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# Qhull

Qhull computes the convex hull, Delaunay triangulation, Voronoi diagram, halfspace intersection about a point, furthest-site Delaunay triangulation, and furthest-site Voronoi diagram. The source code runs in 2-d, 3-d, 4-d, and higher dimensions. Qhull implements the Quickhull algorithm for computing the convex hull. It handles roundoff errors from floating point arithmetic. It computes volumes, surface areas, and approximations to the convex hull.

Qhull does not support triangulation of non-convex surfaces, mesh generation of non-convex objects, medium-sized inputs in 9-D and higher, alpha shapes, weighted Voronoi diagrams, Voronoi volumes, or constrained Delaunay triangulations,

If you call Qhull from your program, please use reentrant Qhull (libqhull_r or libqhullstatic_r). If you use Qhull 2003.1, please upgrade or apply poly.c-qh_gethash.patch.

- News and Bugs about Qhull 2020.2 2020/08/31 (8.0.2)
- Download Qhull (changes)
- GitHub C++ interface to Qhull
- Introduction and manual for Qhull and rbox with a FAQ
- Geomview for 3-D and 4-D visualization of Qhull output
- www.qhull.org

- How is Qhull used?
- Google Scholar, and CiteSeer references to Qhull
- Google Qhull, Images, Books, Patents, Newsgroups, and Who is Qhull?

- MATLAB uses Qhull for their n-d computational geometry functions: convhulln delaunayn griddatan voronoin.
- The geometry package of R provides Qhull in R.
- GNU Octave includes Qhull for computational geometry.
- Mathematica's Delaunay interface qh-math.
- The scipy.spatial of SciPy is implemented with Qhull. It includes support for Delaunay triangulations.

Introduction

- Gregorius' talk on implementing Quickhull, Lloyd's QuickHull3D in Java, and Newbold's Waterman polyhedra
- Fukuda's introduction to convex hulls, Delaunay triangulations, Voronoi diagrams, and linear programming
- LEDA Guide to geometry algorithms
- MathWorld's Computational Geometry from Wolfram Research
- Skiena's Computational Geometry from his Algorithm Design Manual.
- Stony Brook Algorithm Repository, computational geometry

Qhull Documentation and Support

- Manual for Qhull and rbox

  - Quick reference to Programs and Options
  - When to use Qhull
  - Description of Qhull
  - Imprecision in Qhull
  - qconvex -- convex hull
  - qdelaunay -- Delaunay triangulation
  - qvoronoi -- Voronoi diagram
  - qhalf -- halfspace intersection about a point
  - rbox -- generate point distributions

  - README.txt - installation instructions
  - COPYING.txt - copyright notice
  - REGISTER.txt - registration
  - Geomview for visualizing Qhull
  - FAQ - frequently asked questions about Qhull
  - Changes.txt - change history
  - Qhull code
  - Performance of Qhull
  - Index to Qhull Functions

- Qhull build systems
- Send e-mail to qhull@qhull.org
- Report bugs to qhull_bug@qhull.org

Related URLs

- Amenta's directory of computational geometry software
- BGL Boost Graph Library provides C++ classes for graph data structures and algorithms,
- CGAL and Leda libraries for writing computational geometry programs and other combinatorial algorithms
- Clarkson's hull program with exact arithmetic for convex hulls, Delaunay triangulations, Voronoi volumes, and alpha shapes.
- Erickson's Computational Geometry Pages and Software
- Fukuda's cdd program for halfspace intersection and convex hulls (Polco/Java)
- Gartner's Miniball for fast and robust smallest enclosing balls (up to 20-d)
- Mathtools.net of scientific and engineering software
- Owen's International Meshing Roundtable
- Schneiders' Finite Element Mesh Generation page
- Shewchuk's triangle program for 2-d Delaunay
- Teillaud's Computional Geometry Pages for the Society of Computational Geometry and academic conferences.
- Tomilov's quickhull.hpp (doc-ru) implements the Quickhull algorithm for points in general position.
- Voronoi Web Site for all things Voronoi
- Young's Internet Finite Element Resources
- Zolotykh's Skeleton generates all extreme rays of a polyhedral cone using the Double Description Method

FAQs and Newsgroups

- FAQ for computer graphics algorithms
- FAQ for linear programming
- Newsgroup: comp.graphics.algorithms
- Newsgroup: comp.soft-sys.matlab
- Newsgroup: sci.math.num-analysis
- Newsgroup: sci.op-research

The program includes options for input transformations, randomization, tracing, multiple output formats, and execution statistics. The program can be called from within your application.

You can view the results in 2-d, 3-d and 4-d with Geomview. An alternative is VTK.

For an article about Qhull, download from ACM or CiteSeer:

> Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T., "The Quickhull algorithm for convex hulls," ACM Trans. on Mathematical Software, 22(4):469-483, Dec 1996, http://www.qhull.org

Abstract:

> The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull Algorithm with the general dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains non-extreme points, and that it uses less memory.

> Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating point arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of "thick" facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.

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Created: May 17 1995 ---