# Source code for sympy.geometry.ellipse

"""Elliptical geometrical entities.

Contains
* Ellipse
* Circle

"""

from __future__ import division, print_function

from sympy import Expr, Eq
from sympy.core import S, pi, sympify
from sympy.core.evaluate import global_evaluate
from sympy.core.logic import fuzzy_bool
from sympy.core.numbers import Rational, oo
from sympy.core.compatibility import ordered
from sympy.core.symbol import Dummy, _uniquely_named_symbol, _symbol
from sympy.simplify import simplify, trigsimp
from sympy.functions.elementary.miscellaneous import sqrt, Max
from sympy.functions.elementary.trigonometric import cos, sin
from sympy.functions.special.elliptic_integrals import elliptic_e
from sympy.geometry.exceptions import GeometryError
from sympy.geometry.line import Ray2D, Segment2D, Line2D, LinearEntity3D
from sympy.polys import DomainError, Poly, PolynomialError
from sympy.polys.polyutils import _not_a_coeff, _nsort
from sympy.solvers import solve
from sympy.solvers.solveset import linear_coeffs
from sympy.utilities.misc import filldedent, func_name

from .entity import GeometryEntity, GeometrySet
from .point import Point, Point2D, Point3D
from .line import Line, Segment
from .util import idiff

import random

[docs]class Ellipse(GeometrySet):
"""An elliptical GeometryEntity.

Parameters
==========

center : Point, optional
Default value is Point(0, 0)
hradius : number or SymPy expression, optional
vradius : number or SymPy expression, optional
eccentricity : number or SymPy expression, optional
Two of hradius, vradius and eccentricity must be supplied to
create an Ellipse. The third is derived from the two supplied.

Attributes
==========

center
area
circumference
eccentricity
periapsis
apoapsis
focus_distance
foci

Raises
======

GeometryError
When hradius, vradius and eccentricity are incorrectly supplied
as parameters.
TypeError
When center is not a Point.

========

Circle

Notes
-----
Constructed from a center and two radii, the first being the horizontal
radius (along the x-axis) and the second being the vertical radius (along
the y-axis).

When symbolic value for hradius and vradius are used, any calculation that
refers to the foci or the major or minor axis will assume that the ellipse
has its major radius on the x-axis. If this is not true then a manual
rotation is necessary.

Examples
========

>>> from sympy import Ellipse, Point, Rational
>>> e1 = Ellipse(Point(0, 0), 5, 1)
(5, 1)
>>> e2 = Ellipse(Point(3, 1), hradius=3, eccentricity=Rational(4, 5))
>>> e2
Ellipse(Point2D(3, 1), 3, 9/5)

"""

def __contains__(self, o):
if isinstance(o, Point):
x = Dummy('x', real=True)
y = Dummy('y', real=True)

res = self.equation(x, y).subs({x: o.x, y: o.y})
return trigsimp(simplify(res)) is S.Zero
elif isinstance(o, Ellipse):
return self == o
return False

def __eq__(self, o):
"""Is the other GeometryEntity the same as this ellipse?"""
return isinstance(o, Ellipse) and (self.center == o.center and

def __hash__(self):
return super(Ellipse, self).__hash__()

def __new__(

eccentricity = sympify(eccentricity)

if center is None:
center = Point(0, 0)
else:
center = Point(center, dim=2)

if len(center) != 2:
raise ValueError('The center of "{0}" must be a two dimensional point'.format(cls))

if len(list(filter(lambda x: x is not None, (hradius, vradius, eccentricity)))) != 2:
raise ValueError(filldedent('''
"eccentricity" must not be None.'''))

if eccentricity is not None:

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

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".
"""

from sympy.core.evalf import N

c = N(self.center)
return (
'<ellipse fill="{1}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" cx="{2}" cy="{3}" rx="{4}" ry="{5}"/>'
).format(2. * scale_factor, fill_color, c.x, c.y, h, v)

@property
def ambient_dimension(self):
return 2

@property
def apoapsis(self):
"""The apoapsis of the ellipse.

The greatest distance between the focus and the contour.

Returns
=======

apoapsis : number

========

periapsis : Returns shortest distance between foci and contour

Examples
========

>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.apoapsis
2*sqrt(2) + 3

"""
return self.major * (1 + self.eccentricity)

[docs]    def arbitrary_point(self, parameter='t'):
"""A parameterized point on the ellipse.

Parameters
==========

parameter : str, optional
Default value is 't'.

