Sources_For_Win/ParaView/Lib/site-packages/sympy/geometry/entity.py

"""The definition of the base geometrical entity with attributes common to
all derived geometrical entities.

Contains
========

GeometryEntity
GeometricSet

Notes
=====

A GeometryEntity is any object that has special geometric properties.
A GeometrySet is a superclass of any GeometryEntity that can also
be viewed as a sympy.sets.Set.  In particular, points are the only
GeometryEntity not considered a Set.

Rn is a GeometrySet representing n-dimensional Euclidean space. R2 and
R3 are currently the only ambient spaces implemented.

"""

from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.evalf import EvalfMixin, N
from sympy.core.numbers import oo
from sympy.core.symbol import Dummy
from sympy.core.sympify import sympify
from sympy.functions.elementary.trigonometric import cos, sin, atan
from sympy.matrices import eye
from sympy.multipledispatch import dispatch
from sympy.sets import Set, Union, FiniteSet
from sympy.sets.handlers.intersection import intersection_sets
from sympy.sets.handlers.union import union_sets
from sympy.utilities.misc import func_name
from sympy.utilities.iterables import is_sequence


# How entities are ordered; used by __cmp__ in GeometryEntity
ordering_of_classes = [
    "Point2D",
    "Point3D",
    "Point",
    "Segment2D",
    "Ray2D",
    "Line2D",
    "Segment3D",
    "Line3D",
    "Ray3D",
    "Segment",
    "Ray",
    "Line",
    "Plane",
    "Triangle",
    "RegularPolygon",
    "Polygon",
    "Circle",
    "Ellipse",
    "Curve",
    "Parabola"
]


class GeometryEntity(Basic, EvalfMixin):
    """The base class for all geometrical entities.

    This class doesn't represent any particular geometric entity, it only
    provides the implementation of some methods common to all subclasses.

    """

    def __cmp__(self, other):
        """Comparison of two GeometryEntities."""
        n1 = self.__class__.__name__
        n2 = other.__class__.__name__
        c = (n1 > n2) - (n1 < n2)
        if not c:
            return 0

        i1 = -1
        for cls in self.__class__.__mro__:
            try:
                i1 = ordering_of_classes.index(cls.__name__)
                break
            except ValueError:
                i1 = -1
        if i1 == -1:
            return c

        i2 = -1
        for cls in other.__class__.__mro__:
            try:
                i2 = ordering_of_classes.index(cls.__name__)
                break
            except ValueError:
                i2 = -1
        if i2 == -1:
            return c

        return (i1 > i2) - (i1 < i2)

    def __contains__(self, other):
        """Subclasses should implement this method for anything more complex than equality."""
        if type(self) is type(other):
            return self == other
        raise NotImplementedError()

    def __getnewargs__(self):
        """Returns a tuple that will be passed to __new__ on unpickling."""
        return tuple(self.args)

    def __ne__(self, o):
        """Test inequality of two geometrical entities."""
        return not self == o

    def __new__(cls, *args, **kwargs):
        # Points are sequences, but they should not
        # be converted to Tuples, so use this detection function instead.
        def is_seq_and_not_point(a):
            # we cannot use isinstance(a, Point) since we cannot import Point
            if hasattr(a, 'is_Point') and a.is_Point:
                return False
            return is_sequence(a)

        args = [Tuple(*a) if is_seq_and_not_point(a) else sympify(a) for a in args]
        return Basic.__new__(cls, *args)

    def __radd__(self, a):
        """Implementation of reverse add method."""
        return a.__add__(self)

    def __rtruediv__(self, a):
        """Implementation of reverse division method."""
        return a.__truediv__(self)

    def __repr__(self):
        """String representation of a GeometryEntity that can be evaluated
        by sympy."""
        return type(self).__name__ + repr(self.args)

    def __rmul__(self, a):
        """Implementation of reverse multiplication method."""
        return a.__mul__(self)

