# cython: cpow=True
"""
Simple N-D interpolation

.. versionadded:: 0.9

"""
#
# Copyright (C)  Pauli Virtanen, 2010.
#
# Distributed under the same BSD license as Scipy.
#

#
# Note: this file should be run through the Mako template engine before
#       feeding it to Cython.
#
#       Run ``generate_qhull.py`` to regenerate the ``qhull.c`` file
#

cimport cython

from libc.float cimport DBL_EPSILON
from libc.math cimport fabs, sqrt

import numpy as np

import scipy.spatial._qhull as qhull
cimport scipy.spatial._qhull as qhull

import warnings

#------------------------------------------------------------------------------
# Numpy etc.
#------------------------------------------------------------------------------

cdef extern from "numpy/ndarrayobject.h":
    cdef enum:
        NPY_MAXDIMS

ctypedef fused double_or_complex:
    double
    double complex


#------------------------------------------------------------------------------
# Interpolator base class
#------------------------------------------------------------------------------

class NDInterpolatorBase:
    """
    Common routines for interpolators.

    .. versionadded:: 0.9

    """

    def __init__(self, points, values, fill_value=np.nan, ndim=None,
                 rescale=False, need_contiguous=True, need_values=True):
        """
        Check shape of points and values arrays, and reshape values to
        (npoints, nvalues).  Ensure the `points` and values arrays are
        C-contiguous, and of correct type.
        """

        if isinstance(points, qhull.Delaunay):
            # Precomputed triangulation was passed in
            if rescale:
                raise ValueError("Rescaling is not supported when passing "
                                 "a Delaunay triangulation as ``points``.")
            self.tri = points
            points = points.points
        else:
            self.tri = None

        points = _ndim_coords_from_arrays(points)

        if need_contiguous:
            points = np.ascontiguousarray(points, dtype=np.float64)

        if not rescale:
            self.scale = None
            self.points = points
        else:
            # scale to unit cube centered at 0
            self.offset = np.mean(points, axis=0)
            self.points = points - self.offset
            self.scale = np.ptp(points, axis=0)
            self.scale[~(self.scale > 0)] = 1.0  # avoid division by 0
            self.points /= self.scale

        if self.tri is None:
            self._calculate_triangulation(self.points)

        if need_values or values is not None:
            self._set_values(values, fill_value, need_contiguous, ndim)
        else:
            self.values = None

    def _calculate_triangulation(self, points):
        pass

    def _set_values(self, values, fill_value=np.nan, need_contiguous=True, ndim=None):
        values = np.asarray(values)
        _check_init_shape(self.points, values, ndim=ndim)

        self.values_shape = values.shape[1:]
        if values.ndim == 1:
            self.values = values[:,None]
        elif values.ndim == 2:
            self.values = values
        else:
            self.values = values.reshape(values.shape[0],
                                            np.prod(values.shape[1:]))
        
        # Complex or real?
        self.is_complex = np.issubdtype(self.values.dtype, np.complexfloating)
        if self.is_complex:
            if need_contiguous:
                self.values = np.ascontiguousarray(self.values,
                                                    dtype=np.complex128)
            self.fill_value = complex(fill_value)
        else:
            if need_contiguous:
                self.values = np.ascontiguousarray(
                    self.values, dtype=np.float64
                )
            self.fill_value = float(fill_value)

    def _check_call_shape(self, xi):
        xi = np.asanyarray(xi)
        if xi.shape[-1] != self.points.shape[1]:
            raise ValueError("number of dimensions in xi does not match x")
        return xi

    def _scale_x(self, xi):
        if self.scale is None:
            return xi
        else:
            return (xi - self.offset) / self.scale

    def _preprocess_xi(self, *args):
        xi = _ndim_coords_from_arrays(args, ndim=self.points.shape[1])
        xi = self._check_call_shape(xi)
        interpolation_points_shape = xi.shape
        xi = xi.reshape(-1, xi.shape[-1])
        xi = np.ascontiguousarray(xi, dtype=np.float64)
        return self._scale_x(xi), interpolation_points_shape
    
    def __call__(self, *args):
        """
        interpolator(xi)

        Evaluate interpolator at given points.

