# A Walking Method for Non-Decomposition Intersection and Union of Arbitrary Polygons and Polyhedrons

## Abstract

We present a method for computing the intersection and union of non- convex polyhedrons without decomposition in O(n log n) time, where n is the total number of faces of both polyhedrons. We include an accompanying Python package which addresses many of the practical issues associated with implementation and serves as a proof of concept. The key to the method is that by considering the edges of the original ob- jects and the intersections between faces as walking routes, we can e ciently nd the boundary of the intersection of arbitrary objects using directional walks, thus handling the concave case in a natural manner. The method also easily extends to plane slicing and non-convex polyhedron unions, and both the polyhedron and its constituent faces may be non-convex.

- Authors:

- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1389941

- Report Number(s):
- LLNL-TR-737748

- DOE Contract Number:
- AC52-07NA27344

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; non-convex; non-decomposition; intersection; union; directional walk

### Citation Formats

```
Graham, M., and Yao, J.
```*A Walking Method for Non-Decomposition Intersection and Union of Arbitrary Polygons and Polyhedrons*. United States: N. p., 2017.
Web. doi:10.2172/1389941.

```
Graham, M., & Yao, J.
```*A Walking Method for Non-Decomposition Intersection and Union of Arbitrary Polygons and Polyhedrons*. United States. doi:10.2172/1389941.

```
Graham, M., and Yao, J. Mon .
"A Walking Method for Non-Decomposition Intersection and Union of Arbitrary Polygons and Polyhedrons". United States. doi:10.2172/1389941. https://www.osti.gov/servlets/purl/1389941.
```

```
@article{osti_1389941,
```

title = {A Walking Method for Non-Decomposition Intersection and Union of Arbitrary Polygons and Polyhedrons},

author = {Graham, M. and Yao, J.},

abstractNote = {We present a method for computing the intersection and union of non- convex polyhedrons without decomposition in O(n log n) time, where n is the total number of faces of both polyhedrons. We include an accompanying Python package which addresses many of the practical issues associated with implementation and serves as a proof of concept. The key to the method is that by considering the edges of the original ob- jects and the intersections between faces as walking routes, we can e ciently nd the boundary of the intersection of arbitrary objects using directional walks, thus handling the concave case in a natural manner. The method also easily extends to plane slicing and non-convex polyhedron unions, and both the polyhedron and its constituent faces may be non-convex.},

doi = {10.2172/1389941},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2017},

month = {8}

}