    def __rsub__(self, a):
        """Implementation of reverse subtraction method."""
        return a.__sub__(self)

    def __str__(self):
        """String representation of a GeometryEntity."""
        from sympy.printing import sstr
        return type(self).__name__ + sstr(self.args)

    def _eval_subs(self, old, new):
        from sympy.geometry.point import Point, Point3D
        if is_sequence(old) or is_sequence(new):
            if isinstance(self, Point3D):
                old = Point3D(old)
                new = Point3D(new)
            else:
                old = Point(old)
                new = Point(new)
            return  self._subs(old, new)

    def _repr_svg_(self):
        """SVG representation of a GeometryEntity suitable for IPython"""

        try:
            bounds = self.bounds
        except (NotImplementedError, TypeError):
            # if we have no SVG representation, return None so IPython
            # will fall back to the next representation
            return None

        if not all(x.is_number and x.is_finite for x in bounds):
            return None

        svg_top = '''<svg xmlns="http://www.w3.org/2000/svg"
            xmlns:xlink="http://www.w3.org/1999/xlink"
            width="{1}" height="{2}" viewBox="{0}"
            preserveAspectRatio="xMinYMin meet">
            <defs>
                <marker id="markerCircle" markerWidth="8" markerHeight="8"
                    refx="5" refy="5" markerUnits="strokeWidth">
                    <circle cx="5" cy="5" r="1.5" style="stroke: none; fill:#000000;"/>
                </marker>
                <marker id="markerArrow" markerWidth="13" markerHeight="13" refx="2" refy="4"
                       orient="auto" markerUnits="strokeWidth">
                    <path d="M2,2 L2,6 L6,4" style="fill: #000000;" />
                </marker>
                <marker id="markerReverseArrow" markerWidth="13" markerHeight="13" refx="6" refy="4"
                       orient="auto" markerUnits="strokeWidth">
                    <path d="M6,2 L6,6 L2,4" style="fill: #000000;" />
                </marker>
            </defs>'''

        # Establish SVG canvas that will fit all the data + small space
        xmin, ymin, xmax, ymax = map(N, bounds)
        if xmin == xmax and ymin == ymax:
            # This is a point; buffer using an arbitrary size
            xmin, ymin, xmax, ymax = xmin - .5, ymin -.5, xmax + .5, ymax + .5
        else:
            # Expand bounds by a fraction of the data ranges
            expand = 0.1  # or 10%; this keeps arrowheads in view (R plots use 4%)
            widest_part = max([xmax - xmin, ymax - ymin])
            expand_amount = widest_part * expand
            xmin -= expand_amount
            ymin -= expand_amount
            xmax += expand_amount
            ymax += expand_amount
        dx = xmax - xmin
        dy = ymax - ymin
        width = min([max([100., dx]), 300])
        height = min([max([100., dy]), 300])

        scale_factor = 1. if max(width, height) == 0 else max(dx, dy) / max(width, height)
        try:
            svg = self._svg(scale_factor)
        except (NotImplementedError, TypeError):
            # if we have no SVG representation, return None so IPython
            # will fall back to the next representation
            return None

        view_box = "{} {} {} {}".format(xmin, ymin, dx, dy)
        transform = "matrix(1,0,0,-1,0,{})".format(ymax + ymin)
        svg_top = svg_top.format(view_box, width, height)

        return svg_top + (
            '<g transform="{}">{}</g></svg>'
            ).format(transform, svg)

    def _svg(self, scale_factor=1., fill_color="#66cc99"):
        """Returns SVG path element for the GeometryEntity.

        Parameters
        ==========

        scale_factor : float
            Multiplication factor for the SVG stroke-width.  Default is 1.
        fill_color : str, optional
            Hex string for fill color. Default is "#66cc99".
        """
        raise NotImplementedError()

    def _sympy_(self):
        return self

    @property
    def ambient_dimension(self):
        """What is the dimension of the space that the object is contained in?"""
        raise NotImplementedError()

    @property
    def bounds(self):
        """Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
        rectangle for the geometric figure.

