routes in simple polygons is closely related to the problem of finding shortest routes that visit a set of convex polygons in the plane in the presence of obstacles. We show that finding a shortest route that visits a set of convex polygons is NP-hard even when there are no obstacles. We present an O(logn) algorithm to find the shortest route that visits a point and two convex polygons, where n is the total number of vertices.

Abstract: In this paper we present new optimality results for the Delaunay triangulation of a set of points in ?d. These new results are true in all dimensionsd. In particular, we define a power function for a triangulation and show that the Delaunay triangulation minimizes the power function over all triangulations of a point set. We use this result to show that (a) the maximum min-containment radius (the radius of the smallest sphere containing the simplex) of the Delaunay triangulation of a point set in ?d is less than or equal to the maximum min-containment radius of any other triangulation of the point set, (b) the union of circumballs of triangles incident on an interior point in the Delaunay triangulation of a point set lies inside the union of the circumballs of triangles incident on the same point in any other triangulation of the point set, and (c) the weighted sum of squares of the edge lengths is the smallest for Delaunay triangulation, where the weight is the sum of volumes of the triangles incident on the edge. In addition we show that if a triangulation consists of only self-centered triangles (a simplex whose circumcenter falls inside the simplex), then it is the Delaunay triangulation.