"""Elliptical geometrical entities.
Contains
* Ellipse
* Circle
"""
from __future__ import division, print_function
from sympy.core import S, pi, sympify
from sympy.core.logic import fuzzy_bool
from sympy.core.numbers import Rational, oo
from sympy.core.compatibility import range
from sympy.core.symbol import Dummy
from sympy.simplify import simplify, trigsimp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import cos, sin
from sympy.geometry.exceptions import GeometryError
from sympy.polys import DomainError, Poly, PolynomialError
from sympy.polys.polyutils import _not_a_coeff, _nsort
from sympy.solvers import solve
from sympy.utilities.iterables import uniq
from sympy.utilities.misc import filldedent
from sympy.utilities.decorator import doctest_depends_on
from .entity import GeometryEntity, GeometrySet
from .point import Point
from .line import Line, LinearEntity
from .util import _symbol, 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
hradius
vradius
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.
See Also
========
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)
>>> e1.hradius, e1.vradius
(5, 1)
>>> e2 = Ellipse(Point(3, 1), hradius=3, eccentricity=Rational(4, 5))
>>> e2
Ellipse(Point2D(3, 1), 3, 9/5)
Plotting:
>>> from sympy.plotting.pygletplot import PygletPlot as Plot
>>> from sympy import Circle, Segment
>>> c1 = Circle(Point(0,0), 1)
>>> Plot(c1) # doctest: +SKIP
[0]: cos(t), sin(t), 'mode=parametric'
>>> p = Plot() # doctest: +SKIP
>>> p[0] = c1 # doctest: +SKIP
>>> radius = Segment(c1.center, c1.random_point())
>>> p[1] = radius # doctest: +SKIP
>>> p # doctest: +SKIP
[0]: cos(t), sin(t), 'mode=parametric'
[1]: t*cos(1.546086215036205357975518382),
t*sin(1.546086215036205357975518382), 'mode=parametric'
"""
def __new__(
cls, center=None, hradius=None, vradius=None, eccentricity=None,
**kwargs):
hradius = sympify(hradius)
vradius = sympify(vradius)
eccentricity = sympify(eccentricity)
if center is None:
center = Point(0, 0)
else:
center = Point(center)
if len(center) != 2:
raise ValueError('The center of "{0}" must be a two dimensional point'.format(cls))
if len(list(filter(None, (hradius, vradius, eccentricity)))) != 2:
raise ValueError('Exactly two arguments of "hradius", '
'"vradius", and "eccentricity" must not be None."')
if eccentricity is not None:
if hradius is None:
hradius = vradius / sqrt(1 - eccentricity**2)
elif vradius is None:
vradius = hradius * sqrt(1 - eccentricity**2)
if hradius == vradius:
return Circle(center, hradius, **kwargs)
return GeometryEntity.__new__(cls, center, hradius, vradius, **kwargs)
@property
def ambient_dimension(self):
return 2
@property
def center(self):
"""The center of the ellipse.
Returns
=======
center : number
See Also
========
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 hradius(self):
"""The horizontal radius of the ellipse.
Returns
=======
hradius : number
See Also
========
vradius, major, minor
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.hradius
3
"""
return self.args[1]
@property
def vradius(self):
"""The vertical radius of the ellipse.
Returns
=======
vradius : number
See Also
========
hradius, major, minor
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.vradius
1
"""
return self.args[2]
@property
def minor(self):
"""Shorter axis of the ellipse (if it can be determined) else vradius.
Returns
=======
minor : number or expression
See Also
========
hradius, vradius, major
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
return self.vradius
@property
def major(self):
"""Longer axis of the ellipse (if it can be determined) else hradius.
Returns
=======
major : number or expression
See Also
========
hradius, vradius, minor
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
return self.hradius
@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
"""
return simplify(S.Pi * self.hradius * self.vradius)
@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*Integral(sqrt((-8*_x**2/9 + 1)/(-_x**2 + 1)), (_x, 0, 1))
"""
from sympy import Integral
if self.eccentricity == 1:
return 2*pi*self.hradius
else:
x = Dummy('x', real=True)
return 4*self.major*Integral(
sqrt((1 - (self.eccentricity*x)**2)/(1 - x**2)), (x, 0, 1))
@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
@property
def periapsis(self):
"""The periapsis of the ellipse.
