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# Spatial algorithms and data structures (`scipy.spatial`)

## Spatial transformations

These are contained in the `scipy.spatial.transform` submodule.

## Nearest-neighbor queries

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

## Distance metrics

Distance metrics are contained in the `scipy.spatial.distance` submodule.

## Delaunay triangulation, convex hulls, and Voronoi diagrams

`Delaunay`(points[, furthest_site, ...]) | Delaunay tessellation 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, etc.) appearing in the Delaunay tessellation (N-D simplices), convex hull facets, and Voronoi ridges (N-1-D simplices) are represented in the following scheme:

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

    # coordinates of the jth vertex of the ith simplex
    tess.points[tess.simplices[i, j], :]        # tessellation 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 ith 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-D paraboloid.

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

## Miscellaneous 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 pth power of the L**p distance between two arrays.
`procrustes`(data1, data2) | Procrustes analysis, a similarity test for two data sets.
`geometric_slerp`(start, end, t[, tol]) | Geometric spherical linear interpolation.

## Warnings / Errors used in `scipy.spatial`

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