from sympy.vector.coordsysrect import CoordSys3D
from sympy.vector.scalar import BaseScalar
from sympy.vector.vector import Vector, BaseVector
from sympy.vector.operators import gradient, curl, divergence
from sympy import diff, integrate, S, simplify
from sympy.core import sympify
from sympy.vector.dyadic import Dyadic


def express(expr, system, system2=None, variables=False):
    """
    Global function for 'express' functionality.

    Re-expresses a Vector, Dyadic or scalar(sympyfiable) in the given
    coordinate system.

    If 'variables' is True, then the coordinate variables (base scalars)
    of other coordinate systems present in the vector/scalar field or
    dyadic are also substituted in terms of the base scalars of the
    given system.

    Parameters
    ==========

    expr : Vector/Dyadic/scalar(sympyfiable)
        The expression to re-express in CoordSys3D 'system'

    system: CoordSys3D
        The coordinate system the expr is to be expressed in

    system2: CoordSys3D
        The other coordinate system required for re-expression
        (only for a Dyadic Expr)

    variables : boolean
        Specifies whether to substitute the coordinate variables present
        in expr, in terms of those of parameter system

    Examples
    ========

    >>> from sympy.vector import CoordSys3D
    >>> from sympy import Symbol, cos, sin
    >>> N = CoordSys3D('N')
    >>> q = Symbol('q')
    >>> B = N.orient_new_axis('B', q, N.k)
    >>> from sympy.vector import express
    >>> express(B.i, N)
    (cos(q))*N.i + (sin(q))*N.j
    >>> express(N.x, B, variables=True)
    -sin(q)*B.y + cos(q)*B.x
    >>> d = N.i.outer(N.i)
    >>> express(d, B, N) == (cos(q))*(B.i|N.i) + (-sin(q))*(B.j|N.i)
    True

    """

    if expr == 0 or expr == Vector.zero:
        return expr

    if not isinstance(system, CoordSys3D):
        raise TypeError("system should be a CoordSys3D \
                        instance")

    if isinstance(expr, Vector):
        if system2 is not None:
            raise ValueError("system2 should not be provided for \
                                Vectors")
        # Given expr is a Vector
        if variables:
            # If variables attribute is True, substitute
            # the coordinate variables in the Vector
            system_list = []
            for x in expr.atoms(BaseScalar, BaseVector):
                if x.system != system:
                    system_list.append(x.system)
            system_list = set(system_list)
            subs_dict = {}
            for f in system_list:
                subs_dict.update(f.scalar_map(system))
            expr = expr.subs(subs_dict)
        # Re-express in this coordinate system
        outvec = Vector.zero
        parts = expr.separate()
        for x in parts:
            if x != system:
                temp = system.rotation_matrix(x) * parts[x].to_matrix(x)
                outvec += matrix_to_vector(temp, system)
            else:
                outvec += parts[x]
        return outvec

    elif isinstance(expr, Dyadic):
        if system2 is None:
            system2 = system
        if not isinstance(system2, CoordSys3D):
            raise TypeError("system2 should be a CoordSys3D \
                            instance")
        outdyad = Dyadic.zero
        var = variables
        for k, v in expr.components.items():
            outdyad += (express(v, system, variables=var) *
                        (express(k.args[0], system, variables=var) |
                         express(k.args[1], system2, variables=var)))

        return outdyad

    else:
        if system2 is not None:
            raise ValueError("system2 should not be provided for \
                                Vectors")
        if variables:
            # Given expr is a scalar field
            system_set = set([])
            expr = sympify(expr)
            # Subsitute all the coordinate variables
            for x in expr.atoms(BaseScalar):
                if x.system != system:
                    system_set.add(x.system)
            subs_dict = {}
            for f in system_set:
                subs_dict.update(f.scalar_map(system))
            return expr.subs(subs_dict)
        return expr


def directional_derivative(scalar, vect):
    """
    Returns the directional derivative of a scalar field computed along a given vector
    in given coordinate system.

    Parameters
    ==========

    scalar : SymPy Expr
        The scalar field to compute the gradient of

    vect : Vector
        The vector operand

    coord_sys : CoordSys3D
        The coordinate system to calculate the gradient in

    Examples
    ========

    >>> from sympy.vector import CoordSys3D, directional_derivative
    >>> R = CoordSys3D('R')
    >>> f1 = R.x*R.y*R.z
    >>> v1 = 3*R.i + 4*R.j + R.k
    >>> directional_derivative(f1, v1)
    R.x*R.y + 4*R.x*R.z + 3*R.y*R.z
    >>> f2 = 5*R.x**2*R.z
    >>> directional_derivative(f2, v1)
    5*R.x**2 + 30*R.x*R.z

    """
    return gradient(scalar).dot(vect).doit()


def is_conservative(field):
    """
    Checks if a field is conservative.