Returns
=======

arbitrary_point : Point

Raises
======

ValueError
When parameter already appears in the functions.

========

sympy.geometry.point.Point

Examples
========

>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.arbitrary_point()
Point2D(3*cos(t), 2*sin(t))

"""
t = _symbol(parameter, real=True)
if t.name in (f.name for f in self.free_symbols):
raise ValueError(filldedent('Symbol %s already appears in object '
'and cannot be used as a parameter.' % t.name))

@property
def area(self):
"""The area of the ellipse.

Returns
=======

area : number

Examples
========

>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.area
3*pi

"""

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

"""

return (self.center.x - h, self.center.y - v, self.center.x + h, self.center.y + v)

@property
def center(self):
"""The center of the ellipse.

Returns
=======

center : number

========

sympy.geometry.point.Point

Examples
========

>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.center
Point2D(0, 0)

"""
return self.args[0]

@property
def circumference(self):
"""The circumference of the ellipse.

Examples
========

>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.circumference
12*elliptic_e(8/9)

"""
if self.eccentricity == 1:
# degenerate
return 4*self.major
elif self.eccentricity == 0:
# circle
else:
return 4*self.major*elliptic_e(self.eccentricity**2)

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

Returns
=======

eccentricity : number

Examples
========

>>> from sympy import Point, Ellipse, sqrt
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, sqrt(2))
>>> e1.eccentricity
sqrt(7)/3

"""
return self.focus_distance / self.major

[docs]    def encloses_point(self, p):
"""
Return True if p is enclosed by (is inside of) self.

Notes
-----
Being on the border of self is considered False.

Parameters
==========

p : Point

Returns
=======

encloses_point : True, False or None

========

sympy.geometry.point.Point

Examples
========

>>> from sympy import Ellipse, S
>>> from sympy.abc import t
>>> e = Ellipse((0, 0), 3, 2)
>>> e.encloses_point((0, 0))
True
>>> e.encloses_point(e.arbitrary_point(t).subs(t, S.Half))
False
>>> e.encloses_point((4, 0))
False

"""
p = Point(p, dim=2)
if p in self:
return False

if len(self.foci) == 2:
# if the combined distance from the foci to p (h1 + h2) is less
# than the combined distance from the foci to the minor axis
# (which is the same as the major axis length) then p is inside
# the ellipse
h1, h2 = [f.distance(p) for f in self.foci]
test = 2*self.major - (h1 + h2)
else:

return fuzzy_bool(test.is_positive)

[docs]    def equation(self, x='x', y='y', _slope=None):
"""
Returns the equation of an ellipse aligned with the x and y axes;
when slope is given, the equation returned corresponds to an ellipse
with a major axis having that slope.

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'.
_slope : Expr, optional
The slope of the major axis. Ignored when 'None'.

Returns
=======

equation : sympy expression

========

arbitrary_point : Returns parameterized point on ellipse

Examples
========

>>> from sympy import Point, Ellipse, pi
>>> from sympy.abc import x, y
>>> e1 = Ellipse(Point(1, 0), 3, 2)
>>> eq1 = e1.equation(x, y); eq1
y**2/4 + (x/3 - 1/3)**2 - 1
>>> eq2 = e1.equation(x, y, _slope=1); eq2
(-x + y + 1)**2/8 + (x + y - 1)**2/18 - 1

A point on e1 satisfies eq1. Let's use one on the x-axis:

>>> p1 = e1.center + Point(e1.major, 0)
>>> assert eq1.subs(x, p1.x).subs(y, p1.y) == 0

When rotated the same as the rotated ellipse, about the center
point of the ellipse, it will satisfy the rotated ellipse's
equation, too:

>>> r1 = p1.rotate(pi/4, e1.center)
>>> assert eq2.subs(x, r1.x).subs(y, r1.y) == 0

References
==========

.. [1] https://math.stackexchange.com/questions/108270/what-is-the-equation-of-an-ellipse-that-is-not-aligned-with-the-axis
.. [2] https://en.wikipedia.org/wiki/Ellipse#Equation_of_a_shifted_ellipse

"""

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

dx = x - self.center.x
dy = y - self.center.y

if _slope is not None:
L = (dy - _slope*dx)**2
l = (_slope*dy + dx)**2
h = 1 + _slope**2
b = h*self.major**2
a = h*self.minor**2
return l/b + L/a - 1

else:
return t1 + t2 - 1

[docs]    def evolute(self, x='x', y='y'):
"""The equation of evolute of the ellipse.