        Parameters
        ----------
        x1, x2, ... xn: array-like of float
            Points where to interpolate data at.
            x1, x2, ... xn can be array-like of float with broadcastable shape.
            or x1 can be array-like of float with shape ``(..., ndim)``
        """
        xi, interpolation_points_shape = self._preprocess_xi(*args)

        if self.is_complex:
            r = self._evaluate_complex(xi)
        else:
            r = self._evaluate_double(xi)

        return np.asarray(r).reshape(interpolation_points_shape[:-1] + self.values_shape)


cpdef _ndim_coords_from_arrays(points, ndim=None):
    """
    Convert a tuple of coordinate arrays to a (..., ndim)-shaped array.

    """
    cdef ssize_t j, n

    if isinstance(points, tuple) and len(points) == 1:
        # handle argument tuple
        points = points[0]
    if isinstance(points, tuple):
        p = np.broadcast_arrays(*points)
        n = len(p)
        for j in range(1, n):
            if p[j].shape != p[0].shape:
                raise ValueError("coordinate arrays do not have the same shape")
        points = np.empty(p[0].shape + (len(points),), dtype=float)
        for j, item in enumerate(p):
            points[...,j] = item
    else:
        points = np.asanyarray(points)
        if points.ndim == 1:
            if ndim is None:
                points = points.reshape(-1, 1)
            else:
                points = points.reshape(-1, ndim)
    return points


cdef _check_init_shape(points, values, ndim=None):
    """
    Check shape of points and values arrays

    """
    if values.shape[0] != points.shape[0]:
        raise ValueError("different number of values and points")
    if points.ndim != 2:
        raise ValueError("invalid shape for input data points")
    if points.shape[1] < 2:
        raise ValueError("input data must be at least 2-D")
    if ndim is not None and points.shape[1] != ndim:
        raise ValueError("this mode of interpolation available only for "
                         "%d-D data" % ndim)


#------------------------------------------------------------------------------
# Linear interpolation in N-D
#------------------------------------------------------------------------------

class LinearNDInterpolator(NDInterpolatorBase):
    """
    LinearNDInterpolator(points, values, fill_value=np.nan, rescale=False)

    Piecewise linear interpolator in N > 1 dimensions.

    .. versionadded:: 0.9

    Methods
    -------
    __call__

    Parameters
    ----------
    points : ndarray of floats, shape (npoints, ndims); or Delaunay
        2-D array of data point coordinates, or a precomputed Delaunay triangulation.
    values : ndarray of float or complex, shape (npoints, ...), optional
        N-D array of data values at `points`.  The length of `values` along the
        first axis must be equal to the length of `points`. Unlike some
        interpolators, the interpolation axis cannot be changed.
    fill_value : float, optional
        Value used to fill in for requested points outside of the
        convex hull of the input points.  If not provided, then
        the default is ``nan``.
    rescale : bool, optional
        Rescale points to unit cube before performing interpolation.
        This is useful if some of the input dimensions have
        incommensurable units and differ by many orders of magnitude.

    Notes
    -----
    The interpolant is constructed by triangulating the input data
    with Qhull [1]_, and on each triangle performing linear
    barycentric interpolation.

    .. note:: For data on a regular grid use `interpn` instead.

    Examples
    --------
    We can interpolate values on a 2D plane:

    >>> from scipy.interpolate import LinearNDInterpolator
    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> rng = np.random.default_rng()
    >>> x = rng.random(10) - 0.5
    >>> y = rng.random(10) - 0.5
    >>> z = np.hypot(x, y)
    >>> X = np.linspace(min(x), max(x))
    >>> Y = np.linspace(min(y), max(y))
    >>> X, Y = np.meshgrid(X, Y)  # 2D grid for interpolation
    >>> interp = LinearNDInterpolator(list(zip(x, y)), z)
    >>> Z = interp(X, Y)
    >>> plt.pcolormesh(X, Y, Z, shading='auto')
    >>> plt.plot(x, y, "ok", label="input point")
    >>> plt.legend()
    >>> plt.colorbar()
    >>> plt.axis("equal")
    >>> plt.show()

    See also
    --------
    griddata :
        Interpolate unstructured D-D data.
    NearestNDInterpolator :
        Nearest-neighbor interpolator in N dimensions.
    CloughTocher2DInterpolator :
        Piecewise cubic, C1 smooth, curvature-minimizing interpolator in 2D.
    interpn : Interpolation on a regular grid or rectilinear grid.
    RegularGridInterpolator : Interpolator on a regular or rectilinear grid
                              in arbitrary dimensions (`interpn` wraps this
                              class).