The shortest distance between the focus and the contour.
Returns
=======
periapsis : number
See Also
========
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
-2*sqrt(2) + 3
"""
return self.major * (1 - self.eccentricity)
@property
def apoapsis(self):
"""The apoapsis of the ellipse.
The greatest distance between the focus and the contour.
Returns
=======
apoapsis : number
See Also
========
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)
@property
def focus_distance(self):
"""The focale distance of the ellipse.
The distance between the center and one focus.
Returns
=======
focus_distance : number
See Also
========
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
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.
See Also
========
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
hr, vr = self.hradius, self.vradius
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 bounds(self):
"""Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
rectangle for the geometric figure.
"""
h, v = self.hradius, self.vradius
return (self.center.x - h, self.center.y - v, self.center.x + h, self.center.y + v)
[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 self.hradius == self.vradius:
return self.func(*self.args)
if (angle/S.Pi).is_integer:
return super(Ellipse, self).rotate(angle, pt)
if (2*angle/S.Pi).is_integer:
return self.func(self.center.rotate(angle, pt), self.vradius, self.hradius)
# 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)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
h = self.hradius
v = self.vradius
return self.func(c.scale(x, y), hradius=h*x, vradius=v*y)
[docs] def reflect(self, line):
"""Override GeometryEntity.reflect since the radius
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.
"""
from .util import _uniquely_named_symbol
if line.slope in (0, oo):
c = self.center
c = c.reflect(line)
return self.func(c, -self.hradius, self.vradius)
else:
x, y = [_uniquely_named_symbol(name, self, line) 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 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
See Also
========
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)
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:
test = self.radius - self.center.distance(p)
return fuzzy_bool(test.is_positive)
@doctest_depends_on(modules=('pyglet',))
[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.
See Also
========
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))
[Line(Point2D(3, 0), Point2D(3, -12))]
>>> # This will plot an ellipse together with a tangent line.
>>> from sympy.plotting.pygletplot import PygletPlot as Plot
>>> from sympy import Point, Ellipse
>>> e = Ellipse(Point(0,0), 3, 2)
>>> t = e.tangent_lines(e.random_point())
>>> p = Plot()
>>> p[0] = e # doctest: +SKIP
>>> p[1] = t # doctest: +SKIP
"""
p = Point(p)
if self.encloses_point(p):
return []
if p in self:
delta = self.center - p
rise = (self.vradius ** 2)*delta.x
run = -(self.hradius ** 2)*delta.y
p2 = Point(simplify(p.x + run),
simplify(p.y + rise))
return [Line(p, p2)]
else:
if len(self.foci) == 2:
f1, f2 = self.foci
maj = self.hradius
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])]
[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.
See Also
========
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
"""
inter = None
if isinstance(o, Ellipse):
inter = self.intersection(o)
if isinstance(inter, Ellipse):
return False
return (inter is not None and len(inter) == 1
and isinstance(inter[0], Point))
elif isinstance(o, LinearEntity):
inter = self._do_line_intersection(o)
if inter is not None and len(inter) == 1:
return inter[0] in o
else:
return False
elif isinstance(o, Polygon):
c = 0
for seg in o.sides:
inter = self._do_line_intersection(seg)
c += len([True for point in inter if point in seg])
return c == 1
else:
raise NotImplementedError("Unknown argument type")
[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))
[Line(Point2D(0, 0), Point2D(1, 0))]
>>> e.normal_lines(c)
[Line(Point2D(0, 0), Point2D(0, 1)), Line(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)
[Line(Point2D(-38/47, -85/31), Point2D(9/47, -21/17)),
Line(Point2D(19/13, -43/21), Point2D(32/13, -8/3))]
Whereas the above solution has an operation count of 12, the exact
solution has an operation count of 2020.
"""
p = Point(p)
# 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)]
[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.
See Also
========
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)
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))
return Point(self.center.x + self.hradius*cos(t),
self.center.y + self.vradius*sin(t))
[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)
return [t, -S.Pi, S.Pi]
[docs] def random_point(self, seed=None):
"""A random point on the ellipse.