    Paramaters
    ==========

    field : Vector
        The field to check for conservative property

    Examples
    ========

    >>> from sympy.vector import CoordSys3D
    >>> from sympy.vector import is_conservative
    >>> R = CoordSys3D('R')
    >>> is_conservative(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k)
    True
    >>> is_conservative(R.z*R.j)
    False

    """

    # Field is conservative irrespective of system
    # Take the first coordinate system in the result of the
    # separate method of Vector
    if not isinstance(field, Vector):
        raise TypeError("field should be a Vector")
    if field == Vector.zero:
        return True
    return curl(field).simplify() == Vector.zero


def is_solenoidal(field):
    """
    Checks if a field is solenoidal.

    Paramaters
    ==========

    field : Vector
        The field to check for solenoidal property

    Examples
    ========

    >>> from sympy.vector import CoordSys3D
    >>> from sympy.vector import is_solenoidal
    >>> R = CoordSys3D('R')
    >>> is_solenoidal(R.y*R.z*R.i + R.x*R.z*R.j + R.x*R.y*R.k)
    True
    >>> is_solenoidal(R.y * R.j)
    False

    """

    # Field is solenoidal irrespective of system
    # Take the first coordinate system in the result of the
    # separate method in Vector
    if not isinstance(field, Vector):
        raise TypeError("field should be a Vector")
    if field == Vector.zero:
        return True
    return divergence(field).simplify() == S(0)


def scalar_potential(field, coord_sys):
    """
    Returns the scalar potential function of a field in a given
    coordinate system (without the added integration constant).

    Parameters
    ==========

    field : Vector
        The vector field whose scalar potential function is to be
        calculated

    coord_sys : CoordSys3D
        The coordinate system to do the calculation in

    Examples
    ========

    >>> from sympy.vector import CoordSys3D
    >>> from sympy.vector import scalar_potential, gradient
    >>> R = CoordSys3D('R')
    >>> scalar_potential(R.k, R) == R.z
    True
    >>> scalar_field = 2*R.x**2*R.y*R.z
    >>> grad_field = gradient(scalar_field)
    >>> scalar_potential(grad_field, R)
    2*R.x**2*R.y*R.z

    """

    # Check whether field is conservative
    if not is_conservative(field):
        raise ValueError("Field is not conservative")
    if field == Vector.zero:
        return S(0)
    # Express the field exntirely in coord_sys
    # Subsitute coordinate variables also
    if not isinstance(coord_sys, CoordSys3D):
        raise TypeError("coord_sys must be a CoordSys3D")
    field = express(field, coord_sys, variables=True)
    dimensions = coord_sys.base_vectors()
    scalars = coord_sys.base_scalars()
    # Calculate scalar potential function
    temp_function = integrate(field.dot(dimensions[0]), scalars[0])
    for i, dim in enumerate(dimensions[1:]):
        partial_diff = diff(temp_function, scalars[i + 1])
        partial_diff = field.dot(dim) - partial_diff
        temp_function += integrate(partial_diff, scalars[i + 1])
    return temp_function


def scalar_potential_difference(field, coord_sys, point1, point2):
    """
    Returns the scalar potential difference between two points in a
    certain coordinate system, wrt a given field.

    If a scalar field is provided, its values at the two points are
    considered. If a conservative vector field is provided, the values
    of its scalar potential function at the two points are used.

    Returns (potential at point2) - (potential at point1)

    The position vectors of the two Points are calculated wrt the
    origin of the coordinate system provided.

    Parameters
    ==========

    field : Vector/Expr
        The field to calculate wrt

    coord_sys : CoordSys3D
        The coordinate system to do the calculations in

    point1 : Point
        The initial Point in given coordinate system

    position2 : Point
        The second Point in the given coordinate system

    Examples
    ========

    >>> from sympy.vector import CoordSys3D, Point
    >>> from sympy.vector import scalar_potential_difference
    >>> R = CoordSys3D('R')
    >>> P = R.origin.locate_new('P', R.x*R.i + R.y*R.j + R.z*R.k)
    >>> vectfield = 4*R.x*R.y*R.i + 2*R.x**2*R.j
    >>> scalar_potential_difference(vectfield, R, R.origin, P)
    2*R.x**2*R.y
    >>> Q = R.origin.locate_new('O', 3*R.i + R.j + 2*R.k)
    >>> scalar_potential_difference(vectfield, R, P, Q)
    -2*R.x**2*R.y + 18

