Spatial algorithms and data structures (scipy.spatial)

Nearest-neighbor Queries

KDTree(data[, leafsize])	kd-tree for quick nearest-neighbor lookup
cKDTree	kd-tree for quick nearest-neighbor lookup
distance
Rectangle(maxes, mins)	Hyperrectangle class.

Delaunay Triangulation, Convex Hulls and Voronoi Diagrams

Delaunay(points[, furthest_site, ...])	Delaunay tesselation in N dimensions.
ConvexHull(points[, incremental, qhull_options])	Convex hulls in N dimensions.
Voronoi(points[, furthest_site, ...])	Voronoi diagrams in N dimensions.
SphericalVoronoi(points[, radius, center])	Voronoi diagrams on the surface of a sphere.
HalfspaceIntersection(halfspaces, interior_point)	Halfspace intersections in N dimensions.

Plotting Helpers

delaunay_plot_2d(tri[, ax])	Plot the given Delaunay triangulation in 2-D
convex_hull_plot_2d(hull[, ax])	Plot the given convex hull diagram in 2-D
voronoi_plot_2d(vor[, ax])	Plot the given Voronoi diagram in 2-D

See also

Tutorial

Simplex representation

The simplices (triangles, tetrahedra, ...) appearing in the Delaunay tesselation (N-dim simplices), convex hull facets, and Voronoi ridges (N-1 dim simplices) are represented in the following scheme:

tess = Delaunay(points)
hull = ConvexHull(points)
voro = Voronoi(points)

# coordinates of the j-th vertex of the i-th simplex
tess.points[tess.simplices[i, j], :]        # tesselation element
hull.points[hull.simplices[i, j], :]        # convex hull facet
voro.vertices[voro.ridge_vertices[i, j], :] # ridge between Voronoi cells

For Delaunay triangulations and convex hulls, the neighborhood structure of the simplices satisfies the condition:

tess.neighbors[i,j] is the neighboring simplex of the i-th simplex, opposite to the j-vertex. It is -1 in case of no neighbor.

Convex hull facets also define a hyperplane equation:

(hull.equations[i,:-1] * coord).sum() + hull.equations[i,-1] == 0

Similar hyperplane equations for the Delaunay triangulation correspond to the convex hull facets on the corresponding N+1 dimensional paraboloid.

The Delaunay triangulation objects offer a method for locating the simplex containing a given point, and barycentric coordinate computations.

Functions

tsearch(tri, xi)	Find simplices containing the given points.
distance_matrix(x, y[, p, threshold])	Compute the distance matrix.
minkowski_distance(x, y[, p])	Compute the L**p distance between two arrays.
minkowski_distance_p(x, y[, p])	Compute the p-th power of the L**p distance between two arrays.
procrustes(data1, data2)	Procrustes analysis, a similarity test for two data sets.