    References
    ----------
    .. [1] http://www.qhull.org/

    """

    def __init__(self, points, values, fill_value=np.nan, rescale=False):
        NDInterpolatorBase.__init__(self, points, values, fill_value=fill_value,
                rescale=rescale)

    def _calculate_triangulation(self, points):
        self.tri = qhull.Delaunay(points)

    def _evaluate_double(self, xi):
        return self._do_evaluate(xi, 1.0)

    def _evaluate_complex(self, xi):
        return self._do_evaluate(xi, 1.0j)

    @cython.boundscheck(False)
    @cython.wraparound(False)
    def _do_evaluate(self, const double[:,::1] xi, const double_or_complex dummy):
        cdef const double_or_complex[:,::1] values = self.values
        cdef double_or_complex[:,::1] out
        cdef const int[:,::1] simplices = self.tri.simplices
        cdef double_or_complex fill_value
        cdef double c[NPY_MAXDIMS]
        cdef int i, j, k, m, ndim, isimplex, start, nvalues
        cdef qhull.DelaunayInfo_t info
        cdef double eps, eps_broad
        
        ndim = xi.shape[1]
        fill_value = self.fill_value

        qhull._get_delaunay_info(&info, self.tri, 1, 0, 0)

        out = np.empty((xi.shape[0], self.values.shape[1]),
                       dtype=self.values.dtype)
        nvalues = out.shape[1]

        start = 0
        eps = 100 * DBL_EPSILON
        eps_broad = sqrt(DBL_EPSILON)

        # NOTE: a nogil block segfaults here with Python 3.10
        # and 3.11 on x86_64 Ubuntu Linux with gcc 9.x and 11.x
        # and therefore nogil was disabled to fix gh-21885
        for i in range(xi.shape[0]):

            # 1) Find the simplex

            isimplex = qhull._find_simplex(&info, c,
                                           &xi[0,0] + i*ndim,
                                           &start, eps, eps_broad)

            # 2) Linear barycentric interpolation

            if isimplex == -1:
                # don't extrapolate
                for k in range(nvalues):
                    out[i,k] = fill_value
                continue

            for k in range(nvalues):
                out[i,k] = 0

            for j in range(ndim+1):
                for k in range(nvalues):
                    m = simplices[isimplex,j]
                    out[i,k] = out[i,k] + c[j] * values[m,k]

        return out


#------------------------------------------------------------------------------
# Gradient estimation in 2D
#------------------------------------------------------------------------------

class GradientEstimationWarning(Warning):
    pass

@cython.cdivision(True)
cdef int _estimate_gradients_2d_global(const qhull.DelaunayInfo_t *d,
                                       const double *data,
                                       int maxiter,
                                       double tol,
                                       double *y) noexcept nogil:
    """
    Estimate gradients of a function at the vertices of a 2d triangulation.

    Parameters
    ----------
    info : input
        Triangulation in 2D
    data : input
        Function values at the vertices
    maxiter : input
        Maximum number of Gauss-Seidel iterations
    tol : input
        Absolute / relative stop tolerance
    y : output, shape (npoints, 2)
        Derivatives [F_x, F_y] at the vertices

    Returns
    -------
    num_iterations
        Number of iterations if converged, 0 if maxiter reached
        without convergence

    Notes
    -----
    This routine uses a re-implementation of the global approximate
    curvature minimization algorithm described in [Nielson83] and [Renka84].

    References
    ----------
    .. [Nielson83] G. Nielson,
       ''A method for interpolating scattered data based upon a minimum norm
       network''.
       Math. Comp., 40, 253 (1983).
    .. [Renka84] R. J. Renka and A. K. Cline.
       ''A Triangle-based C1 interpolation method.'',
       Rocky Mountain J. Math., 14, 223 (1984).

    """
    cdef double Q[2*2]
    cdef double s[2]
    cdef double r[2]
    cdef int ipoint, iiter, k, ipoint2, jpoint2
    cdef double f1, f2, df2, ex, ey, L, L3, det, err, change