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 Point, Ellipse
>>> e1 = Ellipse(Point(1, 0), 3, 2)
>>> e1.evolute()
2**(2/3)*y**(2/3) + (3*x - 3)**(2/3) - 5**(2/3)
"""
if len(self.args) != 3:
raise NotImplementedError('Evolute of arbitrary Ellipse is not supported.')
x = _symbol(x, real=True)
y = _symbol(y, real=True)
t1 = (self.hradius*(x - self.center.x))**Rational(2, 3)
t2 = (self.vradius*(y - self.center.y))**Rational(2, 3)

@property
def foci(self):
"""The foci of the ellipse.

Notes
-----
The foci can only be calculated if the major/minor axes are known.

Raises
======

ValueError
When the major and minor axis cannot be determined.

========

sympy.geometry.point.Point
focus_distance : Returns the distance between focus and center

Examples
========

>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.foci
(Point2D(-2*sqrt(2), 0), Point2D(2*sqrt(2), 0))

"""
c = self.center
if hr == vr:
return (c, c)

# calculate focus distance manually, since focus_distance calls this
# routine
fd = sqrt(self.major**2 - self.minor**2)
if hr == self.minor:
# foci on the y-axis
return (c + Point(0, -fd), c + Point(0, fd))
elif hr == self.major:
# foci on the x-axis
return (c + Point(-fd, 0), c + Point(fd, 0))

@property
def focus_distance(self):
"""The focal distance of the ellipse.

The distance between the center and one focus.

Returns
=======

focus_distance : number

========

foci

Examples
========

>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.focus_distance
2*sqrt(2)

"""
return Point.distance(self.center, self.foci[0])

@property
"""The horizontal radius of the ellipse.

Returns
=======

========

Examples
========

>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
3

"""
return self.args[1]

[docs]    def intersection(self, o):
"""The intersection of this ellipse and another geometrical entity
o.

Parameters
==========

o : GeometryEntity

Returns
=======

intersection : list of GeometryEntity objects

Notes
-----
Currently supports intersections with Point, Line, Segment, Ray,
Circle and Ellipse types.

========

sympy.geometry.entity.GeometryEntity

Examples
========

>>> from sympy import Ellipse, Point, Line, sqrt
>>> e = Ellipse(Point(0, 0), 5, 7)
>>> e.intersection(Point(0, 0))
[]
>>> e.intersection(Point(5, 0))
[Point2D(5, 0)]
>>> e.intersection(Line(Point(0,0), Point(0, 1)))
[Point2D(0, -7), Point2D(0, 7)]
>>> e.intersection(Line(Point(5,0), Point(5, 1)))
[Point2D(5, 0)]
>>> e.intersection(Line(Point(6,0), Point(6, 1)))
[]
>>> e = Ellipse(Point(-1, 0), 4, 3)
>>> e.intersection(Ellipse(Point(1, 0), 4, 3))
[Point2D(0, -3*sqrt(15)/4), Point2D(0, 3*sqrt(15)/4)]
>>> e.intersection(Ellipse(Point(5, 0), 4, 3))
[Point2D(2, -3*sqrt(7)/4), Point2D(2, 3*sqrt(7)/4)]
>>> e.intersection(Ellipse(Point(100500, 0), 4, 3))
[]
>>> e.intersection(Ellipse(Point(0, 0), 3, 4))
[Point2D(3, 0), Point2D(-363/175, -48*sqrt(111)/175), Point2D(-363/175, 48*sqrt(111)/175)]
>>> e.intersection(Ellipse(Point(-1, 0), 3, 4))
[Point2D(-17/5, -12/5), Point2D(-17/5, 12/5), Point2D(7/5, -12/5), Point2D(7/5, 12/5)]
"""
# TODO: Replace solve with nonlinsolve, when nonlinsolve will be able to solve in real domain
x = Dummy('x', real=True)
y = Dummy('y', real=True)

if isinstance(o, Point):
if o in self:
return [o]
else:
return []

elif isinstance(o, (Segment2D, Ray2D)):
ellipse_equation = self.equation(x, y)
result = solve([ellipse_equation, Line(o.points[0], o.points[1]).equation(x, y)], [x, y])
return list(ordered([Point(i) for i in result if i in o]))

elif isinstance(o, Polygon):
return o.intersection(self)

elif isinstance(o, (Ellipse, Line2D)):
if o == self:
return self
else:
ellipse_equation = self.equation(x, y)
return list(ordered([Point(i) for i in solve([ellipse_equation, o.equation(x, y)], [x, y])]))
elif isinstance(o, LinearEntity3D):
raise TypeError('Entity must be two dimensional, not three dimensional')
else:
raise TypeError('Intersection not handled for %s' % func_name(o))

[docs]    def is_tangent(self, o):
"""Is o tangent to the ellipse?

