"""Geometrical Planes.

Contains
========
Plane

"""
from __future__ import print_function, division

from sympy.core import S, C, sympify, Dummy, nan, Eq, symbols, Symbol, Rational
from sympy.core.function import expand_mul
from sympy.functions.elementary.trigonometric import _pi_coeff as pi_coeff, \
    sqrt
from sympy.core.logic import fuzzy_and
from sympy.core.exprtools import factor_terms
from sympy.simplify.simplify import simplify
from sympy.solvers import solve
from sympy.polys.polytools import cancel
from sympy.geometry.exceptions import GeometryError
from .entity import GeometryEntity
from .point3d import Point3D
from .point import Point
from .line3d import LinearEntity3D, Line3D, Segment3D, Ray3D
from .line import Line, Segment, Ray
from sympy.matrices import Matrix

class Plane(GeometryEntity):
    """
    A plane is a flat, two-dimensional surface. A plane is the two-dimensional
    analogue of a point (zero-dimensions), a line (one-dimension) and a solid
    (three-dimensions). A plane can generally be constructed by two types of
    inputs. They are three non-collinear points and a point and the plane's
    normal vector.

    Attributes
    ==========

    p1
    normal_vector

    Examples
    ========

    >>> from sympy import Plane, Point3D
    >>> from sympy.abc import x
    >>> Plane(Point3D(1, 1, 1), Point3D(2, 3, 4), Point3D(2, 2, 2))
    Plane(Point3D(1, 1, 1), (-1, 2, -1))
    >>> Plane((1, 1, 1), (2, 3, 4), (2, 2, 2))
    Plane(Point3D(1, 1, 1), (-1, 2, -1))
    >>> Plane(Point3D(1, 1, 1), normal_vector=(1,4,7))
    Plane(Point3D(1, 1, 1), (1, 4, 7))

    """
    def __new__(cls, p1, a=None, b=None, **kwargs):
        p1 = Point3D(p1)
        if not a and not b and kwargs.get('normal_vector', None):
            a = kwargs.pop('normal_vector')

        if not b and not isinstance(a, Point3D) and \
                len(a) == 3:
            normal_vector = a
        elif a and b:
            p2 = Point3D(a)
            p3 = Point3D(b)
            if Point3D.are_collinear(p1, p2, p3):
                raise NotImplementedError('Enter three non-collinear points')
            a = p1.direction_ratio(p2)
            b = p1.direction_ratio(p3)
            normal_vector = tuple(Matrix(a).cross(Matrix(b)))
        else:
            raise ValueError('Either provide 3 3D points or a point with a '
            'normal vector')
        return GeometryEntity.__new__(cls, p1, normal_vector, **kwargs)

    @property
    def p1(self):
        """The only defining point of the plane. Others can be obtained from the
        arbitrary_point method.

        See Also
        ========

        sympy.geometry.point3d.Point3D

        Examples
        ========

        >>> from sympy import Point3D, Plane
        >>> a = Plane(Point3D(1, 1, 1), Point3D(2, 3, 4), Point3D(2, 2, 2))
        >>> a.p1
        Point3D(1, 1, 1)

        """
        return self.args[0]

    @property
    def normal_vector(self):
        """Normal vector of the given plane.

        Examples
        ========

        >>> from sympy import Point3D, Plane
        >>> a = Plane(Point3D(1, 1, 1), Point3D(2, 3, 4), Point3D(2, 2, 2))
        >>> a.normal_vector
        (-1, 2, -1)
        >>> a = Plane(Point3D(1, 1, 1), normal_vector=(1, 4, 7))
        >>> a.normal_vector
        (1, 4, 7)

        """
        return self.args[1]

    def equation(self, x=None, y=None, z=None):
        """The equation of the Plane.