    # initialize
    for ipoint in range(2*d.npoints):
        y[ipoint] = 0

    #
    # Main point:
    #
    #    Z = sum_T sum_{E in T} int_E |W''|^2 = min!
    #
    # where W'' is the second derivative of the Clough-Tocher
    # interpolant to the direction of the edge E in triangle T.
    #
    # The minimization is done iteratively: for each vertex V,
    # the sum
    #
    #    Z_V = sum_{E connected to V} int_E |W''|^2
    #
    # is minimized separately, using existing values at other V.
    #
    # Since the interpolant can be written as
    #
    #     W(x) = f(x) + w(x)^T y
    #
    # where y = [ F_x(V); F_y(V) ], it is clear that the solution to
    # the local problem is given as a solution of the 2x2 matrix
    # equation.
    #
    # Here, we use the Clough-Tocher interpolant, which restricted to
    # a single edge is
    #
    #     w(x) = (1 - x)**3   * f1
    #          + x*(1 - x)**2 * (df1 + 3*f1)
    #          + x**2*(1 - x) * (df2 + 3*f2)
    #          + x**3         * f2
    #
    # where f1, f2 are values at the vertices, and df1 and df2 are
    # derivatives along the edge (away from the vertices).
    #
    # As a consequence, one finds
    #
    #     L^3 int_{E} |W''|^2 = y^T A y + 2 B y + C
    #
    # with
    #
    #     A   = [4, -2; -2, 4]
    #     B   = [6*(f1 - f2), 6*(f2 - f1)]
    #     y   = [df1, df2]
    #     L   = length of edge E
    #
    # and C is not needed for minimization. Since df1 = dF1.E, df2 = -dF2.E,
    # with dF1 = [F_x(V_1), F_y(V_1)], and the edge vector E = V2 - V1,
    # we have
    #
    #     Z_V = dF1^T Q dF1 + 2 s.dF1 + const.
    #
    # which is minimized by
    #
    #     dF1 = -Q^{-1} s
    #
    # where
    #
    #     Q = sum_E [A_11 E E^T]/L_E^3 = 4 sum_E [E E^T]/L_E^3
    #     s = sum_E [ B_1 + A_21 df2] E /L_E^3
    #       = sum_E [ 6*(f1 - f2) + 2*(E.dF2)] E / L_E^3
    #

    # Gauss-Seidel
    for iiter in range(maxiter):
        err = 0
        for ipoint in range(d.npoints):
            for k in range(2*2):
                Q[k] = 0
            for k in range(2):
                s[k] = 0

            # walk over neighbours of given point
            for jpoint2 in range(d.vertex_neighbors_indptr[ipoint],
                                  d.vertex_neighbors_indptr[ipoint+1]):
                ipoint2 = d.vertex_neighbors_indices[jpoint2]

                # edge
                ex = d.points[2*ipoint2 + 0] - d.points[2*ipoint + 0]
                ey = d.points[2*ipoint2 + 1] - d.points[2*ipoint + 1]
                L = sqrt(ex**2 + ey**2)
                L3 = L*L*L

                # data at vertices
                f1 = data[ipoint]
                f2 = data[ipoint2]

                # scaled gradient projections on the edge
                df2 = -ex*y[2*ipoint2 + 0] - ey*y[2*ipoint2 + 1]

                # edge sum
                Q[0] += 4*ex*ex / L3
                Q[1] += 4*ex*ey / L3
                Q[3] += 4*ey*ey / L3

                s[0] += (6*(f1 - f2) - 2*df2) * ex / L3
                s[1] += (6*(f1 - f2) - 2*df2) * ey / L3

            Q[2] = Q[1]

            # solve

            det = Q[0]*Q[3] - Q[1]*Q[2]
            r[0] = ( Q[3]*s[0] - Q[1]*s[1])/det
            r[1] = (-Q[2]*s[0] + Q[0]*s[1])/det

            change = max(fabs(y[2*ipoint + 0] + r[0]),
                         fabs(y[2*ipoint + 1] + r[1]))

            y[2*ipoint + 0] = -r[0]
            y[2*ipoint + 1] = -r[1]

            # relative/absolute error
            change /= max(1.0, max(fabs(r[0]), fabs(r[1])))
            err = max(err, change)

        if err < tol:
            return iiter + 1

    # Didn't converge before maxiter
    return 0

@cython.boundscheck(False)
@cython.wraparound(False)
cpdef estimate_gradients_2d_global(tri, y, int maxiter=400, double tol=1e-6):
    cdef const double[:,::1] data
    cdef double[:,:,::1] grad
    cdef qhull.DelaunayInfo_t info
    cdef int k, ret, nvalues

    y = np.asanyarray(y)

    if y.shape[0] != tri.npoints:
        raise ValueError("'y' has a wrong number of items")