Parameters
==========

o : GeometryEntity
An Ellipse, LinearEntity or Polygon

Raises
======

NotImplementedError
When the wrong type of argument is supplied.

Returns
=======

is_tangent: boolean
True if o is tangent to the ellipse, False otherwise.

========

tangent_lines

Examples
========

>>> from sympy import Point, Ellipse, Line
>>> p0, p1, p2 = Point(0, 0), Point(3, 0), Point(3, 3)
>>> e1 = Ellipse(p0, 3, 2)
>>> l1 = Line(p1, p2)
>>> e1.is_tangent(l1)
True

"""
if isinstance(o, Point2D):
return False
elif isinstance(o, Ellipse):
intersect = self.intersection(o)
if isinstance(intersect, Ellipse):
return True
elif intersect:
return all((self.tangent_lines(i)[0]).equals((o.tangent_lines(i)[0])) for i in intersect)
else:
return False
elif isinstance(o, Line2D):
hit = self.intersection(o)
if not hit:
return False
if len(hit) == 1:
return True
# might return None if it can't decide
return hit[0].equals(hit[1])
elif isinstance(o, Ray2D):
intersect = self.intersection(o)
if len(intersect) == 1:
return intersect[0] != o.source and not self.encloses_point(o.source)
else:
return False
elif isinstance(o, (Segment2D, Polygon)):
all_tangents = False
segments = o.sides if isinstance(o, Polygon) else [o]
for segment in segments:
intersect = self.intersection(segment)
if len(intersect) == 1:
if not any(intersect[0] in i for i in segment.points) \
and all(not self.encloses_point(i) for i in segment.points):
all_tangents = True
continue
else:
return False
else:
return all_tangents
return all_tangents
elif isinstance(o, (LinearEntity3D, Point3D)):
raise TypeError('Entity must be two dimensional, not three dimensional')
else:
raise TypeError('Is_tangent not handled for %s' % func_name(o))

@property
def major(self):
"""Longer axis of the ellipse (if it can be determined) else hradius.

Returns
=======

major : number or expression

========

Examples
========

>>> from sympy import Point, Ellipse, Symbol
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.major
3

>>> a = Symbol('a')
>>> b = Symbol('b')
>>> Ellipse(p1, a, b).major
a
>>> Ellipse(p1, b, a).major
b

>>> m = Symbol('m')
>>> M = m + 1
>>> Ellipse(p1, m, M).major
m + 1

"""
ab = self.args[1:3]
if len(ab) == 1:
return ab[0]
a, b = ab
o = b - a < 0
if o == True:
return a
elif o == False:
return b

@property
def minor(self):
"""Shorter axis of the ellipse (if it can be determined) else vradius.

Returns
=======

minor : number or expression

========

Examples
========

>>> from sympy import Point, Ellipse, Symbol
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.minor
1

>>> a = Symbol('a')
>>> b = Symbol('b')
>>> Ellipse(p1, a, b).minor
b
>>> Ellipse(p1, b, a).minor
a

>>> m = Symbol('m')
>>> M = m + 1
>>> Ellipse(p1, m, M).minor
m

"""
ab = self.args[1:3]
if len(ab) == 1:
return ab[0]
a, b = ab
o = a - b < 0
if o == True:
return a
elif o == False:
return b

[docs]    def normal_lines(self, p, prec=None):
"""Normal lines between p and the ellipse.

Parameters
==========

p : Point

Returns
=======

normal_lines : list with 1, 2 or 4 Lines

Examples
========

>>> from sympy import Line, Point, Ellipse
>>> e = Ellipse((0, 0), 2, 3)
>>> c = e.center
>>> e.normal_lines(c + Point(1, 0))
[Line2D(Point2D(0, 0), Point2D(1, 0))]
>>> e.normal_lines(c)
[Line2D(Point2D(0, 0), Point2D(0, 1)), Line2D(Point2D(0, 0), Point2D(1, 0))]

Off-axis points require the solution of a quartic equation. This
often leads to very large expressions that may be of little practical
use. An approximate solution of prec digits can be obtained by
passing in the desired value:

>>> e.normal_lines((3, 3), prec=2)
[Line2D(Point2D(-0.81, -2.7), Point2D(0.19, -1.2)),
Line2D(Point2D(1.5, -2.0), Point2D(2.5, -2.7))]

Whereas the above solution has an operation count of 12, the exact
solution has an operation count of 2020.
"""
p = Point(p, dim=2)

# XXX change True to something like self.angle == 0 if the arbitrarily
# rotated ellipse is introduced.
# https://github.com/sympy/sympy/issues/2815)
if True:
rv = []
if p.x == self.center.x:
rv.append(Line(self.center, slope=oo))
if p.y == self.center.y:
rv.append(Line(self.center, slope=0))
if rv:
# at these special orientations of p either 1 or 2 normals
# exist and we are done
return rv

# find the 4 normal points and construct lines through them with
# the corresponding slope
x, y = Dummy('x', real=True), Dummy('y', real=True)
eq = self.equation(x, y)
dydx = idiff(eq, y, x)
norm = -1/dydx
slope = Line(p, (x, y)).slope
seq = slope - norm

# TODO: Replace solve with solveset, when this line is tested
yis = solve(seq, y)[0]
xeq = eq.subs(y, yis).as_numer_denom()[0].expand()
if len(xeq.free_symbols) == 1:
try:
# this is so much faster, it's worth a try
xsol = Poly(xeq, x).real_roots()
except (DomainError, PolynomialError, NotImplementedError):
# TODO: Replace solve with solveset, when these lines are tested
xsol = _nsort(solve(xeq, x), separated=True)[0]
points = [Point(i, solve(eq.subs(x, i), y)[0]) for i in xsol]
else:
raise NotImplementedError(
'intersections for the general ellipse are not supported')
slopes = [norm.subs(zip((x, y), pt.args)) for pt in points]
if prec is not None:
points = [pt.n(prec) for pt in points]
slopes = [i if _not_a_coeff(i) else i.n(prec) for i in slopes]
return [Line(pt, slope=s) for pt, s in zip(points, slopes)]

@property
def periapsis(self):
"""The periapsis of the ellipse.

The shortest distance between the focus and the contour.

Returns
=======

periapsis : number

========

apoapsis : Returns greatest distance between focus and contour

Examples
========

>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.periapsis
3 - 2*sqrt(2)

"""
return self.major * (1 - self.eccentricity)

@property
def semilatus_rectum(self):
"""
Calculates the semi-latus rectum of the Ellipse.

Semi-latus rectum is defined as one half of the the chord through a
focus parallel to the conic section directrix of a conic section.

Returns
=======

semilatus_rectum : number

========

apoapsis : Returns greatest distance between focus and contour

periapsis : The shortest distance between the focus and the contour

Examples
========

>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.semilatus_rectum
1/3

References
==========

[1] http://mathworld.wolfram.com/SemilatusRectum.html
[2] https://en.wikipedia.org/wiki/Ellipse#Semi-latus_rectum

"""
return self.major * (1 - self.eccentricity ** 2)

[docs]    def auxiliary_circle(self):
"""Returns a Circle whose diameter is the major axis of the ellipse.

Examples
========

>>> from sympy import Circle, Ellipse, Point, symbols
>>> c = Point(1, 2)
>>> Ellipse(c, 8, 7).auxiliary_circle()
Circle(Point2D(1, 2), 8)
>>> a, b = symbols('a b')
>>> Ellipse(c, a, b).auxiliary_circle()
Circle(Point2D(1, 2), Max(a, b))
"""

[docs]    def director_circle(self):
"""
Returns a Circle consisting of all points where two perpendicular
tangent lines to the ellipse cross each other.

Returns
=======

Circle
A director circle returned as a geometric object.

Examples
========

>>> from sympy import Circle, Ellipse, Point, symbols
>>> c = Point(3,8)
>>> Ellipse(c, 7, 9).director_circle()
Circle(Point2D(3, 8), sqrt(130))
>>> a, b = symbols('a b')
>>> Ellipse(c, a, b).director_circle()
Circle(Point2D(3, 8), sqrt(a**2 + b**2))

References
==========

.. [1] https://en.wikipedia.org/wiki/Director_circle

"""

[docs]    def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Ellipse.

Parameters
==========

parameter : str, optional
Default value is 't'.