        Examples
        ========

        >>> from sympy import Point3D, Plane
        >>> a = Plane(Point3D(1, 1, 2), Point3D(2, 4, 7), Point3D(3, 5, 1))
        >>> a.equation()
        -23*x + 11*y - 2*z + 16
        >>> a = Plane(Point3D(1, 4, 2), normal_vector=(6, 6, 6))
        >>> a.equation()
        6*x + 6*y + 6*z - 42

        """
        x, y, z = [i if i else Symbol(j, real=True) for i, j in zip((x, y, z), 'xyz')]
        a = Point3D(x, y, z)
        b = self.p1.direction_ratio(a)
        c = self.normal_vector
        return (sum(i*j for i, j in zip(b, c)))

    def projection(self, pt):
        """Project the given point onto the plane along the plane normal.

        Parameters
        ==========

        Point or Point3D

        Returns
        =======

        Point3D

        Examples
        ========

        >>> from sympy import Plane, Point, Point3D
        >>> A = Plane(Point3D(1, 1, 2), normal_vector=(1, 1, 1))

        The projection is along the normal vector direction, not the z
        axis, so (1, 1) does not project to (1, 1, 2) on the plane A:

        >>> b = Point(1, 1)
        >>> A.projection(b)
        Point3D(5/3, 5/3, 2/3)
        >>> _ in A
        True

        But the point (1, 1, 2) projects to (1, 1) on the XY-plane:

        >>> XY = Plane((0, 0, 0), (0, 0, 1))
        >>> XY.projection((1, 1, 2))
        Point3D(1, 1, 0)
        """
        rv = Point3D(pt)
        if rv in self:
            return rv
        return self.intersection(Line3D(rv, rv + Point3D(self.normal_vector)))[0]


    def projection_line(self, line):
        """Project the given line onto the plane through the normal plane
        containing the line.

        Parameters
        ==========

        LinearEntity or LinearEntity3D

        Returns
        =======

        Point3D, Line3D, Ray3D or Segment3D

        Notes
        =====

        For the interaction between 2D and 3D lines(segments, rays), you should
        convert the line to 3D by using this method. For example for finding the
        intersection between a 2D and a 3D line, convert the 2D line to a 3D line
        by projecting it on a required plane and then proceed to find the
        intersection between those lines.

        Examples
        ========

        >>> from sympy import Plane, Line, Line3D, Point, Point3D
        >>> a = Plane(Point3D(1, 1, 1), normal_vector=(1, 1, 1))
        >>> b = Line(Point(1, 1), Point(2, 2))
        >>> a.projection_line(b)
        Line3D(Point3D(4/3, 4/3, 1/3), Point3D(5/3, 5/3, -1/3))
        >>> c = Line3D(Point3D(1, 1, 1), Point3D(2, 2, 2))
        >>> a.projection_line(c)
        Point3D(1, 1, 1)

        """
        from sympy.geometry.line import LinearEntity
        from sympy.geometry.line3d import LinearEntity3D
        if not isinstance(line, (LinearEntity, LinearEntity3D)):
            raise NotImplementedError('Enter a linear entity only')
        a, b = self.projection(line.p1), self.projection(line.p2)
        if a == b:
            # projection does not imply intersection so for
            # this case (line parallel to plane's normal) we
            # return the projection point
            return a
        if isinstance(line, (Line, Line3D)):
            return Line3D(a, b)
        if isinstance(line, (Ray, Ray3D)):
            return Ray3D(a, b)
        if isinstance(line, (Segment, Segment3D)):
            return Segment3D(a, b)

    def is_parallel(self, l):
        """Is the given geometric entity parallel to the plane?

        Parameters
        ==========

        LinearEntity3D or Plane

        Returns
        =======

        Boolean

        Examples
        ========

        >>> from sympy import Plane, Point3D
        >>> a = Plane(Point3D(1,4,6), normal_vector=(2, 4, 6))
        >>> b = Plane(Point3D(3,1,3), normal_vector=(4, 8, 12))
        >>> a.is_parallel(b)
        True

        """
        from sympy.geometry.line3d import LinearEntity3D
        if isinstance(l, LinearEntity3D):
            a = l.direction_ratio
            b = self.normal_vector
            c = sum([i*j for i, j in zip(a, b)])
            if c == 0:
                return True
            else:
                return False
        elif isinstance(l, Plane):
            a = Matrix(l.normal_vector)
            b = Matrix(self.normal_vector)
            if a.cross(b).is_zero:
                return True
            else:
                return False

    def is_perpendicular(self, l):
        """is the given geometric entity perpendicualar to the given plane?