    if np.issubdtype(y.dtype, np.complexfloating):
        rg = estimate_gradients_2d_global(tri, y.real, maxiter=maxiter, tol=tol)
        ig = estimate_gradients_2d_global(tri, y.imag, maxiter=maxiter, tol=tol)
        r = np.zeros(rg.shape, dtype=complex)
        r.real = rg
        r.imag = ig
        return r

    y_shape = y.shape

    if y.ndim == 1:
        y = y[:,None]

    y = y.reshape(tri.npoints, -1).T
    y = np.ascontiguousarray(y, dtype=np.float64)
    yi = np.empty((y.shape[0], y.shape[1], 2))

    data = y
    grad = yi

    qhull._get_delaunay_info(&info, tri, 0, 0, 1)
    nvalues = data.shape[0]

    for k in range(nvalues):
        with nogil:
            ret = _estimate_gradients_2d_global(
                &info,
                &data[k,0],
                maxiter,
                tol,
                &grad[k,0,0])

        if ret == 0:
            warnings.warn("Gradient estimation did not converge, "
                          "the results may be inaccurate",
                          GradientEstimationWarning)

    return yi.transpose(1, 0, 2).reshape(y_shape + (2,))


#------------------------------------------------------------------------------
# Cubic interpolation in 2D
#------------------------------------------------------------------------------


@cython.cdivision(True)
cdef double_or_complex _clough_tocher_2d_single(const qhull.DelaunayInfo_t *d,
                                                int isimplex,
                                                double *b,
                                                double_or_complex *f,
                                                double_or_complex *df) noexcept nogil:
    """
    Evaluate Clough-Tocher interpolant on a 2D triangle.

    Parameters
    ----------
    d :
        Delaunay info
    isimplex : int
        Triangle to evaluate on
    b : shape (3,)
        Barycentric coordinates of the point on the triangle
    f : shape (3,)
        Function values at vertices
    df : shape (3, 2)
        Gradient values at vertices

    Returns
    -------
    w :
        Value of the interpolant at the given point

    References
    ----------
    .. [CT] See, for example,
       P. Alfeld,
       ''A trivariate Clough-Tocher scheme for tetrahedral data''.
       Computer Aided Geometric Design, 1, 169 (1984);
       G. Farin,
       ''Triangular Bernstein-Bezier patches''.
       Computer Aided Geometric Design, 3, 83 (1986).

    """
    cdef double_or_complex \
         c3000, c0300, c0030, c0003, \
         c2100, c2010, c2001, c0210, c0201, c0021, \
         c1200, c1020, c1002, c0120, c0102, c0012, \
         c1101, c1011, c0111
    cdef double_or_complex \
         f1, f2, f3, df12, df13, df21, df23, df31, df32
    cdef double g[3]
    cdef double e12x, e12y, e23x, e23y, e31x, e31y
    cdef double_or_complex w
    cdef double minval
    cdef double b1, b2, b3, b4
    cdef int k, itri
    cdef double c[3]
    cdef double y[2]

    # XXX: optimize + refactor this!

    e12x = (+ d.points[0 + 2*d.simplices[3*isimplex + 1]]
            - d.points[0 + 2*d.simplices[3*isimplex + 0]])
    e12y = (+ d.points[1 + 2*d.simplices[3*isimplex + 1]]
            - d.points[1 + 2*d.simplices[3*isimplex + 0]])

    e23x = (+ d.points[0 + 2*d.simplices[3*isimplex + 2]]
            - d.points[0 + 2*d.simplices[3*isimplex + 1]])
    e23y = (+ d.points[1 + 2*d.simplices[3*isimplex + 2]]
            - d.points[1 + 2*d.simplices[3*isimplex + 1]])

    e31x = (+ d.points[0 + 2*d.simplices[3*isimplex + 0]]
            - d.points[0 + 2*d.simplices[3*isimplex + 2]])
    e31y = (+ d.points[1 + 2*d.simplices[3*isimplex + 0]]
            - d.points[1 + 2*d.simplices[3*isimplex + 2]])

    f1 = f[0]
    f2 = f[1]
    f3 = f[2]

    df12 = +(df[2*0+0]*e12x + df[2*0+1]*e12y)
    df21 = -(df[2*1+0]*e12x + df[2*1+1]*e12y)
    df23 = +(df[2*1+0]*e23x + df[2*1+1]*e23y)
    df32 = -(df[2*2+0]*e23x + df[2*2+1]*e23y)
    df31 = +(df[2*2+0]*e31x + df[2*2+1]*e31y)
    df13 = -(df[2*0+0]*e31x + df[2*0+1]*e31y)

