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 constrained Delaunay triangulations, triangulation of non-convex surfaces, mesh generation of non-convex objects, or medium-sized inputs in 9-D and higher.

Introduction

• Fukuda's introduction to convex hulls, Delaunay triangulations, Voronoi diagrams, and linear programming
• Lambert's Java visualization of convex hull algorithms
• 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

  Programs and Options qconvex -- convex hull qdelaunay -- Delaunay triangulation qvoronoi -- Voronoi diagram qhalf -- halfspace intersection about a point rbox -- generate point distributions

  Description of Qhull COPYING.txt - copyright notice REGISTER.txt - registration README.txt - installation instructions Changes.txt - change history Qhull functions, macros, and data structures with source

• Frequently asked questions about Qhull
• 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,
• 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
• Gartner's Miniball for fast and robust smallest enclosing balls (up to 20-d)
• Google's directory for Science > Math > Geometry > Computational Geometry > Software
• Leda and CGAL libraries for writing computational geometry programs and other combinatorial algorithms
• Magic Software source code for computer graphics, image analysis, and numerical methods
• Mathtools.net of scientific and engineering software
• Owen's Meshing Research Corner
• Schneiders' Finite Element Mesh Generation page
• Shewchuk's triangle program for 2-d Delaunay
• Skorobohatyj's Mathprog@ZIB for mathematical software
• Voronoi Web Site for all things Voronoi
• Zolotykh's Skeleton generates all extreme rays of a polyhedral cone using the Double Description Method

FAQs and Newsgroups

• FAQ for computer graphics algorithms (geometric structures)
• 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 CiteSeer or www.acm.org:

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 ---