        Parameters
        ==========

        LinearEntity3D or Plane

        Returns
        =======

        Boolean

        Examples
        ========

        >>> from sympy import Plane, Point3D
        >>> a = Plane(Point3D(1,4,6), normal_vector=(2, 4, 6))
        >>> b = Plane(Point3D(2, 2, 2), normal_vector=(-1, 2, -1))
        >>> a.is_perpendicular(b)
        True

        """
        from sympy.geometry.line3d import LinearEntity3D
        if isinstance(l, LinearEntity3D):
            a = Matrix(l.direction_ratio)
            b = Matrix(self.normal_vector)
            if a.cross(b).is_zero:
                return True
            else:
                return False
        elif isinstance(l, Plane):
           a = Matrix(l.normal_vector)
           b = Matrix(self.normal_vector)
           if a.dot(b) == 0:
               return True
           else:
               return False
        else:
            return False

    def distance(self, o):
        """Distance beteen the plane and another geometric entity.

        Parameters
        ==========

        Point3D, LinearEntity3D, Plane.

        Returns
        =======

        distance

        Notes
        =====

        This method accepts only 3D entities as it's parameter, but if you want
        to calculate the distance between a 2D entity and a plane you should
        first convert to a 3D entity by projecting onto a desired plane and
        then proceed to calculate the distance.

        Examples
        ========

        >>> from sympy import Point, Point3D, Line, Line3D, Plane
        >>> a = Plane(Point3D(1, 1, 1), normal_vector=(1, 1, 1))
        >>> b = Point3D(1, 2, 3)
        >>> a.distance(b)
        sqrt(3)
        >>> c = Line3D(Point3D(2, 3, 1), Point3D(1, 2, 2))
        >>> a.distance(c)
        0

        """
        from sympy.geometry.line3d import LinearEntity3D
        x, y, z = map(Dummy, 'xyz')
        if self.intersection(o) != []:
            return S.Zero

        if isinstance(o, Point3D):
           x, y, z = map(Dummy, 'xyz')
           k = self.equation(x, y, z)
           a, b, c = [k.coeff(i) for i in (x, y, z)]
           d = k.xreplace({x: o.args[0], y: o.args[1], z: o.args[2]})
           t = abs(d/sqrt(a**2 + b**2 + c**2))
           return t
        if isinstance(o, LinearEntity3D):
            a, b = o.p1, self.p1
            c = Matrix(a.direction_ratio(b))
            d = Matrix(self.normal_vector)
            e = c.dot(d)
            f = sqrt(sum([i**2 for i in self.normal_vector]))
            return abs(e / f)
        if isinstance(o, Plane):
            a, b = o.p1, self.p1
            c = Matrix(a.direction_ratio(b))
            d = Matrix(self.normal_vector)
            e = c.dot(d)
            f = sqrt(sum([i**2 for i in self.normal_vector]))
            return abs(e / f)

    def angle_between(self, o):
        """Angle between the plane and other geometric entity.

        Parameters
        ==========

        LinearEntity3D, Plane.

        Returns
        =======

        angle : angle in radians

        Notes
        =====

        This method accepts only 3D entities as it's parameter, but if you want
        to calculate the angle between a 2D entity and a plane you should
        first convert to a 3D entity by projecting onto a desired plane and
        then proceed to calculate the angle.