    c3000 = f1
    c2100 = (df12 + 3*c3000)/3
    c2010 = (df13 + 3*c3000)/3
    c0300 = f2
    c1200 = (df21 + 3*c0300)/3
    c0210 = (df23 + 3*c0300)/3
    c0030 = f3
    c1020 = (df31 + 3*c0030)/3
    c0120 = (df32 + 3*c0030)/3

    c2001 = (c2100 + c2010 + c3000)/3
    c0201 = (c1200 + c0300 + c0210)/3
    c0021 = (c1020 + c0120 + c0030)/3

    #
    # Now, we need to impose the condition that the gradient of the spline
    # to some direction `w` is a linear function along the edge.
    #
    # As long as two neighbouring triangles agree on the choice of the
    # direction `w`, this ensures global C1 differentiability.
    # Otherwise, the choice of the direction is arbitrary (except that
    # it should not point along the edge, of course).
    #
    # In [CT]_, it is suggested to pick `w` as the normal of the edge.
    # This choice is given by the formulas
    #
    #    w_12 = E_24 + g[0] * E_23
    #    w_23 = E_34 + g[1] * E_31
    #    w_31 = E_14 + g[2] * E_12
    #
    #    g[0] = -(e24x*e23x + e24y*e23y) / (e23x**2 + e23y**2)
    #    g[1] = -(e34x*e31x + e34y*e31y) / (e31x**2 + e31y**2)
    #    g[2] = -(e14x*e12x + e14y*e12y) / (e12x**2 + e12y**2)
    #
    # However, this choice gives an interpolant that is *not*
    # invariant under affine transforms. This has some bad
    # consequences: for a very narrow triangle, the spline can
    # develops huge oscillations. For instance, with the input data
    #
    #     [(0, 0), (0, 1), (eps, eps)],   eps = 0.01
    #     F  = [0, 0, 1]
    #     dF = [(0,0), (0,0), (0,0)]
    #
    # one observes that as eps -> 0, the absolute maximum value of the
    # interpolant approaches infinity.
    #
    # So below, we aim to pick affine invariant `g[k]`.
    # We choose
    #
    #     w = V_4' - V_4
    #
    # where V_4 is the centroid of the current triangle, and V_4' the
    # centroid of the neighbour. Since this quantity transforms similarly
    # as the gradient under affine transforms, the resulting interpolant
    # is affine-invariant. Moreover, two neighbouring triangles clearly
    # always agree on the choice of `w` (sign is unimportant), and so
    # this choice also makes the interpolant C1.
    #
    # The drawback here is a performance penalty, since we need to
    # peek into neighbouring triangles.
    #

    for k in range(3):
        itri = d.neighbors[3*isimplex + k]

        if itri == -1:
            # No neighbour.
            # Compute derivative to the centroid direction (e_12 + e_13)/2.
            g[k] = -1./2
            continue

        # Centroid of the neighbour, in our local barycentric coordinates

        y[0] = (+ d.points[0 + 2*d.simplices[3*itri + 0]]
                + d.points[0 + 2*d.simplices[3*itri + 1]]
                + d.points[0 + 2*d.simplices[3*itri + 2]]) / 3

        y[1] = (+ d.points[1 + 2*d.simplices[3*itri + 0]]
                + d.points[1 + 2*d.simplices[3*itri + 1]]
                + d.points[1 + 2*d.simplices[3*itri + 2]]) / 3

        qhull._barycentric_coordinates(2, d.transform + isimplex*2*3, y, c)

        # Rewrite V_4'-V_4 = const*[(V_4-V_2) + g_i*(V_3 - V_2)]

        # Now, observe that the results can be written *in terms of
        # barycentric coordinates*. Barycentric coordinates stay
        # invariant under affine transformations, so we can directly
        # conclude that the choice below is affine-invariant.

