# Quadtree Decomposition, Steiner Triangulation, and Ray Shooting

@inproceedings{Cheng1998QuadtreeDS, title={Quadtree Decomposition, Steiner Triangulation, and Ray Shooting}, author={Siu-Wing Cheng and Kam-Hing Lee}, booktitle={ISAAC}, year={1998} }

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…

## 11 Citations

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## References

SHOWING 1-10 OF 12 REFERENCES

### Approximating the minimum weight steiner triangulation

- Computer ScienceDiscret. Comput. Geom.
- 1994

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

- MathematicsSCG '97
- 1997

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

- Computer ScienceSCG '94
- 1994

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

- Computer ScienceInf. Process. Lett.
- 1990

### Provably good mesh generation

- Computer ScienceProceedings [1990] 31st Annual Symposium on Foundations of Computer Science
- 1990

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

- Materials ScienceSCG '98
- 1998

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

- Physics
- 1985

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

- Computer ScienceSCG '97
- 1997

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