Classe « Delaunay »

Informations générales

Héritage

builtins.object
    _QhullUser
        Delaunay

Définition

class Delaunay(_QhullUser):

help(Delaunay)

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

    Delaunay tessellation in N dimensions.

    .. versionadded:: 0.9

    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

        .. versionadded:: 0.12.0
    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".

        .. versionadded:: 0.12.0

    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 <http://www.qhull.org/>`__ for more).
    paraboloid_scale, paraboloid_shift : float
        Scale and shift for the extra paraboloid dimension
        (see `Qhull documentation <http://www.qhull.org/>`__ 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.

        .. note::

           Computing convex hulls via the Delaunay triangulation is
           inefficient and subject to increased numerical instability.
           Use `ConvexHull` instead.
    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.

        .. versionadded:: 0.12.0
    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.

        .. versionadded:: 1.4.0

    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 <http://www.qhull.org/>`__.

    .. note::

       Unless you pass in the Qhull option "QJ", Qhull does not
       guarantee that each input point appears as a vertex in the
       Delaunay triangulation. Omitted points are listed in the
       `coplanar` attribute.

    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()

    Point indices and coordinates for the two triangles forming the
    triangulation:

    >>> tri.simplices
    array([[2, 3, 0],                 # may vary
           [3, 1, 0]], dtype=int32)

    Note that depending on how rounding errors go, the simplices may
    be in a different order than above.

    >>> points[tri.simplices]
    array([[[ 1. ,  0. ],            # may vary
            [ 1. ,  1. ],
            [ 0. ,  0. ]],
           [[ 1. ,  1. ],
            [ 0. ,  1.1],
            [ 0. ,  0. ]]])

    Triangle 0 is the only neighbor of triangle 1, and it's opposite to
    vertex 1 of triangle 1:

    >>> tri.neighbors[1]
    array([-1,  0, -1], dtype=int32)
    >>> points[tri.simplices[1,1]]
    array([ 0. ,  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)
    array([ 1, -1, 1], dtype=int32)

    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)]
    array([[3, 1, 0],                 # may vary
           [3, 1, 0]], dtype=int32)

    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)]
    array([[ 0.1       ,  0.09090909,  0.80909091],
           [ 1.5       , -0.90909091,  0.40909091],
           [ 0.5       ,  0.5       ,  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.