# Source code for ufl.algorithms.change_to_reference

    """Algorithm for replacing gradients in an expression."""

    # Copyright (C) 2008-2016 Martin Sandve Alnæs
    #
    # This file is part of UFL (https://www.fenicsproject.org)
    #
    # SPDX-License-Identifier:    LGPL-3.0-or-later

    from ufl.algorithms.apply_function_pullbacks import apply_function_pullbacks
    from ufl.algorithms.apply_geometry_lowering import apply_geometry_lowering
    from ufl.checks import is_cellwise_constant
    from ufl.classes import Grad, JacobianInverse, ReferenceGrad, ReferenceValue, Restricted
    from ufl.core.multiindex import indices
    from ufl.corealg.map_dag import map_expr_dag
    from ufl.corealg.multifunction import MultiFunction
    from ufl.domain import extract_unique_domain
    from ufl.tensors import as_tensor

    """
    # Some notes:
    # Below, let v_i mean physical coordinate of vertex i and V_i mean the
    reference cell coordinate of the same vertex.

    # Add a type for CellVertices? Note that vertices must be computed in
    linear cell cases! triangle_vertices[i,j] = component j of vertex i,
    following ufc numbering conventions

    # DONE Add a type for CellEdgeLengths? Note that these are only easy to
    define in the linear cell case!
    triangle_edge_lengths    = [v1v2, v0v2, v0v1] # shape (3,)
    tetrahedron_edge_lengths = [v0v1, v0v2, v0v3, v1v2, v1v3, v2v3] # shape (6,)

    # DONE Here's how to compute edge lengths from the Jacobian:
    J =[ [dx0/dX0, dx0/dX1],
         [dx1/dX0, dx1/dX1] ]
    # First compute the edge vector, which is constant for each edge: the
    vector from one vertex to the other
    reference_edge_vector_0 = V2 - V1 # Example! Add a type ReferenceEdgeVectors?
    # Then apply J to it and take the length of the resulting vector, this is generic for affine cells
    edge_length_i = || dot(J, reference_edge_vector_i) ||

    e2 = || J[:,0] . < 1, 0> || = || J[:,0] || = || dx/dX0 || = edge length of edge 2 (v0-v1)
    e1 = || J[:,1] . < 0, 1> || = || J[:,1] || = || dx/dX1 || = edge length of edge 1 (v0-v2)
    e0 = || J[:,:] . <-1, 1> || = || < J[0,1]-J[0,0], J[1,1]-J[1,0] > || = || dx/dX <-1,1> ||
        = edge length of edge 0 (v1-v2)

    trev = triangle_reference_edge_vector
    evec0 = J00 * trev[edge][0] + J01 * trev[edge][1]  =  J*trev[edge]
    evec1 = J10 * trev[edge][0] + J11 * trev[edge][1]
    elen[edge] = sqrt(evec0*evec0 + evec1*evec1)  = sqrt((J*trev[edge])**2)

    trev = triangle_reference_edge_vector
    evec0 = J00 * trev[edge][0] + J01 * trev[edge][1]  =  J*trev
    evec1 = J10 * trev[edge][0] + J11 * trev[edge][1]
    evec2 = J20 * trev[edge][0] + J21 * trev[edge][1] # Manifold: triangle in 3D
    elen[edge] = sqrt(evec0*evec0 + evec1*evec1 + evec2*evec2)  = sqrt((J*trev[edge])**2)

    trev = tetrahedron_reference_edge_vector
    evec0 = sum(J[0,k] * trev[edge][k] for k in range(3))
    evec1 = sum(J[1,k] * trev[edge][k] for k in range(3))
    evec2 = sum(J[2,k] * trev[edge][k] for k in range(3))
    elen[edge] = sqrt(evec0*evec0 + evec1*evec1 + evec2*evec2)  = sqrt((J*trev[edge])**2)

    # DONE Here's how to compute min/max facet edge length:
    triangle:
      r = facetarea
    tetrahedron:
      min(elen[edge] for edge in range(6))
    or
      min( min(elen[0], min(elen[1], elen[2])), min(elen[3], min(elen[4], elen[5])) )
    or
      min1 = min_value(elen[0], min_value(elen[1], elen[2]))
      min2 = min_value(elen[3], min_value(elen[4], elen[5]))
      r = min_value(min1, min2)
    (want proper Min/Max types for this!)

