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ellipse.py
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ellipse.py
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"""Elliptical geometrical entities.
Contains
* Ellipse
* Circle
"""
from sympy.core import S, C, sympify, pi, Dummy
from sympy.core.logic import fuzzy_bool
from sympy.core.numbers import oo
from sympy.simplify import simplify, trigsimp
from sympy.functions.elementary.miscellaneous import sqrt, Max, Min
from sympy.functions.elementary.complexes import im
from sympy.geometry.exceptions import GeometryError
from sympy.solvers import solve
from entity import GeometryEntity
from point import Point
from line import LinearEntity, Line
from util import _symbol, idiff
import random
from sympy.utilities.decorator import doctest_depends_on, no_attrs_in_subclass
class Ellipse(GeometryEntity):
"""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(Point(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'
"""
_doctest_depends_on = {'modules': ('numpy', 'matplotlib')}
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(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 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
Point(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
"""
rv = Min(*self.args[1:3])
if rv.func is Min:
return self.vradius
return rv
@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
"""
rv = Max(*self.args[1:3])
if rv.func is Max:
return self.hradius
return rv
@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))
"""
if self.eccentricity == 1:
return 2*pi*self.hradius
else:
x = C.Dummy('x', real=True)
return 4*self.major*C.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
(Point(-2*sqrt(2), 0), Point(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))
def rotate(self, angle=0, pt=None):
"""Rotate ``angle`` radians counterclockwise about Point ``pt``.
Note: since the general ellipse is not supported, the axes of
the ellipse will not be rotated. Only the center is rotated to
a new position.
Examples
========
>>> from sympy import Ellipse, pi
>>> Ellipse((1, 0), 2, 1).rotate(pi/2)
Ellipse(Point(0, 1), 2, 1)
"""
return super(Ellipse, self).rotate(angle, pt)
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(Point(0, 0), 4)
>>> Ellipse((0, 0), 2, 1).scale(2)
Ellipse(Point(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)
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(Point(1, 0), -1)
"""
if line.slope in (0, oo):
c = self.center
c = c.reflect(line)
return self.func(c, -self.hradius, self.vradius)
raise NotImplementedError('reflection line not horizontal | vertical.')
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',))
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(Point(3, 0), Point(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
"""
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
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])]
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 isinstance(inter[0], Point)
and len(inter) == 1)
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")
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()
Point(3*cos(t), 2*sin(t))
"""
t = _symbol(parameter)
if t.name in (f.name for f in self.free_symbols):
raise ValueError('Symbol %s already appears in object and cannot be used as a parameter.' % t.name)
return Point(self.center.x + self.hradius*C.cos(t),
self.center.y + self.vradius*C.sin(t))
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]
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
Point(...)
>>> p1 = e1.random_point(seed=0); p1.n(2)
Point(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)
Point(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.')
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)
else:
is_good = True
try:
is_good = (det > 0)
except NotImplementedError: # symbolic, allow
is_good = True
if is_good:
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_circle_intersection(self, o):
"""The intersection of an Ellipse and a Circle.
Private helper method for `intersection`.
"""
variables = self.equation().atoms(C.Symbol)
if len(variables) > 2:
return None
if self.center == o.center:
a, b, r = o.major, o.minor, self.radius
x = a*sqrt(simplify((r**2 - b**2)/(a**2 - b**2)))
y = b*sqrt(simplify((a**2 - r**2)/(a**2 - b**2)))
return list(set([Point(x, y), Point(x, -y), Point(-x, y),
Point(-x, -y)]))
else:
x, y = variables
xx = solve(self.equation(), x)
intersect = []
for xi in xx:
yy = solve(o.equation().subs(x, xi), y)
for yi in yy:
intersect.append(Point(xi, yi))
return list(set(intersect))
def _do_ellipse_intersection(self, o):
"""The intersection of two ellipses.
Private helper method for `intersection`.
"""
x = Dummy('x')
y = Dummy('y')
seq = self.equation(x, y)