Showing papers on "Constrained Delaunay triangulation published in 1987"

A faster divide-and-conquer algorithm for constructing delaunay triangulations

Abstract: An easily implemented modification to the divide-and-conquer algorithm for computing the Delaunay triangulation ofn sites in the plane is presented. The change reduces its ź(n logn) expected running time toO(n log logn) for a large class of distributions that includes the uniform distribution in the unit square. Experimental evidence presented demonstrates that the modified algorithm performs very well forn≤216, the range of the experiments. It is conjectured that the average number of edges it creates--a good measure of its efficiency--is no more than twice optimal forn less than seven trillion. The improvement is shown to extend to the computation of the Delaunay triangulation in theLp metric for 1

Constrained Delaunay triangulations

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.

An optimal algorithm for constructing the Delaunay triangulation of a set of line segments

Abstract: In this paper, we first define a new Voronoi diagram for the endpoints of a set of line segments in the plane which do not intersect (except possibly at their endpoints), which is called a bounded Voronoi diagram. In this Voronoi diagram, the line segments themselves are regarded as obstacles. We present an optimal T(n log n) algorithm to construct it, where n is the number of input line segments.We then show that the straight-line dual of the bounded Voronoi diagram of a set of non-intersecting line segments is the Delaunay triangulation of that set. And the straight-line dual can be obtained in time proportional to the number of input line segments when the corresponding bounded Voronoi diagram is available. Consequently, we obtain an optimal T(n log n) algorithm to construct the Delaunay triangulation of a set of n non-intersecting line segments in the plane. Our algorithm improves the time bound O(n2) of the previous best algorithm.

Spatially ordered networks and topographic reconstructions

Abstract: A technique is discussed for obtaining a contour tree efficiently as a byproduct of an operational contouring system. This tree may then be used to obtain contour symbolism or interval statistics as well as for further geomorphological study. Alternatively, the tree may be obtained without the computational expense of detailed contour interpolation. The contouring system proceeds by assuming a Voronoi neighbourhood or domain about each data point and generating a dual-graph Delaunay triangulation accordingly. Since a triangulation may be traversed in a tree order, individual triangles may be processed in a guaranteed top-to-bottom sequence on the map. At the active edge of the map under construction a linked list is maintained of the contour ‘stubs’ available to be updated by the next triangle processed. Any new contour segment may extend an existing stub, open two new stubs or close (connect) two previous stubs. Extending this list of edge links backwards into the existing map permits storage of...

On Approximation Behavior of the Greedy Triangulation for Convex Polygons

Abstract: We prove that the greedy triangulation heuristic for minimum weight triangulation of convex polygons yields solutions within a constant factor from the optimum. For interesting classes of convex polygons, we derive small upper bounds on the constant approximation factor. Our results contrast with Kirkpatrick's Ω(n) bound on the approximation factor of the Delaunay triangulation heuristic for minimum weight triangulation of convexn-vertex polygons. On the other hand, we present a straightforward implementation of the greedy triangulation heuristic for ann-vertex convex point set or a convex polygon takingO(n 2) time andO(n) space. To derive the latter result, we show that given a convex polygonP, one can find for all verticesv ofP a shortest diagonal ofP incident tov in linear time. Finally, we observe that the greedy triangulation for convex polygons having so-called semicircular property can be constructed in timeO(n logn).