Quadtree Decomposition, Steiner Triangulation, and Ray Shooting

  title={Quadtree Decomposition, Steiner Triangulation, and Ray Shooting},
  author={Siu-Wing Cheng and Kam-Hing Lee},
We present a new quadtree-based decomposition of a polygon possibly with holes. For a polygon of n vertices, a truncated decomposition can be computed in O(n log n) time which yields a Steiner triangulation of the interior of the polygon that has O(n log n) size and approximates the minimum weight Steiner triangulation (MWST) to within a constant factor. An approximate MWST is good for ray shooting in the average case as defined by Aronov and Fortune. The untruncated decomposition also yields… 

Approximate minimum weight Steiner triangulation in three dimensions

The minimum weight triangulation problem for p olyhedra and general ob-stacle set in three dimensions is studied and it is shown that minimumweighttriangulation is shown to be a considerable improvement over the O(n6) b ound known for this case.

A ug 2 00 1 Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented. [SIGACT News, 32(3) Issue, 120 Sep. 2001, 63–72.] The computational geometry community has made many

Choosing R-tree or Quadtree Spatial Data Indexing in One Oracle Spatial Database

This paper focuses on the decisions to choose R-tree and quadtree spatial indexing using Oracle spatial database in mobile GIS application and the result can save the time until 42.5%.

Computational geometry column 44

The open problem of whether or not every pair of equal-area polygons has a hinged dissection is discussed.

Choosing R-tree or Quadtree Spatial DataIndexing in One Oracle Spatial Database System to Make Faster Showing Geographical Map in Mobile Geographical Information System Technology

From the research condition, the result of using Quadtree and R-tree spatial data indexing method in one single spatial database can save the time until 42.5%.

A Benchmark and analysis of spatial data structures for physical simulations

This project seeks to compare spatial data structures in systems with uniformly and non-uniformly distributed particles, while varying the number of particles and the filling factor.

Computational geometry column 47

A remarkable theorem is described: "It is possible to tile the plane with non-overlapping squares using exactly one square of each integral dimension." Thus, one can "square the plane."

Computational geometry

A compendium of thirty previously published open problems in computational geometry is presented and three new problems are added to the list.

Study on Mobile Spatial Service Optimization and Its Application in Forest Intelligent Management System

The mobile space data optimization model is constructed in order to maximize the advantages of mobile spatial information service, the mobile space service optimization method such as spatial data multi-scale representation, compression storage, index optimization and progressive transmission is studied, and the model is applied into the forest intelligent management system.



Approximating the minimum weight steiner triangulation

We show that the length of the minimum weight Steiner triangulation (MWST) of a point set can be approximated within a constant factor by a triangulation algorithm based on quadtrees. InO(n logn)

Average-case ray shooting and minimum weight triangulations

A polynomial-time algorithm is given that computes a triangulations compatible with a set of polyhedral obstacles; the area of the triangulation is within a multiplicative constant of the smallest possible.

Query-sensitive ray shooting

A query-sensitive data structure for ray shooting is proposed, which means that the performance of the data structure depends on the local geometry of obstacles near the query segment, and the complexity of theLocal geometry near the segment is measured by a parameter called the simple cover complexity.

Simplified Linear-Time Jordan Sorting and Polygon Clipping

Provably good mesh generation

It is shown how to triangulate a planar point set or a polygonally bounded domain with triangles of bounded aspect ratio, and how to produce a linear-size Delaunay triangulation of a multidimensional point set by adding a linear number of extra points.

Approximation algorithms for multiple-tool miling

An efficient approximation algorithm is presented for the multiple-tool milling problem, where continuous tool movement is possible in one plane and the direction normal to it is used only for retracting the tool.

Computational geometry: an introduction

This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.

Realistic input models for geometric algorithms

The traditional worst-case analysis often fails to predict the actual behavior of the running time of geometric algorithms in practical situations, so models that describe the properties that realistic inputs have are needed so that the analysis can take these properties into account.

Theorem 3. Given a polygon of n vertices, there is an approximate MWST of

  • Theorem 3. Given a polygon of n vertices, there is an approximate MWST of

Approximation algorithms for multipletool milling

  • Proc. 14th ACM Symposium on Computational Geometry
  • 1998