Convex Hull

Consider a set $S$ of $n$ points fixed in $d$-dimensional space. The Convex Hull of this set is the largest convex polytope whose vertices are drawn from $S$.

Directory Structure

The directory structure is detailed below. Test cases will be added in the near future.

├── convex-hull
│   ├── qhull
│   │   ├── __init__.py
│   │   ├── qhull.py
│   │   ├── qhull2d.py
│   ├── tests
│   │   ├── __init__.py
├── LICENSE
├── README.md
└── .gitignore

Example Usage

import numpy as np
from qhull import qhull

points = np.random.rand(100, 2)
qhull(points, tolerance=1e-10, verbose=True)

The output for qhull2d.py is a np.ndarray of points. Each pair of points is an edge in the Convex Hull.

> qhull2d(points, tolerance=1e-10, verbose=True)
array([[0.07103606, 0.0871293 ],
       [0.0202184 , 0.83261985],
       [0.77815675, 0.87001215],
       [0.56804456, 0.92559664],
       [0.0202184 , 0.83261985],
       [0.56804456, 0.92559664],
       [0.96366276, 0.38344152],
       [0.77815675, 0.87001215],
       [0.96366276, 0.38344152],
       [0.07103606, 0.0871293 ],
       [0.96366276, 0.38344152]])

The output for qhull.py is a set of Facet objects. You may directly access Facet.coordinates and Facet.normal to retrieve the coordinates and normal vector defining the orientation of the facet.

> facets = qhull(points, tolerance=1e-10, verbose=True)
> facets
{<__main__.Facet at 0x7ff6506205e0>,
 <__main__.Facet at 0x7ff650654d90>,
 <__main__.Facet at 0x7ff6802ed6d0>,
 <__main__.Facet at 0x7ff6802f9e80>,
 <__main__.Facet at 0x7ff690850f10>,
 <__main__.Facet at 0x7ff69085c040>,
 <__main__.Facet at 0x7ff69090d160>,
 <__main__.Facet at 0x7ff690994a30>,
 <__main__.Facet at 0x7ff690994a60>,
 <__main__.Facet at 0x7ff6a257ba60>}

> facets.pop().coordinates
array([[0.0871293 , 0.0202184 , 0.83261985],
       [0.94466892, 0.52184832, 0.41466194],
       [0.11827443, 0.63992102, 0.14335329]])

If verbose is set to True then each intermediate state of the hull is shown.

2 Dimensional Case	3 Dimensional Case

Dependencies

This project uses NumPy for numerical computations and matplotlib for visualization.

Performance

Dim	Number of Points	qhull Average Time	ConvexHull Average Time
2	10	0.000474	0.000350
2	100	0.000949	0.000384
2	1000	0.001692	0.000479
2	10000	0.003080	0.001744

Dim	Number of Points	qhull Average Time	ConvexHull Average Time
3	10	0.002348	0.000334
3	100	0.014820	0.000430
3	1000	0.055613	0.000902
3	10000	0.291249	0.003224

Notes

The 2D case accepts inputs in all dimensions but the output is not guranteed to be the Convex Hull. The 3D case does not perform DFS in CCW order. The implementation is not robust and suffers from numerical precision issues arising from coplanar facets. I do not have plans to write proper documentation, but you can visit my blog post on this project for theoretic details.

The following resources were invaluable to the development:

• https://en.wikipedia.org/wiki/Quickhull
• https://algolist.ru/maths/geom/convhull/qhull3d.php
• https://media.steampowered.com/apps/valve/2014/DirkGregorius_ImplementingQuickHull.pdf