        Examples
        ========

        >>> from sympy import Point3D, Line3D, Plane
        >>> a = Plane(Point3D(1, 2, 2), normal_vector=(1, 2, 3))
        >>> b = Line3D(Point3D(1, 3, 4), Point3D(2, 2, 2))
        >>> a.angle_between(b)
        -asin(sqrt(21)/6)

        """
        from sympy.geometry.line3d import LinearEntity3D
        if isinstance(o, LinearEntity3D):
            a = Matrix(self.normal_vector)
            b = Matrix(o.direction_ratio)
            c = a.dot(b)
            d = sqrt(sum([i**2 for i in self.normal_vector]))
            e = sqrt(sum([i**2 for i in o.direction_ratio]))
            return C.asin(c/(d*e))
        if isinstance(o, Plane):
            a = Matrix(self.normal_vector)
            b = Matrix(o.normal_vector)
            c = a.dot(b)
            d = sqrt(sum([i**2 for i in self.normal_vector]))
            e = sqrt(sum([i**2 for i in o.normal_vector]))
            return C.acos(c/(d*e))


    @staticmethod
    def are_concurrent(*planes):
        """Is a sequence of Planes concurrent?

        Two or more Planes are concurrent if their intersections
        are a common line.

        Parameters
        ==========

        planes: list

        Returns
        =======

        Boolean

        Examples
        ========

        >>> from sympy import Plane, Point3D
        >>> a = Plane(Point3D(5, 0, 0), normal_vector=(1, -1, 1))
        >>> b = Plane(Point3D(0, -2, 0), normal_vector=(3, 1, 1))
        >>> c = Plane(Point3D(0, -1, 0), normal_vector=(5, -1, 9))
        >>> Plane.are_concurrent(a, b)
        True
        >>> Plane.are_concurrent(a, b, c)
        False

        """
        planes = set(planes)
        if len(planes) < 2:
            return False
        for i in planes:
            if not isinstance(i, Plane):
                raise ValueError('All objects should be Planes but got %s' % i.func)
        planes = list(planes)
        first = planes.pop(0)
        sol = first.intersection(planes[0])
        if sol == []:
            return False
        else:
            line = sol[0]
            for i in planes[1:]:
                l = first.intersection(i)
                if not l or not l[0] in line:
                    return False
            return True

    def perpendicular_line(self, pt):
        """A line perpendicular to the given plane.

        Parameters
        ==========

        pt: Point3D

        Returns
        =======

        Line3D

        Examples
        ========

        >>> from sympy import Plane, Point3D, Line3D
        >>> a = Plane(Point3D(1,4,6), normal_vector=(2, 4, 6))
        >>> a.perpendicular_line(Point3D(9, 8, 7))
        Line3D(Point3D(9, 8, 7), Point3D(11, 12, 13))

        """
        a = self.normal_vector
        return Line3D(pt, direction_ratio=a)

    def parallel_plane(self, pt):
        """
        Plane parallel to the given plane and passing through the point pt.

        Parameters
        ==========

        pt: Point3D

        Returns
        =======

        Plane

        Examples
        ========

        >>> from sympy import Plane, Point3D
        >>> a = Plane(Point3D(1, 4, 6), normal_vector=(2, 4, 6))
        >>> a.parallel_plane(Point3D(2, 3, 5))
        Plane(Point3D(2, 3, 5), (2, 4, 6))

        """
        a = self.normal_vector
        return Plane(pt, normal_vector=a)

    def perpendicular_plane(self, *pts):
        """
        Return a perpendicular passing through the given points. If the
        direction ratio between the points is the same as the Plane's normal
        vector then, to select from the infinite number of possible planes,
        a third point will be chosen on the z-axis (or the y-axis
        if the normal vector is already parallel to the z-axis). If less than
        two points are given they will be supplied as follows: if no point is
        given then pt1 will be self.p1; if a second point is not given it will
        be a point through pt1 on a line parallel to the z-axis (if the normal
        is not already the z-axis, otherwise on the line parallel to the
        y-axis).

