Delaunay triangulations partition a space into regions. Delaunay triangulations can be useful for interpolation and visualizing spatial data.

Qhull is a program which can generate tesselations, convex hulls, and vonoroi diagrams from a set of points. This program is available as a precompiled executable and source code. By interfacing to the command line version of this program, a Delaunay triangulation can be generated.

## Sample CodE

To use this code, download Qhull and copy the ‘qhull.exe’ to the working directory. Make sure the code has read, write, and execute privileges in the working directory. Make sure there are not files named ‘data.txt’ or ‘results.txt’ which need to be preserved.

''' This module generates 2D tesselations for a set of points. The vornoi cells can be filtered by supplying a value associated with each node. ''' __author__ = 'Ed Tate' __email__ = 'edtate-at-gmail-dot-com' def delaunay2D(xpt,ypt,cpt=None,threshold=0): if cpt is None: cpt = [-1 for x in xpt] # write the data file pts_filename = 'data.txt' pts_F = open(pts_filename,'w') #print pts_F pts_F.write('2 # this is a 2-D input set\n') pts_F.write('%i # number of points\n' % len(xpt)) for i,(x,y) in enumerate(zip(xpt,ypt)): pts_F.write('%f %f # data point %i\n' % (x,y,i)) pts_F.close() # trigger the shell command import subprocess p = subprocess.Popen('qhull TI data.txt TO results.txt d i Qc Qt Qbb', shell=True) p.wait() # open the results file and parse results results = open('results.txt','r') print results # get 'i' results data = results.readline() tri_list = [] numLines = int(data) print numLines for i in range(numLines): # load each triplet of indexes to x,y points data = results.readline() idx1,idx2,idx3,dummy = data.split(' ') idx1 = int(idx1) idx2 = int(idx2) idx3 = int(idx3) tri_list.append([idx1,idx2,idx3]) ################# #this generates a fillable collection of tesselations x_list = [] y_list = [] for t in tri_list: if all([ cpt[t[i]]<threshold for i in range(3)]): short_x_list = [ xpt[t[i]] for i in range(3)] short_y_list = [ ypt[t[i]] for i in range(3)] x_list.extend(short_x_list) x_list.append(xpt[t[0]]) x_list.append(None) y_list.extend(short_y_list) y_list.append(ypt[t[0]]) y_list.append(None) return (x_list,y_list) if __name__=='__main__': import random N=100 xpt = [random.random()-0.5 for i in range(0,N)] ypt = [random.random()-0.5 for i in range(0,N)] cpt = [random.random()-0.5 for i in range(0,N)] import matplotlib.pyplot as pp pp.figure() (x_list,y_list) = delaunay2D(xpt,ypt,cpt) pp.plot(xpt,ypt,'b.') pp.plot(x_list,y_list,'k-') pp.fill(x_list,y_list,'g',alpha=0.25,edgecolor='none') pp.show()

When run, this module will produce a diagram with the cells with filled triangulation where the vertices have a random value greater than 0.

## Test Conditions

## Answers

- How to tesselate a set of points
- How to generate a Delaunay Triangulation
- How to use Qhull with Python

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