        """

        raise NotImplementedError()

    def encloses(self, o):
        """
        Return True if o is inside (not on or outside) the boundaries of self.

        The object will be decomposed into Points and individual Entities need
        only define an encloses_point method for their class.

        See Also
        ========

        sympy.geometry.ellipse.Ellipse.encloses_point
        sympy.geometry.polygon.Polygon.encloses_point

        Examples
        ========

        >>> from sympy import RegularPolygon, Point, Polygon
        >>> t  = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
        >>> t2 = Polygon(*RegularPolygon(Point(0, 0), 2, 3).vertices)
        >>> t2.encloses(t)
        True
        >>> t.encloses(t2)
        False

        """

        from sympy.geometry.point import Point
        from sympy.geometry.line import Segment, Ray, Line
        from sympy.geometry.ellipse import Ellipse
        from sympy.geometry.polygon import Polygon, RegularPolygon

        if isinstance(o, Point):
            return self.encloses_point(o)
        elif isinstance(o, Segment):
            return all(self.encloses_point(x) for x in o.points)
        elif isinstance(o, (Ray, Line)):
            return False
        elif isinstance(o, Ellipse):
            return self.encloses_point(o.center) and \
                self.encloses_point(
                Point(o.center.x + o.hradius, o.center.y)) and \
                not self.intersection(o)
        elif isinstance(o, Polygon):
            if isinstance(o, RegularPolygon):
                if not self.encloses_point(o.center):
                    return False
            return all(self.encloses_point(v) for v in o.vertices)
        raise NotImplementedError()

    def equals(self, o):
        return self == o

    def intersection(self, o):
        """
        Returns a list of all of the intersections of self with o.

        Notes
        =====

        An entity is not required to implement this method.

        If two different types of entities can intersect, the item with
        higher index in ordering_of_classes should implement
        intersections with anything having a lower index.

        See Also
        ========

        sympy.geometry.util.intersection

        """
        raise NotImplementedError()

    def is_similar(self, other):
        """Is this geometrical entity similar to another geometrical entity?

        Two entities are similar if a uniform scaling (enlarging or
        shrinking) of one of the entities will allow one to obtain the other.

        Notes
        =====

        This method is not intended to be used directly but rather
        through the `are_similar` function found in util.py.
        An entity is not required to implement this method.
        If two different types of entities can be similar, it is only
        required that one of them be able to determine this.

        See Also
        ========

        scale

        """
        raise NotImplementedError()

    def reflect(self, line):
        """
        Reflects an object across a line.

        Parameters
        ==========

        line: Line

        Examples
        ========

        >>> from sympy import pi, sqrt, Line, RegularPolygon
        >>> l = Line((0, pi), slope=sqrt(2))
        >>> pent = RegularPolygon((1, 2), 1, 5)
        >>> rpent = pent.reflect(l)
        >>> rpent
        RegularPolygon(Point2D(-2*sqrt(2)*pi/3 - 1/3 + 4*sqrt(2)/3, 2/3 + 2*sqrt(2)/3 + 2*pi/3), -1, 5, -atan(2*sqrt(2)) + 3*pi/5)

        >>> from sympy import pi, Line, Circle, Point
        >>> l = Line((0, pi), slope=1)
        >>> circ = Circle(Point(0, 0), 5)
        >>> rcirc = circ.reflect(l)
        >>> rcirc
        Circle(Point2D(-pi, pi), -5)