        Parameters
        ==========

        pts: 0, 1 or 2 Point3D

        Returns
        =======

        Plane

        Examples
        ========

        >>> from sympy import Plane, Point3D, Line3D
        >>> a, b = Point3D(0, 0, 0), Point3D(0, 1, 0)
        >>> Z = (0, 0, 1)
        >>> p = Plane(a, normal_vector=Z)
        >>> p.perpendicular_plane(a, b)
        Plane(Point3D(0, 0, 0), (1, 0, 0))
        """
        if len(pts) > 2:
            raise ValueError('No more than 2 pts should be provided.')

        pts = list(pts)
        if len(pts) == 0:
            pts.append(self.p1)
        if len(pts) == 1:
            x, y, z = self.normal_vector
            if x == y == 0:
                dir = (0, 1, 0)
            else:
                dir = (0, 0, 1)
            pts.append(pts[0] + Point3D(*dir))

        p1, p2 = [Point3D(i) for i in pts]
        l = Line3D(p1, p2)
        n = Line3D(p1, direction_ratio=self.normal_vector)
        if l in n:  # XXX should an error be raised instead?
            # there are infinitely many perpendicular planes;
            x, y, z = self.normal_vector
            if x == y == 0:
                # the z axis is the normal so pick a pt on the y-axis
                p3 = Point3D(0, 1, 0)  # case 1
            else:
                # else pick a pt on the z axis
                p3 = Point3D(0, 0, 1)  # case 2
            # in case that point is already given, move it a bit
            if p3 in l:
                p3 *= 2  # case 3
        else:
            p3 = p1 + Point3D(*self.normal_vector)  # case 4
        return Plane(p1, p2, p3)

    def random_point(self, seed=None):
        """ Returns a random point on the Plane.

        Returns
        =======

        Point3D

        """
        import random
        if seed is not None:
            rng = random.Random(seed)
        else:
            rng = random
        t = Dummy('t')
        return self.arbitrary_point(t).subs(t, Rational(rng.random()))

    def arbitrary_point(self, t=None):
        """ Returns an arbitrary point on the Plane; varying `t` from 0 to 2*pi
        will move the point in a circle of radius 1 about p1 of the Plane.

        Examples
        ========

        >>> from sympy.geometry.plane import Plane
        >>> from sympy.abc import t
        >>> p = Plane((0, 0, 0), (0, 0, 1), (0, 1, 0))
        >>> p.arbitrary_point(t)
        Point3D(0, cos(t), sin(t))
        >>> _.distance(p.p1).simplify()
        1

        Returns
        =======

        Point3D

        """
        from sympy import cos, sin
        t = t or Dummy('t')
        x, y, z = self.normal_vector
        a, b, c = self.p1.args
        if x == y == 0:
            return Point3D(a + cos(t), b + sin(t), c)
        elif x == z == 0:
            return Point3D(a + cos(t), b, c + sin(t))
        elif y == z == 0:
            return Point3D(a, b + cos(t), c + sin(t))
        m = Dummy()
        p = self.projection(Point3D(self.p1.x + cos(t), self.p1.y + sin(t), 0)*m)
        return p.xreplace({m: solve(p.distance(self.p1) - 1, m)[0]})

    def intersection(self, o):
        """ The intersection with other geometrical entity.

        Parameters
        ==========

        Point, Point3D, LinearEntity, LinearEntity3D, Plane

        Returns
        =======

        List

        Examples
        ========

        >>> from sympy import Point, Point3D, Line, Line3D, Plane
        >>> a = Plane(Point3D(1, 2, 3), normal_vector=(1, 1, 1))
        >>> b = Point3D(1, 2, 3)
        >>> a.intersection(b)
        [Point3D(1, 2, 3)]
        >>> c = Line3D(Point3D(1, 4, 7), Point3D(2, 2, 2))
        >>> a.intersection(c)
        [Point3D(2, 2, 2)]
        >>> d = Plane(Point3D(6, 0, 0), normal_vector=(2, -5, 3))
        >>> e = Plane(Point3D(2, 0, 0), normal_vector=(3, 4, -3))
        >>> d.intersection(e)
        [Line3D(Point3D(78/23, -24/23, 0), Point3D(147/23, 321/23, 23))]