Returns
=======
point : Point
See Also
========
sympy.geometry.point.Point
arbitrary_point : Returns parameterized point on ellipse
Notes
-----
A random point may not appear to be on the ellipse, ie, `p in e` may
return False. This is because the coordinates of the point will be
floating point values, and when these values are substituted into the
equation for the ellipse the result may not be zero because of floating
point rounding error.
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)
The random_point method assures that the point will test as being
in the ellipse:
>>> p1 in e1
True
Notes
=====
An arbitrary_point with a random value of t substituted into it may
not test as being on the ellipse because the expression tested that
a point is on the ellipse doesn't simplify to zero and doesn't evaluate
exactly to zero:
>>> from sympy.abc import t
>>> e1.arbitrary_point(t)
Point2D(3*cos(t), 2*sin(t))
>>> p2 = _.subs(t, 0.1)
>>> p2 in e1
False
Note that arbitrary_point routine does not take this approach. A value
for cos(t) and sin(t) (not t) is substituted into the arbitrary point.
There is a small chance that this will give a point that will not
test as being in the ellipse, so the process is repeated (up to 10
times) until a valid point is obtained.
"""
from sympy import sin, cos, Rational
t = _symbol('t')
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
for i in range(10): # should be enough?
# simplify this now or else the Float will turn s into a Float
c = 2*Rational(rng.random()) - 1
s = sqrt(1 - c**2)
p1 = Point(x.subs(cos(t), c), y.subs(sin(t), s))
if p1 in self:
return p1
raise GeometryError(
'Having problems generating a point in the ellipse.')
[docs] def equation(self, x='x', y='y'):
"""The equation 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
See Also
========
arbitrary_point : Returns parameterized point on ellipse
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(1, 0), 3, 2)
>>> e1.equation()
y**2/4 + (x/3 - 1/3)**2 - 1
"""
x = _symbol(x)
y = _symbol(y)
t1 = ((x - self.center.x) / self.hradius)**2
t2 = ((y - self.center.y) / self.vradius)**2
return t1 + t2 - 1
def _do_line_intersection(self, o):
"""
Find the intersection of a LinearEntity and the ellipse.
All LinearEntities are treated as a line and filtered at
the end to see that they lie in o.
"""
hr_sq = self.hradius ** 2
vr_sq = self.vradius ** 2
lp = o.points
ldir = lp[1] - lp[0]
diff = lp[0] - self.center
mdir = Point(ldir.x/hr_sq, ldir.y/vr_sq)
mdiff = Point(diff.x/hr_sq, diff.y/vr_sq)
a = ldir.dot(mdir)
b = ldir.dot(mdiff)
c = diff.dot(mdiff) - 1
det = simplify(b*b - a*c)
result = []
if det == 0:
t = -b / a
result.append(lp[0] + (lp[1] - lp[0]) * t)
# Definite and potential symbolic intersections are allowed.
elif (det > 0) != False:
root = sqrt(det)
t_a = (-b - root) / a
t_b = (-b + root) / a
result.append( lp[0] + (lp[1] - lp[0]) * t_a )
result.append( lp[0] + (lp[1] - lp[0]) * t_b )
return [r for r in result if r in o]
def _do_ellipse_intersection(self, o):
"""The intersection of an ellipse with another ellipse or a circle.
Private helper method for `intersection`.
"""
x = Dummy('x', real=True)
y = Dummy('y', real=True)
seq = self.equation(x, y)
oeq = o.equation(x, y)
# TODO: Replace solve with solveset, when this line is tested
result = solve([seq, oeq], [x, y])
return [Point(*r) for r in list(uniq(result))]
[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.