        if k == 0:
            g[k] = (2*c[2] + c[1] - 1) / (2 - 3*c[2] - 3*c[1])
        elif k == 1:
            g[k] = (2*c[0] + c[2] - 1) / (2 - 3*c[0] - 3*c[2])
        elif k == 2:
            g[k] = (2*c[1] + c[0] - 1) / (2 - 3*c[1] - 3*c[0])

    c0111 = (g[0]*(-c0300 + 3*c0210 - 3*c0120 + c0030)
             + (-c0300 + 2*c0210 - c0120 + c0021 + c0201))/2
    c1011 = (g[1]*(-c0030 + 3*c1020 - 3*c2010 + c3000)
             + (-c0030 + 2*c1020 - c2010 + c2001 + c0021))/2
    c1101 = (g[2]*(-c3000 + 3*c2100 - 3*c1200 + c0300)
             + (-c3000 + 2*c2100 - c1200 + c2001 + c0201))/2

    c1002 = (c1101 + c1011 + c2001)/3
    c0102 = (c1101 + c0111 + c0201)/3
    c0012 = (c1011 + c0111 + c0021)/3

    c0003 = (c1002 + c0102 + c0012)/3

    # extended barycentric coordinates
    minval = b[0]
    for k in range(3):
        if b[k] < minval:
            minval = b[k]

    b1 = b[0] - minval
    b2 = b[1] - minval
    b3 = b[2] - minval
    b4 = 3*minval

    # evaluate the polynomial -- the stupid and ugly way to do it,
    # one of the 4 coordinates is in fact zero
    w = (b1**3*c3000 + 3*b1**2*b2*c2100 + 3*b1**2*b3*c2010 +
         3*b1**2*b4*c2001 + 3*b1*b2**2*c1200 +
         6*b1*b2*b4*c1101 + 3*b1*b3**2*c1020 + 6*b1*b3*b4*c1011 +
         3*b1*b4**2*c1002 + b2**3*c0300 + 3*b2**2*b3*c0210 +
         3*b2**2*b4*c0201 + 3*b2*b3**2*c0120 + 6*b2*b3*b4*c0111 +
         3*b2*b4**2*c0102 + b3**3*c0030 + 3*b3**2*b4*c0021 +
         3*b3*b4**2*c0012 + b4**3*c0003)

    return w

class CloughTocher2DInterpolator(NDInterpolatorBase):
    """CloughTocher2DInterpolator(points, values, tol=1e-6).

    Piecewise cubic, C1 smooth, curvature-minimizing interpolator in 2D.

    .. versionadded:: 0.9

    Methods
    -------
    __call__

    Parameters
    ----------
    points : ndarray of floats, shape (npoints, ndims); or Delaunay
        2-D array of data point coordinates, or a precomputed Delaunay triangulation.
    values : ndarray of float or complex, shape (npoints, ...)
        N-D array of data values at `points`. The length of `values` along the
        first axis must be equal to the length of `points`. Unlike some
        interpolators, the interpolation axis cannot be changed.
    fill_value : float, optional
        Value used to fill in for requested points outside of the
        convex hull of the input points.  If not provided, then
        the default is ``nan``.
    tol : float, optional
        Absolute/relative tolerance for gradient estimation.
    maxiter : int, optional
        Maximum number of iterations in gradient estimation.
    rescale : bool, optional
        Rescale points to unit cube before performing interpolation.
        This is useful if some of the input dimensions have
        incommensurable units and differ by many orders of magnitude.

    Notes
    -----
    The interpolant is constructed by triangulating the input data
    with Qhull [1]_, and constructing a piecewise cubic
    interpolating Bezier polynomial on each triangle, using a
    Clough-Tocher scheme [CT]_.  The interpolant is guaranteed to be
    continuously differentiable.

    The gradients of the interpolant are chosen so that the curvature
    of the interpolating surface is approximatively minimized. The
    gradients necessary for this are estimated using the global
    algorithm described in [Nielson83]_ and [Renka84]_.

    .. note:: For data on a regular grid use `interpn` instead.

    Examples
    --------
    We can interpolate values on a 2D plane:

    >>> from scipy.interpolate import CloughTocher2DInterpolator
    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> rng = np.random.default_rng()
    >>> x = rng.random(10) - 0.5
    >>> y = rng.random(10) - 0.5
    >>> z = np.hypot(x, y)
    >>> X = np.linspace(min(x), max(x))
    >>> Y = np.linspace(min(y), max(y))
    >>> X, Y = np.meshgrid(X, Y)  # 2D grid for interpolation
    >>> interp = CloughTocher2DInterpolator(list(zip(x, y)), z)
    >>> Z = interp(X, Y)
    >>> plt.pcolormesh(X, Y, Z, shading='auto')
    >>> plt.plot(x, y, "ok", label="input point")
    >>> plt.legend()
    >>> plt.colorbar()
    >>> plt.axis("equal")
    >>> plt.show()

    See also
    --------
    griddata :
        Interpolate unstructured D-D data.
    LinearNDInterpolator :
        Piecewise linear interpolator in N > 1 dimensions.
    NearestNDInterpolator :
        Nearest-neighbor interpolator in N > 1 dimensions.
    interpn : Interpolation on a regular grid or rectilinear grid.
    RegularGridInterpolator : Interpolator on a regular or rectilinear grid
                              in arbitrary dimensions (`interpn` wraps this
                              class).