        """
        from sympy.geometry.line3d import LinearEntity3D
        from sympy.geometry.line import LinearEntity
        if isinstance(o, (Point, Point3D)):
            if o in self:
                return [Point3D(o)]
            else:
                return []
        if isinstance(o, (LinearEntity, LinearEntity3D)):
            if o in self:
                p1, p2 = o.p1, o.p2
                if isinstance(o, Segment):
                    o = Segment3D(p1, p2)
                elif isinstance(o, Ray):
                    o = Ray3D(p1, p2)
                elif isinstance(o, Line):
                    o = Line3D(p1, p2)
                else:
                    raise ValueError('unhandled linear entity: %s' % o.func)
                return [o]
            else:
                x, y, z = map(Dummy, 'xyz')
                t = Dummy()  # unnamed else it may clash with a symbol in o
                a = Point3D(o.arbitrary_point(t))
                b = self.equation(x, y, z)
                c = solve(b.subs(list(zip((x, y, z), a.args))), t)
                if not c:
                    return []
                else:
                    p = a.subs(t, c[0])
                    if p not in self:
                        return []  # e.g. a segment might not intersect a plane
                    return [p]
        if isinstance(o, Plane):
            if o == self:
                return [self]
            if self.is_parallel(o):
                return []
            else:
                x, y, z = map(Dummy, 'xyz')
                a, b = Matrix([self.normal_vector]), Matrix([o.normal_vector])
                c = list(a.cross(b))
                d = self.equation(x, y, z)
                e = o.equation(x, y, z)
                f = solve((d.subs(z, 0), e.subs(z, 0)), [x, y])
                if len(f) == 2:
                    return [Line3D(Point3D(f[x], f[y], 0), direction_ratio=c)]
                g = solve((d.subs(y, 0), e.subs(y, 0)),[x, z])
                if len(g) == 2:
                    return [Line3D(Point3D(g[x], 0, g[z]), direction_ratio=c)]
                h = solve((d.subs(x, 0), e.subs(x, 0)),[y, z])
                if len(h) == 2:
                    return [Line3D(Point3D(0, h[y], h[z]), direction_ratio=c)]

    def __contains__(self, o):
        from sympy.geometry.line3d import LinearEntity3D
        from sympy.geometry.line import LinearEntity
        x, y, z = map(Dummy, 'xyz')
        k = self.equation(x, y, z)
        if isinstance(o, Point):
            o = Point3D(o)
        if isinstance(o, Point3D):
            d = k.xreplace(dict(zip((x, y, z), o.args)))
            return d.equals(0)
        elif isinstance(o, (LinearEntity, LinearEntity3D)):
            t = Dummy()
            d = Point3D(o.arbitrary_point(t))
            e = k.subs([(x, d.x), (y, d.y), (z, d.z)])
            return e.equals(0)
        else:
            return False

    def is_coplanar(self, o):
        """ Returns True if `o` is coplanar with self, else False.

        Examples
        ========

        >>> from sympy import Plane, Point3D
        >>> o = (0, 0, 0)
        >>> p = Plane(o, (1, 1, 1))
        >>> p2 = Plane(o, (2, 2, 2))
        >>> p == p2
        False
        >>> p.is_coplanar(p2)
        True
        """
        if isinstance(o, Plane):
            x, y, z = map(Dummy, 'xyz')
            return not cancel(self.equation(x, y, z)/o.equation(x, y, z)).has(x, y, z)
        if isinstance(o, Point3D):
            return o in self
        elif isinstance(o, LinearEntity3D):
            return all(i in self for i in self)
        elif isinstance(o, GeometryEntity):  # XXX should only be handling 2D objects now
            return all(i == 0 for i in self.normal_vector[:2])
