"""Parabolic geometrical entity.

Contains
* Parabola

"""

from sympy.core import S
from sympy.core.compatibility import ordered
from sympy.core.symbol import _symbol
from sympy import symbols, simplify, solve  # type:ignore
from sympy.geometry.entity import GeometryEntity, GeometrySet
from sympy.geometry.point import Point, Point2D
from sympy.geometry.line import Line, Line2D, Ray2D, Segment2D, LinearEntity3D
from sympy.geometry.ellipse import Ellipse
from sympy.functions import sign

class Parabola(GeometrySet):
    """A parabolic GeometryEntity.

    A parabola is declared with a point, that is called 'focus', and
    a line, that is called 'directrix'.
    Only vertical or horizontal parabolas are currently supported.

    Parameters
    ==========

    focus : Point
        Default value is Point(0, 0)
    directrix : Line

    Attributes
    ==========

    focus
    directrix
    axis of symmetry
    focal length
    p parameter
    vertex
    eccentricity

    Raises
    ======
    ValueError
        When `focus` is not a two dimensional point.
        When `focus` is a point of directrix.
    NotImplementedError
        When `directrix` is neither horizontal nor vertical.

    Examples
    ========

    >>> from sympy import Parabola, Point, Line
    >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7,8)))
    >>> p1.focus
    Point2D(0, 0)
    >>> p1.directrix
    Line2D(Point2D(5, 8), Point2D(7, 8))

    """

    def __new__(cls, focus=None, directrix=None, **kwargs):

        if focus:
            focus = Point(focus, dim=2)
        else:
            focus = Point(0, 0)

        directrix = Line(directrix)

        if (directrix.slope != 0 and directrix.slope != S.Infinity):
            raise NotImplementedError('The directrix must be a horizontal'
                                      ' or vertical line')
        if directrix.contains(focus):
            raise ValueError('The focus must not be a point of directrix')

        return GeometryEntity.__new__(cls, focus, directrix, **kwargs)

    @property
    def ambient_dimension(self):
        """Returns the ambient dimension of parabola.

        Returns
        =======

        ambient_dimension : integer

        Examples
        ========

        >>> from sympy import Parabola, Point, Line
        >>> f1 = Point(0, 0)
        >>> p1 = Parabola(f1, Line(Point(5, 8), Point(7, 8)))
        >>> p1.ambient_dimension
        2

        """
        return S(2)

    @property
    def axis_of_symmetry(self):
        """The axis of symmetry of the parabola.

        Returns
        =======

        axis_of_symmetry : Line

        See Also
        ========

        sympy.geometry.line.Line

        Examples
        ========

        >>> from sympy import Parabola, Point, Line
        >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
        >>> p1.axis_of_symmetry
        Line2D(Point2D(0, 0), Point2D(0, 1))

        """
        return self.directrix.perpendicular_line(self.focus)

    @property
    def directrix(self):
        """The directrix of the parabola.

        Returns
        =======

        directrix : Line

        See Also
        ========

        sympy.geometry.line.Line

        Examples
        ========

        >>> from sympy import Parabola, Point, Line
        >>> l1 = Line(Point(5, 8), Point(7, 8))
        >>> p1 = Parabola(Point(0, 0), l1)
        >>> p1.directrix
        Line2D(Point2D(5, 8), Point2D(7, 8))

        """
        return self.args[1]

    @property
    def eccentricity(self):
        """The eccentricity of the parabola.

        Returns
        =======

        eccentricity : number

        A parabola may also be characterized as a conic section with an
        eccentricity of 1. As a consequence of this, all parabolas are
        similar, meaning that while they can be different sizes,
        they are all the same shape.

        See Also
        ========

        https://en.wikipedia.org/wiki/Parabola


        Examples
        ========

        >>> from sympy import Parabola, Point, Line
        >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
        >>> p1.eccentricity
        1

        Notes
        -----
        The eccentricity for every Parabola is 1 by definition.

        """
        return S.One

    def equation(self, x='x', y='y'):
        """The equation of the parabola.

        Parameters
        ==========
        x : str, optional
            Label for the x-axis. Default value is 'x'.
        y : str, optional
            Label for the y-axis. Default value is 'y'.

        Returns
        =======
        equation : sympy expression

        Examples
        ========

        >>> from sympy import Parabola, Point, Line
        >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
        >>> p1.equation()
        -x**2 - 16*y + 64
        >>> p1.equation('f')
        -f**2 - 16*y + 64
        >>> p1.equation(y='z')
        -x**2 - 16*z + 64

        """
        x = _symbol(x, real=True)
        y = _symbol(y, real=True)

        if (self.axis_of_symmetry.slope == 0):
            t1 = 4 * (self.p_parameter) * (x - self.vertex.x)
            t2 = (y - self.vertex.y)**2
        else:
            t1 = 4 * (self.p_parameter) * (y - self.vertex.y)
            t2 = (x - self.vertex.x)**2

        return t1 - t2

    @property
    def focal_length(self):
        """The focal length of the parabola.