    # DONE Here's how to compute circumradius for an interval:
    circumradius_interval = cellvolume / 2

    # DONE Here's how to compute circumradius for a triangle:
    e0 = elen[0]
    e1 = elen[1]
    e2 = elen[2]
    circumradius_triangle = (e0*e1*e2) / (4*cellvolume)

    # DONE Here's how to compute circumradius for a tetrahedron:
    # v1v2 = edge length between vertex 1 and 2
    # la,lb,lc = lengths of the sides of an intermediate triangle
    la = v1v2 * v0v3
    lb = v0v2 * v1v3
    lc = v0v1 * v2v3
    # p = perimeter
    p = (la + lb + lc)
    # s = semiperimeter
    s = p / 2
    # area of intermediate triangle with Herons formula
    tmp_area = sqrt(s * (s - la) * (s - lb) * (s - lc))
    circumradius_tetrahedron = tmp_area / (6*cellvolume)

    """

    class ChangeToReferenceGrad(MultiFunction):
        """Change to reference grad."""

        def __init__(self):
            """Initalise."""
            MultiFunction.__init__(self)

        expr = MultiFunction.reuse_if_untouched

        def terminal(self, o):
            """Apply to terminal."""
            return o

        def grad(self, o):
            """Apply to grad."""
            # Peel off the Grads and count them, and get restriction if
            # it's between the grad and the terminal
            ngrads = 0
            restricted = ""
            rv = False
            while not o._ufl_is_terminal_:
                if isinstance(o, Grad):
                    (o,) = o.ufl_operands
                    ngrads += 1
                elif isinstance(o, Restricted):
                    restricted = o.side()
                    (o,) = o.ufl_operands
                elif isinstance(o, ReferenceValue):
                    rv = True
                    (o,) = o.ufl_operands
                else:
                    raise ValueError(f"Invalid type {o._ufl_class_.__name__}")
            f = o
            if rv:
                f = ReferenceValue(f)

            # Get domain and create Jacobian inverse object
            domain = extract_unique_domain(o)
            Jinv = JacobianInverse(domain)

            if is_cellwise_constant(Jinv):
                # Optimise slightly by turning Grad(Grad(...)) into
                # J^(-T)J^(-T)RefGrad(RefGrad(...))
                # rather than J^(-T)RefGrad(J^(-T)RefGrad(...))

                # Create some new indices
                ii = indices(len(f.ufl_shape))  # Indices to get to the scalar component of f
                jj = indices(
                    ngrads
                )  # Indices to sum over the local gradient axes with the inverse Jacobian
                kk = indices(ngrads)  # Indices for the leftover inverse Jacobian axes

                # Preserve restricted property
                if restricted:
                    Jinv = Jinv(restricted)
                    f = f(restricted)

                # Apply the same number of ReferenceGrad without mappings
                lgrad = f
                for i in range(ngrads):
                    lgrad = ReferenceGrad(lgrad)

                # Apply mappings with scalar indexing operations (assumes
                # ReferenceGrad(Jinv) is zero)
                jinv_lgrad_f = lgrad[ii + jj]
                for j, k in zip(jj, kk):
                    jinv_lgrad_f = Jinv[j, k] * jinv_lgrad_f

                # Wrap back in tensor shape, derivative axes at the end
                jinv_lgrad_f = as_tensor(jinv_lgrad_f, ii + kk)

            else:
                # J^(-T)RefGrad(J^(-T)RefGrad(...))

                # Preserve restricted property
                if restricted:
                    Jinv = Jinv(restricted)
                    f = f(restricted)

                jinv_lgrad_f = f
                for foo in range(ngrads):
                    ii = indices(
                        len(jinv_lgrad_f.ufl_shape)
                    )  # Indices to get to the scalar component of f
                    j, k = indices(2)

                    lgrad = ReferenceGrad(jinv_lgrad_f)
                    jinv_lgrad_f = Jinv[j, k] * lgrad[ii + (j,)]

                    # Wrap back in tensor shape, derivative axes at the end
                    jinv_lgrad_f = as_tensor(jinv_lgrad_f, ii + (k,))

            return jinv_lgrad_f

        def reference_grad(self, o):
            """Apply to reference_grad."""
            raise ValueError("Not expecting reference grad while applying change to reference grad.")

        def coefficient_derivative(self, o):
            """Apply to coefficient_derivative."""
            raise ValueError(
                "Coefficient derivatives should be expanded before applying change to reference grad."
            )

    def change_to_reference_grad(e):
        """Change Grad objects in expression to products of JacobianInverse and ReferenceGrad.

        Assumes the expression is preprocessed or at least that derivatives
        have been expanded.

        Args:
            e: An Expr or Form.
        """
        mf = ChangeToReferenceGrad()
        return map_expr_dag(mf, e)

    def change_integrand_geometry_representation(integrand, scale, integral_type):
        """Change integrand geometry to the right representations."""
        integrand = apply_function_pullbacks(integrand)

        integrand = change_to_reference_grad(integrand)

        integrand = integrand * scale

        if integral_type == "quadrature":
            physical_coordinates_known = True
        else:
            physical_coordinates_known = False
        integrand = apply_geometry_lowering(integrand, physical_coordinates_known)

        return integrand