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bundles / scipy 1.17.1 / scipy / spatial / _qhull / Delaunay

class

scipy.spatial._qhull:Delaunay

source: /scipy/spatial/_qhull.cpython-314-x86_64-linux-gnu.so

Signature

class   Delaunay ( points ,  furthest_site = False ,  incremental = False ,  qhull_options = None )

Members

Summary

Delaunay tessellation in N dimensions.

Extended Summary

Parameters

points : ndarray of floats, shape (npoints, ndim)

Coordinates of points to triangulate

furthest_site : bool, optional

Whether to compute a furthest-site Delaunay triangulation. Default: False

incremental : bool, optional

Allow adding new points incrementally. This takes up some additional resources.

qhull_options : str, optional

Additional options to pass to Qhull. See Qhull manual for details. Option "Qt" is always enabled. Default:"Qbb Qc Qz Qx Q12" for ndim > 4 and "Qbb Qc Qz Q12" otherwise. Incremental mode omits "Qz".

Attributes

points : ndarray of double, shape (npoints, ndim)

Coordinates of input points.

simplices : ndarray of ints, shape (nsimplex, ndim+1)

Indices of the points forming the simplices in the triangulation. For 2-D, the points are oriented counterclockwise.

neighbors : ndarray of ints, shape (nsimplex, ndim+1)

Indices of neighbor simplices for each simplex. The kth neighbor is opposite to the kth vertex. For simplices at the boundary, -1 denotes no neighbor.

equations : ndarray of double, shape (nsimplex, ndim+2)

[normal, offset] forming the hyperplane equation of the facet on the paraboloid (see Qhull documentation for more).

paraboloid_scale, paraboloid_shift : float

Scale and shift for the extra paraboloid dimension (see Qhull documentation for more).

transform : ndarray of double, shape (nsimplex, ndim+1, ndim)

Affine transform from x to the barycentric coordinates c. This is defined by 

T c = x - r

At vertex j, c_j = 1 and the other coordinates zero.

For simplex i, transform[i,:ndim,:ndim] contains inverse of the matrix T, and transform[i,ndim,:] contains the vector r.

If the simplex is degenerate or nearly degenerate, its barycentric transform contains NaNs.

vertex_to_simplex : ndarray of int, shape (npoints,)

Lookup array, from a vertex, to some simplex which it is a part of. If qhull option "Qc" was not specified, the list will contain -1 for points that are not vertices of the tessellation.

convex_hull : ndarray of int, shape (nfaces, ndim)

Vertices of facets forming the convex hull of the point set. The array contains the indices of the points belonging to the (N-1)-dimensional facets that form the convex hull of the triangulation.

coplanar : ndarray of int, shape (ncoplanar, 3)

Indices of coplanar points and the corresponding indices of the nearest facet and the nearest vertex. Coplanar points are input points which were not included in the triangulation due to numerical precision issues.

If option "Qc" is not specified, this list is not computed.

vertex_neighbor_vertices : tuple of two ndarrays of int; (indptr, indices)

Neighboring vertices of vertices. The indices of neighboring vertices of vertex k are indices[indptr[k]:indptr[k+1]].

furthest_site

True if this was a furthest site triangulation and False if not.

Raises

: QhullError

Raised when Qhull encounters an error condition, such as geometrical degeneracy when options to resolve are not enabled.

: ValueError

Raised if an incompatible array is given as input.

Notes

The tessellation is computed using the Qhull library Qhull library.

Examples

Triangulation of a set of points: 
import numpy as np
points = np.array([[0, 0], [0, 1.1], [1, 0], [1, 1]])
from scipy.spatial import Delaunay
tri = Delaunay(points)
✓
We can plot it: 
import matplotlib.pyplot as plt
✓
plt.triplot(points[:,0], points[:,1], tri.simplices)
plt.plot(points[:,0], points[:,1], 'o')
✗
plt.show()
✓
fig-cbfad6cb9e7eb4c3.png
Point indices and coordinates for the two triangles forming the triangulation: 
tri.simplices
✗
Note that depending on how rounding errors go, the simplices may be in a different order than above. 
points[tri.simplices]
✗
Triangle 0 is the only neighbor of triangle 1, and it's opposite to vertex 1 of triangle 1: 
tri.neighbors[1]
✓
points[tri.simplices[1,1]]
✗
We can find out which triangle points are in: 
p = np.array([(0.1, 0.2), (1.5, 0.5), (0.5, 1.05)])
✓
tri.find_simplex(p)
✗
The returned integers in the array are the indices of the simplex the corresponding point is in. If -1 is returned, the point is in no simplex. Be aware that the shortcut in the following example only works correctly for valid points as invalid points result in -1 which is itself a valid index for the last simplex in the list. 
p_valids = np.array([(0.1, 0.2), (0.5, 1.05)])
✓
tri.simplices[tri.find_simplex(p_valids)]
✗
We can also compute barycentric coordinates in triangle 1 for these points: 
b = tri.transform[1,:2].dot(np.transpose(p - tri.transform[1,2]))
✓
np.c_[np.transpose(b), 1 - b.sum(axis=0)]
✗
The coordinates for the first point are all positive, meaning it is indeed inside the triangle. The third point is on an edge, hence its null third coordinate.

Aliases

  • scipy.spatial.Delaunay

Referenced by

This package