        Returns
        =======

        focal_lenght : number or symbolic expression

        Notes
        =====

        The distance between the vertex and the focus
        (or the vertex and directrix), measured along the axis
        of symmetry, is the "focal length".

        See Also
        ========

        https://en.wikipedia.org/wiki/Parabola

        Examples
        ========

        >>> from sympy import Parabola, Point, Line
        >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
        >>> p1.focal_length
        4

        """
        distance = self.directrix.distance(self.focus)
        focal_length = distance/2

        return focal_length

    @property
    def focus(self):
        """The focus of the parabola.

        Returns
        =======

        focus : Point

        See Also
        ========

        sympy.geometry.point.Point

        Examples
        ========

        >>> from sympy import Parabola, Point, Line
        >>> f1 = Point(0, 0)
        >>> p1 = Parabola(f1, Line(Point(5, 8), Point(7, 8)))
        >>> p1.focus
        Point2D(0, 0)

        """
        return self.args[0]

    def intersection(self, o):
        """The intersection of the parabola and another geometrical entity `o`.

        Parameters
        ==========

        o : GeometryEntity, LinearEntity

        Returns
        =======

        intersection : list of GeometryEntity objects

        Examples
        ========

        >>> from sympy import Parabola, Point, Ellipse, Line, Segment
        >>> p1 = Point(0,0)
        >>> l1 = Line(Point(1, -2), Point(-1,-2))
        >>> parabola1 = Parabola(p1, l1)
        >>> parabola1.intersection(Ellipse(Point(0, 0), 2, 5))
        [Point2D(-2, 0), Point2D(2, 0)]
        >>> parabola1.intersection(Line(Point(-7, 3), Point(12, 3)))
        [Point2D(-4, 3), Point2D(4, 3)]
        >>> parabola1.intersection(Segment((-12, -65), (14, -68)))
        []

        """
        x, y = symbols('x y', real=True)
        parabola_eq = self.equation()
        if isinstance(o, Parabola):
            if o in self:
                return [o]
            else:
                return list(ordered([Point(i) for i in solve([parabola_eq, o.equation()], [x, y])]))
        elif isinstance(o, Point2D):
            if simplify(parabola_eq.subs([(x, o._args[0]), (y, o._args[1])])) == 0:
                return [o]
            else:
                return []
        elif isinstance(o, (Segment2D, Ray2D)):
            result = solve([parabola_eq, Line2D(o.points[0], o.points[1]).equation()], [x, y])
            return list(ordered([Point2D(i) for i in result if i in o]))
        elif isinstance(o, (Line2D, Ellipse)):
            return list(ordered([Point2D(i) for i in solve([parabola_eq, o.equation()], [x, y])]))
        elif isinstance(o, LinearEntity3D):
            raise TypeError('Entity must be two dimensional, not three dimensional')
        else:
            raise TypeError('Wrong type of argument were put')

    @property
    def p_parameter(self):
        """P is a parameter of parabola.

        Returns
        =======

        p : number or symbolic expression

        Notes
        =====

        The absolute value of p is the focal length. The sign on p tells
        which way the parabola faces. Vertical parabolas that open up
        and horizontal that open right, give a positive value for p.
        Vertical parabolas that open down and horizontal that open left,
        give a negative value for p.


        See Also
        ========

        http://www.sparknotes.com/math/precalc/conicsections/section2.rhtml

        Examples
        ========

        >>> from sympy import Parabola, Point, Line
        >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
        >>> p1.p_parameter
        -4

        """
        if self.axis_of_symmetry.slope == 0:
            x = self.directrix.coefficients[2]
            p = sign(self.focus.args[0] + x)
        else:
            y = self.directrix.coefficients[2]
            p = sign(self.focus.args[1] + y)
        return p * self.focal_length

    @property
    def vertex(self):
        """The vertex of the parabola.

        Returns
        =======

        vertex : Point

        See Also
        ========

        sympy.geometry.point.Point

        Examples
        ========

        >>> from sympy import Parabola, Point, Line
        >>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
        >>> p1.vertex
        Point2D(0, 4)

        """
        focus = self.focus
        if (self.axis_of_symmetry.slope == 0):
            vertex = Point(focus.args[0] - self.p_parameter, focus.args[1])
        else:
            vertex = Point(focus.args[0], focus.args[1] - self.p_parameter)

        return vertex