Returns
=======

plot_interval : list
[parameter, lower_bound, upper_bound]

Examples
========

>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.plot_interval()
[t, -pi, pi]

"""
t = _symbol(parameter, real=True)
return [t, -S.Pi, S.Pi]

[docs]    def random_point(self, seed=None):
"""A random point on the ellipse.

Returns
=======

point : Point

Examples
========

>>> from sympy import Point, Ellipse, Segment
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.random_point() # gives some random point
Point2D(...)
>>> p1 = e1.random_point(seed=0); p1.n(2)
Point2D(2.1, 1.4)

Notes
=====

When creating a random point, one may simply replace the
parameter with a random number. When doing so, however, the
random number should be made a Rational or else the point
may not test as being in the ellipse:

>>> from sympy.abc import t
>>> from sympy import Rational
>>> arb = e1.arbitrary_point(t); arb
Point2D(3*cos(t), 2*sin(t))
>>> arb.subs(t, .1) in e1
False
>>> arb.subs(t, Rational(.1)) in e1
True
>>> arb.subs(t, Rational('.1')) in e1
True

========
sympy.geometry.point.Point
arbitrary_point : Returns parameterized point on ellipse
"""
from sympy import sin, cos, Rational
t = _symbol('t', real=True)
x, y = self.arbitrary_point(t).args
# get a random value in [-1, 1) corresponding to cos(t)
# and confirm that it will test as being in the ellipse
if seed is not None:
rng = random.Random(seed)
else:
rng = random
# simplify this now or else the Float will turn s into a Float
r = Rational(rng.random())
c = 2*r - 1
s = sqrt(1 - c**2)
return Point(x.subs(cos(t), c), y.subs(sin(t), s))

[docs]    def reflect(self, line):
is not a GeometryEntity.

Examples
========

>>> from sympy import Circle, Line
>>> Circle((0, 1), 1).reflect(Line((0, 0), (1, 1)))
Circle(Point2D(1, 0), -1)
>>> from sympy import Ellipse, Line, Point
>>> Ellipse(Point(3, 4), 1, 3).reflect(Line(Point(0, -4), Point(5, 0)))
Traceback (most recent call last):
...
NotImplementedError:
General Ellipse is not supported but the equation of the reflected
Ellipse is given by the zeros of: f(x, y) = (9*x/41 + 40*y/41 +
37/41)**2 + (40*x/123 - 3*y/41 - 364/123)**2 - 1

Notes
=====

Until the general ellipse (with no axis parallel to the x-axis) is
supported a NotImplemented error is raised and the equation whose
zeros define the rotated ellipse is given.

"""

if line.slope in (0, oo):
c = self.center
c = c.reflect(line)
else:
x, y = [_uniquely_named_symbol(
name, (self, line), real=True) for name in 'xy']
expr = self.equation(x, y)
p = Point(x, y).reflect(line)
result = expr.subs(zip((x, y), p.args
), simultaneous=True)
raise NotImplementedError(filldedent(
'General Ellipse is not supported but the equation '
'of the reflected Ellipse is given by the zeros of: ' +
"f(%s, %s) = %s" % (str(x), str(y), str(result))))

[docs]    def rotate(self, angle=0, pt=None):
"""Rotate angle radians counterclockwise about Point pt.

Note: since the general ellipse is not supported, only rotations that
are integer multiples of pi/2 are allowed.

Examples
========

>>> from sympy import Ellipse, pi
>>> Ellipse((1, 0), 2, 1).rotate(pi/2)
Ellipse(Point2D(0, 1), 1, 2)
>>> Ellipse((1, 0), 2, 1).rotate(pi)
Ellipse(Point2D(-1, 0), 2, 1)
"""
if (angle/S.Pi).is_integer:
return super(Ellipse, self).rotate(angle, pt)
if (2*angle/S.Pi).is_integer:
# XXX see https://github.com/sympy/sympy/issues/2815 for general ellipes
raise NotImplementedError('Only rotations of pi/2 are currently supported for Ellipse.')

[docs]    def scale(self, x=1, y=1, pt=None):
"""Override GeometryEntity.scale since it is the major and minor
axes which must be scaled and they are not GeometryEntities.

Examples
========

>>> from sympy import Ellipse
>>> Ellipse((0, 0), 2, 1).scale(2, 4)
Circle(Point2D(0, 0), 4)
>>> Ellipse((0, 0), 2, 1).scale(2)
Ellipse(Point2D(0, 0), 4, 1)
"""
c = self.center
if pt:
pt = Point(pt, dim=2)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)

[docs]    def tangent_lines(self, p):
"""Tangent lines between p and the ellipse.