    References
    ----------
    .. [1] http://www.qhull.org/

    .. [CT] See, for example,
       P. Alfeld,
       ''A trivariate Clough-Tocher scheme for tetrahedral data''.
       Computer Aided Geometric Design, 1, 169 (1984);
       G. Farin,
       ''Triangular Bernstein-Bezier patches''.
       Computer Aided Geometric Design, 3, 83 (1986).

    .. [Nielson83] G. Nielson,
       ''A method for interpolating scattered data based upon a minimum norm
       network''.
       Math. Comp., 40, 253 (1983).

    .. [Renka84] R. J. Renka and A. K. Cline.
       ''A Triangle-based C1 interpolation method.'',
       Rocky Mountain J. Math., 14, 223 (1984).

    """

    def __init__(self, points, values, fill_value=np.nan,
                 tol=1e-6, maxiter=400, rescale=False):
        self._tol = tol
        self._maxiter = maxiter
        NDInterpolatorBase.__init__(self, points, values, ndim=2,
                                    fill_value=fill_value, rescale=rescale,
                                    need_values=False)
    
    def _set_values(self, values, fill_value=np.nan, need_contiguous=True, ndim=None):
        """
        Sets the values of the interpolation points.

        Parameters
        ----------
        values : ndarray of float or complex, shape (npoints, ...)
            Data values.
        """
        NDInterpolatorBase._set_values(self, values, fill_value=fill_value, need_contiguous=need_contiguous, ndim=ndim)
        if self.values is not None:
            self.grad = estimate_gradients_2d_global(self.tri, self.values,
                                                    tol=self._tol, maxiter=self._maxiter)
    
    def _calculate_triangulation(self, points):
        self.tri = qhull.Delaunay(points)

    def _evaluate_double(self, xi):
        return self._do_evaluate(xi, 1.0)

    def _evaluate_complex(self, xi):
        return self._do_evaluate(xi, 1.0j)

    @cython.boundscheck(False)
    @cython.wraparound(False)
    def _do_evaluate(self, const double[:,::1] xi, const double_or_complex dummy):
        cdef const double_or_complex[:,::1] values = self.values
        cdef const double_or_complex[:,:,:] grad = self.grad
        cdef double_or_complex[:,::1] out
        cdef const int[:,::1] simplices = self.tri.simplices
        cdef double c[NPY_MAXDIMS]
        cdef double_or_complex f[NPY_MAXDIMS+1]
        cdef double_or_complex df[2*NPY_MAXDIMS+2]
        cdef double_or_complex w
        cdef double_or_complex fill_value
        cdef int i, j, k, ndim, isimplex, start, nvalues
        cdef qhull.DelaunayInfo_t info
        cdef double eps, eps_broad

        ndim = xi.shape[1]
        fill_value = self.fill_value

        qhull._get_delaunay_info(&info, self.tri, 1, 1, 0)

        out = np.zeros((xi.shape[0], self.values.shape[1]),
                       dtype=self.values.dtype)
        nvalues = out.shape[1]

        start = 0
        eps = 100 * DBL_EPSILON
        eps_broad = sqrt(eps)

        with nogil:
            for i in range(xi.shape[0]):
                # 1) Find the simplex

                isimplex = qhull._find_simplex(&info, c,
                                               &xi[i,0],
                                               &start, eps, eps_broad)
                
                # 2) Clough-Tocher interpolation

                if isimplex == -1:
                    # outside triangulation
                    for k in range(nvalues):
                        out[i,k] = fill_value
                    continue

                for k in range(nvalues):
                    for j in range(ndim+1):
                        f[j] = values[simplices[isimplex,j],k]
                        df[2*j] = grad[simplices[isimplex,j],k,0]
                        df[2*j+1] = grad[simplices[isimplex,j],k,1]

                    w = _clough_tocher_2d_single(&info, isimplex, c, f, df)
                    out[i,k] = w

        return out