    """

    if not isinstance(coord_sys, CoordSys3D):
        raise TypeError("coord_sys must be a CoordSys3D")
    if isinstance(field, Vector):
        # Get the scalar potential function
        scalar_fn = scalar_potential(field, coord_sys)
    else:
        # Field is a scalar
        scalar_fn = field
    # Express positions in required coordinate system
    origin = coord_sys.origin
    position1 = express(point1.position_wrt(origin), coord_sys,
                        variables=True)
    position2 = express(point2.position_wrt(origin), coord_sys,
                        variables=True)
    # Get the two positions as substitution dicts for coordinate variables
    subs_dict1 = {}
    subs_dict2 = {}
    scalars = coord_sys.base_scalars()
    for i, x in enumerate(coord_sys.base_vectors()):
        subs_dict1[scalars[i]] = x.dot(position1)
        subs_dict2[scalars[i]] = x.dot(position2)
    return scalar_fn.subs(subs_dict2) - scalar_fn.subs(subs_dict1)


def matrix_to_vector(matrix, system):
    """
    Converts a vector in matrix form to a Vector instance.

    It is assumed that the elements of the Matrix represent the
    measure numbers of the components of the vector along basis
    vectors of 'system'.

    Parameters
    ==========

    matrix : SymPy Matrix, Dimensions: (3, 1)
        The matrix to be converted to a vector

    system : CoordSys3D
        The coordinate system the vector is to be defined in

    Examples
    ========

    >>> from sympy import ImmutableMatrix as Matrix
    >>> m = Matrix([1, 2, 3])
    >>> from sympy.vector import CoordSys3D, matrix_to_vector
    >>> C = CoordSys3D('C')
    >>> v = matrix_to_vector(m, C)
    >>> v
    C.i + 2*C.j + 3*C.k
    >>> v.to_matrix(C) == m
    True

    """

    outvec = Vector.zero
    vects = system.base_vectors()
    for i, x in enumerate(matrix):
        outvec += x * vects[i]
    return outvec


def _path(from_object, to_object):
    """
    Calculates the 'path' of objects starting from 'from_object'
    to 'to_object', along with the index of the first common
    ancestor in the tree.

    Returns (index, list) tuple.
    """

    if from_object._root != to_object._root:
        raise ValueError("No connecting path found between " +
                         str(from_object) + " and " + str(to_object))

    other_path = []
    obj = to_object
    while obj._parent is not None:
        other_path.append(obj)
        obj = obj._parent
    other_path.append(obj)
    object_set = set(other_path)
    from_path = []
    obj = from_object
    while obj not in object_set:
        from_path.append(obj)
        obj = obj._parent
    index = len(from_path)
    i = other_path.index(obj)
    while i >= 0:
        from_path.append(other_path[i])
        i -= 1
    return index, from_path


def orthogonalize(*vlist, **kwargs):
    """
    Takes a sequence of independent vectors and orthogonalizes them
    using the Gram - Schmidt process. Returns a list of
    orthogonal or orthonormal vectors.

    Parameters
    ==========

    vlist : sequence of independent vectors to be made orthogonal.

    orthonormal : Optional parameter
                  Set to True if the the vectors returned should be
                  orthonormal.
                  Default: False

    Examples
    ========

    >>> from sympy.vector.coordsysrect import CoordSys3D
    >>> from sympy.vector.vector import Vector, BaseVector
    >>> from sympy.vector.functions import orthogonalize
    >>> C = CoordSys3D('C')
    >>> i, j, k = C.base_vectors()
    >>> v1 = i + 2*j
    >>> v2 = 2*i + 3*j
    >>> orthogonalize(v1, v2)
    [C.i + 2*C.j, 2/5*C.i + (-1/5)*C.j]

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Gram-Schmidt_process

    """
    orthonormal = kwargs.get('orthonormal', False)

    if not all(isinstance(vec, Vector) for vec in vlist):
        raise TypeError('Each element must be of Type Vector')

    ortho_vlist = []
    for i, term in enumerate(vlist):
        for j in range(i):
            term -= ortho_vlist[j].projection(vlist[i])
        # TODO : The following line introduces a performance issue
        # and needs to be changed once a good solution for issue #10279 is
        # found.
        if simplify(term).equals(Vector.zero):
            raise ValueError("Vector set not linearly independent")
        ortho_vlist.append(term)

    if orthonormal:
        ortho_vlist = [vec.normalize() for vec in ortho_vlist]

    return ortho_vlist