If p is on the ellipse, returns the tangent line through point p.
Otherwise, returns the tangent line(s) from p to the ellipse, or
None if no tangent line is possible (e.g., p inside ellipse).

Parameters
==========

p : Point

Returns
=======

tangent_lines : list with 1 or 2 Lines

Raises
======

NotImplementedError
Can only find tangent lines for a point, p, on the ellipse.

========

sympy.geometry.point.Point, sympy.geometry.line.Line

Examples
========

>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.tangent_lines(Point(3, 0))
[Line2D(Point2D(3, 0), Point2D(3, -12))]

"""
p = Point(p, dim=2)
if self.encloses_point(p):
return []

if p in self:
delta = self.center - p
p2 = Point(simplify(p.x + run),
simplify(p.y + rise))
return [Line(p, p2)]
else:
if len(self.foci) == 2:
f1, f2 = self.foci
test = (2*maj -
Point.distance(f1, p) -
Point.distance(f2, p))
else:
test = self.radius - Point.distance(self.center, p)
if test.is_number and test.is_positive:
return []
# else p is outside the ellipse or we can't tell. In case of the
# latter, the solutions returned will only be valid if
# the point is not inside the ellipse; if it is, nan will result.
x, y = Dummy('x'), Dummy('y')
eq = self.equation(x, y)
dydx = idiff(eq, y, x)
slope = Line(p, Point(x, y)).slope

# TODO: Replace solve with solveset, when this line is tested
tangent_points = solve([slope - dydx, eq], [x, y])

# handle horizontal and vertical tangent lines
if len(tangent_points) == 1:
assert tangent_points[0][
0] == p.x or tangent_points[0][1] == p.y
return [Line(p, p + Point(1, 0)), Line(p, p + Point(0, 1))]

# others
return [Line(p, tangent_points[0]), Line(p, tangent_points[1])]

@property
"""The vertical radius of the ellipse.

Returns
=======

========

Examples
========

>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
1

"""
return self.args[2]

[docs]    def second_moment_of_area(self, point=None):
"""Returns the second moment and product moment area of an ellipse.

Parameters
==========

point : Point, two-tuple of sympifiable objects, or None(default=None)
point is the point about which second moment of area is to be found.
If "point=None" it will be calculated about the axis passing through the
centroid of the ellipse.

Returns
=======

I_xx, I_yy, I_xy : number or sympy expression
I_xx, I_yy are second moment of area of an ellise.
I_xy is product moment of area of an ellipse.

Examples
========

>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.second_moment_of_area()
(3*pi/4, 27*pi/4, 0)

References
==========

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

"""

I_xy = 0

if point is None:
return I_xx, I_yy, I_xy

# parallel axis theorem
I_xx = I_xx + self.area*((point[1] - self.center.y)**2)
I_yy = I_yy + self.area*((point[0] - self.center.x)**2)
I_xy = I_xy + self.area*(point[0] - self.center.x)*(point[1] - self.center.y)

return I_xx, I_yy, I_xy

[docs]class Circle(Ellipse):
"""A circle in space.

Constructed simply from a center and a radius, from three
non-collinear points, or the equation of a circle.

Parameters
==========

center : Point
radius : number or sympy expression
points : sequence of three Points
equation : equation of a circle

Attributes
==========

circumference
equation

Raises
======

GeometryError
When the given equation is not that of a circle.
When trying to construct circle from incorrect parameters.

========

Ellipse, sympy.geometry.point.Point

Examples
========

>>> from sympy import Eq
>>> from sympy.geometry import Point, Circle
>>> from sympy.abc import x, y, a, b

A circle constructed from a center and radius:

>>> c1 = Circle(Point(0, 0), 5)
(5, 5, 5)

A circle constructed from three points:

>>> c2 = Circle(Point(0, 0), Point(1, 1), Point(1, 0))
(sqrt(2)/2, sqrt(2)/2, sqrt(2)/2, Point2D(1/2, 1/2))

A circle can be constructed from an equation in the form
a*x**2 + by**2 + gx + hy + c = 0, too:

>>> Circle(x**2 + y**2 - 25)
Circle(Point2D(0, 0), 5)

If the variables corresponding to x and y are named something
else, their name or symbol can be supplied:

>>> Circle(Eq(a**2 + b**2, 25), x='a', y=b)
Circle(Point2D(0, 0), 5)
"""

def __new__(cls, *args, **kwargs):
from sympy.geometry.util import find
from .polygon import Triangle
evaluate = kwargs.get('evaluate', global_evaluate[0])
if len(args) == 1 and isinstance(args[0], Expr):
x = kwargs.get('x', 'x')
y = kwargs.get('y', 'y')
equation = args[0]
if isinstance(equation, Eq):
equation = equation.lhs - equation.rhs
x = find(x, equation)
y = find(y, equation)

try:
a, b, c, d, e = linear_coeffs(equation, x**2, y**2, x, y)
except ValueError:
raise GeometryError("The given equation is not that of a circle.")

if a == 0 or b == 0 or a != b:
raise GeometryError("The given equation is not that of a circle.")

center_x = -c/a/2
center_y = -d/b/2
r2 = (center_x**2) + (center_y**2) - e

return Circle((center_x, center_y), sqrt(r2), evaluate=evaluate)

else:
c, r = None, None
if len(args) == 3:
args = [Point(a, dim=2, evaluate=evaluate) for a in args]
t = Triangle(*args)
if not isinstance(t, Triangle):
return t
c = t.circumcenter
elif len(args) == 2:
c = Point(args[0], dim=2, evaluate=evaluate)
r = args[1]
# this will prohibit imaginary radius
try:
r = Point(r, 0, evaluate=evaluate).x
except:
raise GeometryError("Circle with imaginary radius is not permitted")

if not (c is None or r is None):
if r == 0:
return c
return GeometryEntity.__new__(cls, c, r, **kwargs)

@property
def circumference(self):
"""The circumference of the circle.

Returns
=======

circumference : number or SymPy expression

Examples
========

>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)
>>> c1.circumference
12*pi

"""
return 2 * S.Pi * self.radius

[docs]    def equation(self, x='x', y='y'):
"""The equation of the circle.

Parameters
==========

x : str or Symbol, optional
Default value is 'x'.
y : str or Symbol, optional
Default value is 'y'.

Returns
=======

equation : SymPy expression

Examples
========

>>> from sympy import Point, Circle
>>> c1 = Circle(Point(0, 0), 5)
>>> c1.equation()
x**2 + y**2 - 25

"""
x = _symbol(x, real=True)
y = _symbol(y, real=True)
t1 = (x - self.center.x)**2
t2 = (y - self.center.y)**2
return t1 + t2 - self.major**2

[docs]    def intersection(self, o):
"""The intersection of this circle with another geometrical entity.

Parameters
==========

o : GeometryEntity

Returns
=======

intersection : list of GeometryEntities

Examples
========

>>> from sympy import Point, Circle, Line, Ray
>>> p1, p2, p3 = Point(0, 0), Point(5, 5), Point(6, 0)
>>> p4 = Point(5, 0)
>>> c1 = Circle(p1, 5)
>>> c1.intersection(p2)
[]
>>> c1.intersection(p4)
[Point2D(5, 0)]
>>> c1.intersection(Ray(p1, p2))
[Point2D(5*sqrt(2)/2, 5*sqrt(2)/2)]
>>> c1.intersection(Line(p2, p3))
[]

"""
return Ellipse.intersection(self, o)

@property

Returns
=======

radius : number or sympy expression

========

Examples
========

>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)
6

"""
return self.args[1]

[docs]    def reflect(self, line):
is not a GeometryEntity.

Examples
========

>>> from sympy import Circle, Line
>>> Circle((0, 1), 1).reflect(Line((0, 0), (1, 1)))
Circle(Point2D(1, 0), -1)
"""
c = self.center
c = c.reflect(line)

[docs]    def scale(self, x=1, y=1, pt=None):
is not a GeometryEntity.

Examples
========

>>> from sympy import Circle
>>> Circle((0, 0), 1).scale(2, 2)
Circle(Point2D(0, 0), 2)
>>> Circle((0, 0), 1).scale(2, 4)
Ellipse(Point2D(0, 0), 2, 4)
"""
c = self.center
if pt:
pt = Point(pt, dim=2)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
c = c.scale(x, y)
x, y = [abs(i) for i in (x, y)]
if x == y:

@property
"""
This Ellipse property is an alias for the Circle's radius.

Whereas hradius, major and minor can use Ellipse's conventions,
the vradius does not exist for a circle. It is always a positive
value in order that the Circle, like Polygons, will have an
area that can be positive or negative as determined by the sign

Examples
========

>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)