See Also
========
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(-363/175, -48*sqrt(111)/175), Point2D(-363/175, 48*sqrt(111)/175), Point2D(3, 0)]
>>> 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)]
"""
if isinstance(o, Point):
if o in self:
return [o]
else:
return []
elif isinstance(o, LinearEntity):
# LinearEntity may be a ray/segment, so check the points
# of intersection for coincidence first
return self._do_line_intersection(o)
elif isinstance(o, Circle):
return self._do_ellipse_intersection(o)
elif isinstance(o, Ellipse):
if o == self:
return self
else:
return self._do_ellipse_intersection(o)
return o.intersection(self)
[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)
y = _symbol(y)
t1 = (self.hradius*(x - self.center.x))**Rational(2, 3)
t2 = (self.vradius*(y - self.center.y))**Rational(2, 3)
return t1 + t2 - (self.hradius**2 - self.vradius**2)**Rational(2, 3)
def __eq__(self, o):
"""Is the other GeometryEntity the same as this ellipse?"""
return isinstance(o, GeometryEntity) and (self.center == o.center and
self.hradius == o.hradius and
self.vradius == o.vradius)
def __hash__(self):
return super(Ellipse, self).__hash__()
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 _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)
h, v = N(self.hradius), N(self.vradius)
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)
[docs]class Circle(Ellipse):
"""A circle in space.
Constructed simply from a center and a radius, or from three
non-collinear points.
Parameters
==========
center : Point
radius : number or sympy expression
points : sequence of three Points
Attributes
==========
radius (synonymous with hradius, vradius, major and minor)
circumference
equation
Raises
======
GeometryError
When trying to construct circle from three collinear points.
When trying to construct circle from incorrect parameters.
See Also
========
Ellipse, sympy.geometry.point.Point
Examples
========
>>> from sympy.geometry import Point, Circle
>>> # a circle constructed from a center and radius
>>> c1 = Circle(Point(0, 0), 5)
>>> c1.hradius, c1.vradius, c1.radius
(5, 5, 5)
>>> # a circle costructed from three points
>>> c2 = Circle(Point(0, 0), Point(1, 1), Point(1, 0))
>>> c2.hradius, c2.vradius, c2.radius, c2.center
(sqrt(2)/2, sqrt(2)/2, sqrt(2)/2, Point2D(1/2, 1/2))
"""
def __new__(cls, *args, **kwargs):
c, r = None, None
if len(args) == 3:
args = [Point(a) for a in args]
if Point.is_collinear(*args):
raise GeometryError(
"Cannot construct a circle from three collinear points")
from .polygon import Triangle
t = Triangle(*args)
c = t.circumcenter
r = t.circumradius
elif len(args) == 2:
# Assume (center, radius) pair
c = Point(args[0])
r = sympify(args[1])
if not (c is None or r is None):
return GeometryEntity.__new__(cls, c, r, **kwargs)
raise GeometryError("Circle.__new__ received unknown arguments")
@property
def radius(self):
"""The radius of the circle.
Returns
=======
radius : number or sympy expression
See Also
========
Ellipse.major, Ellipse.minor, Ellipse.hradius, Ellipse.vradius
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)
>>> c1.radius
6
"""
return self.args[1]
@property
def vradius(self):
"""
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
of the hradius.
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)
>>> c1.vradius
6
"""
return abs(self.radius)
@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)
y = _symbol(y)
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))
[]
"""
if isinstance(o, Circle):
if o.center == self.center:
if o.radius == self.radius:
return o
return []
dx, dy = (o.center - self.center).args
d = sqrt(simplify(dy**2 + dx**2))
R = o.radius + self.radius
if d > R or d < abs(self.radius - o.radius):
return []
a = simplify((self.radius**2 - o.radius**2 + d**2) / (2*d))
x2 = self.center.x + (dx * a/d)
y2 = self.center.y + (dy * a/d)
h = sqrt(simplify(self.radius**2 - a**2))
rx = -dy * (h/d)
ry = dx * (h/d)
xi_1 = simplify(x2 + rx)
xi_2 = simplify(x2 - rx)
yi_1 = simplify(y2 + ry)
yi_2 = simplify(y2 - ry)
ret = [Point(xi_1, yi_1)]
if xi_1 != xi_2 or yi_1 != yi_2:
ret.append(Point(xi_2, yi_2))
return ret
return Ellipse.intersection(self, o)
[docs] def scale(self, x=1, y=1, pt=None):
"""Override GeometryEntity.scale since the radius
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)
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:
return self.func(c, x*self.radius)
h = v = self.radius
return Ellipse(c, hradius=h*x, vradius=v*y)
[docs] def reflect(self, line):
"""Override GeometryEntity.reflect since the radius
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)
return self.func(c, -self.radius)
from .polygon import Polygon