        """
        from sympy.geometry.point import Point

        g = self
        l = line
        o = Point(0, 0)
        if l.slope.is_zero:
            y = l.args[0].y
            if not y:  # x-axis
                return g.scale(y=-1)
            reps = [(p, p.translate(y=2*(y - p.y))) for p in g.atoms(Point)]
        elif l.slope is oo:
            x = l.args[0].x
            if not x:  # y-axis
                return g.scale(x=-1)
            reps = [(p, p.translate(x=2*(x - p.x))) for p in g.atoms(Point)]
        else:
            if not hasattr(g, 'reflect') and not all(
                    isinstance(arg, Point) for arg in g.args):
                raise NotImplementedError(
                    'reflect undefined or non-Point args in %s' % g)
            a = atan(l.slope)
            c = l.coefficients
            d = -c[-1]/c[1]  # y-intercept
            # apply the transform to a single point
            x, y = Dummy(), Dummy()
            xf = Point(x, y)
            xf = xf.translate(y=-d).rotate(-a, o).scale(y=-1
                ).rotate(a, o).translate(y=d)
            # replace every point using that transform
            reps = [(p, xf.xreplace({x: p.x, y: p.y})) for p in g.atoms(Point)]
        return g.xreplace(dict(reps))

    def rotate(self, angle, pt=None):
        """Rotate ``angle`` radians counterclockwise about Point ``pt``.

        The default pt is the origin, Point(0, 0)

        See Also
        ========

        scale, translate

        Examples
        ========

        >>> from sympy import Point, RegularPolygon, Polygon, pi
        >>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
        >>> t # vertex on x axis
        Triangle(Point2D(1, 0), Point2D(-1/2, sqrt(3)/2), Point2D(-1/2, -sqrt(3)/2))
        >>> t.rotate(pi/2) # vertex on y axis now
        Triangle(Point2D(0, 1), Point2D(-sqrt(3)/2, -1/2), Point2D(sqrt(3)/2, -1/2))

        """
        newargs = []
        for a in self.args:
            if isinstance(a, GeometryEntity):
                newargs.append(a.rotate(angle, pt))
            else:
                newargs.append(a)
        return type(self)(*newargs)

    def scale(self, x=1, y=1, pt=None):
        """Scale the object by multiplying the x,y-coordinates by x and y.

        If pt is given, the scaling is done relative to that point; the
        object is shifted by -pt, scaled, and shifted by pt.

        See Also
        ========

        rotate, translate

        Examples
        ========

        >>> from sympy import RegularPolygon, Point, Polygon
        >>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
        >>> t
        Triangle(Point2D(1, 0), Point2D(-1/2, sqrt(3)/2), Point2D(-1/2, -sqrt(3)/2))
        >>> t.scale(2)
        Triangle(Point2D(2, 0), Point2D(-1, sqrt(3)/2), Point2D(-1, -sqrt(3)/2))
        >>> t.scale(2, 2)
        Triangle(Point2D(2, 0), Point2D(-1, sqrt(3)), Point2D(-1, -sqrt(3)))

        """
        from sympy.geometry.point import Point
        if pt:
            pt = Point(pt, dim=2)
            return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
        return type(self)(*[a.scale(x, y) for a in self.args])  # if this fails, override this class

    def translate(self, x=0, y=0):
        """Shift the object by adding to the x,y-coordinates the values x and y.

        See Also
        ========

        rotate, scale

        Examples
        ========

        >>> from sympy import RegularPolygon, Point, Polygon
        >>> t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)
        >>> t
        Triangle(Point2D(1, 0), Point2D(-1/2, sqrt(3)/2), Point2D(-1/2, -sqrt(3)/2))
        >>> t.translate(2)
        Triangle(Point2D(3, 0), Point2D(3/2, sqrt(3)/2), Point2D(3/2, -sqrt(3)/2))
        >>> t.translate(2, 2)
        Triangle(Point2D(3, 2), Point2D(3/2, sqrt(3)/2 + 2), Point2D(3/2, 2 - sqrt(3)/2))

        """
        newargs = []
        for a in self.args:
            if isinstance(a, GeometryEntity):
                newargs.append(a.translate(x, y))
            else:
                newargs.append(a)
        return self.func(*newargs)

    def parameter_value(self, other, t):
        """Return the parameter corresponding to the given point.
        Evaluating an arbitrary point of the entity at this parameter
        value will return the given point.

        Examples
        ========

        >>> from sympy import Line, Point
        >>> from sympy.abc import t
        >>> a = Point(0, 0)
        >>> b = Point(2, 2)
        >>> Line(a, b).parameter_value((1, 1), t)
        {t: 1/2}
        >>> Line(a, b).arbitrary_point(t).subs(_)
        Point2D(1, 1)
        """
        from sympy.geometry.point import Point
        from sympy.solvers.solvers import solve
        if not isinstance(other, GeometryEntity):
            other = Point(other, dim=self.ambient_dimension)
        if not isinstance(other, Point):
            raise ValueError("other must be a point")
        T = Dummy('t', real=True)
        sol = solve(self.arbitrary_point(T) - other, T, dict=True)
        if not sol:
            raise ValueError("Given point is not on %s" % func_name(self))
        return {t: sol[0][T]}


class GeometrySet(GeometryEntity, Set):
    """Parent class of all GeometryEntity that are also Sets
    (compatible with sympy.sets)
    """
    def _contains(self, other):
        """sympy.sets uses the _contains method, so include it for compatibility."""

        if isinstance(other, Set) and other.is_FiniteSet:
            return all(self.__contains__(i) for i in other)

        return self.__contains__(other)

@dispatch(GeometrySet, Set)  # type:ignore # noqa:F811
def union_sets(self, o): # noqa:F811
    """ Returns the union of self and o
    for use with sympy.sets.Set, if possible. """


    # if its a FiniteSet, merge any points
    # we contain and return a union with the rest
    if o.is_FiniteSet:
        other_points = [p for p in o if not self._contains(p)]
        if len(other_points) == len(o):
            return None
        return Union(self, FiniteSet(*other_points))
    if self._contains(o):
        return self
    return None


@dispatch(GeometrySet, Set)  # type: ignore # noqa:F811
def intersection_sets(self, o): # noqa:F811
    """ Returns a sympy.sets.Set of intersection objects,
    if possible. """

    from sympy.geometry import Point

    try:
        # if o is a FiniteSet, find the intersection directly
        # to avoid infinite recursion
        if o.is_FiniteSet:
            inter = FiniteSet(*(p for p in o if self.contains(p)))
        else:
            inter = self.intersection(o)
    except NotImplementedError:
        # sympy.sets.Set.reduce expects None if an object
        # doesn't know how to simplify
        return None

    # put the points in a FiniteSet
    points = FiniteSet(*[p for p in inter if isinstance(p, Point)])
    non_points = [p for p in inter if not isinstance(p, Point)]

    return Union(*(non_points + [points]))

def translate(x, y):
    """Return the matrix to translate a 2-D point by x and y."""
    rv = eye(3)
    rv[2, 0] = x
    rv[2, 1] = y
    return rv


def scale(x, y, pt=None):
    """Return the matrix to multiply a 2-D point's coordinates by x and y.

    If pt is given, the scaling is done relative to that point."""
    rv = eye(3)
    rv[0, 0] = x
    rv[1, 1] = y
    if pt:
        from sympy.geometry.point import Point
        pt = Point(pt, dim=2)
        tr1 = translate(*(-pt).args)
        tr2 = translate(*pt.args)
        return tr1*rv*tr2
    return rv


def rotate(th):
    """Return the matrix to rotate a 2-D point about the origin by ``angle``.

    The angle is measured in radians. To Point a point about a point other
    then the origin, translate the Point, do the rotation, and
    translate it back:

    >>> from sympy.geometry.entity import rotate, translate
    >>> from sympy import Point, pi
    >>> rot_about_11 = translate(-1, -1)*rotate(pi/2)*translate(1, 1)
    >>> Point(1, 1).transform(rot_about_11)
    Point2D(1, 1)
    >>> Point(0, 0).transform(rot_about_11)
    Point2D(2, 0)
    """
    s = sin(th)
    rv = eye(3)*cos(th)
    rv[0, 1] = s
    rv[1, 0] = -s
    rv[2, 2] = 1
    return rv