# Ellipses¶

class sympy.geometry.ellipse.Ellipse(center=None, hradius=None, vradius=None, eccentricity=None, **kwargs)[source]

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.

Raises

GeometryError

When $$hradius$$, $$vradius$$ and $$eccentricity$$ are incorrectly supplied as parameters.

TypeError

When $$center$$ is not a Point.

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)


Attributes

property apoapsis

The apoapsis of the ellipse.

The greatest distance between the focus and the contour.

Returns

apoapsis : number

Examples

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


periapsis

Returns shortest distance between foci and contour

arbitrary_point(parameter='t')[source]

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.

Examples

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

property area

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

auxiliary_circle()[source]

Returns a Circle whose diameter is the major axis of the ellipse.

Examples

>>> from sympy import 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))

property bounds

Return a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure.

property center

The center of the ellipse.

Returns

center : number

Examples

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

property circumference

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)

director_circle()[source]

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 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

R483

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

property eccentricity

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

encloses_point(p)[source]

Return True if p is enclosed by (is inside of) self.

Parameters

p : Point

Returns

encloses_point : True, False or None

Notes

Being on the border of self is considered False.

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

equation(x='x', y='y', _slope=None)[source]

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

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


arbitrary_point

Returns parameterized point on ellipse

References

R484

https://math.stackexchange.com/questions/108270/what-is-the-equation-of-an-ellipse-that-is-not-aligned-with-the-axis

R485

https://en.wikipedia.org/wiki/Ellipse#Equation_of_a_shifted_ellipse

evolute(x='x', y='y')[source]

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)

property foci

The foci of the ellipse.

Raises

ValueError

When the major and minor axis cannot be determined.

Notes

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

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))


sympy.geometry.point.Point

focus_distance

Returns the distance between focus and center

property focus_distance

The focal distance of the ellipse.

The distance between the center and one focus.

Returns

focus_distance : number

Examples

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

property hradius

The horizontal radius of the ellipse.

Returns

Examples

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


intersection(o)[source]

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.

Examples

>>> from sympy import Ellipse, Point, Line
>>> 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)]

is_tangent(o)[source]

Is $$o$$ tangent to the ellipse?

Parameters

o : GeometryEntity

An Ellipse, LinearEntity or Polygon

Returns

is_tangent: boolean

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

Raises

NotImplementedError

When the wrong type of argument is supplied.

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

property major

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

property minor

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

normal_lines(p, prec=None)[source]

Normal lines between $$p$$ and the ellipse.

Parameters

p : Point

Returns

normal_lines : list with 1, 2 or 4 Lines

Examples

>>> from sympy import 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.

property periapsis

The periapsis of the ellipse.

The shortest distance between the focus and the contour.

Returns

periapsis : number

Examples

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


apoapsis

Returns greatest distance between focus and contour

plot_interval(parameter='t')[source]

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]

polar_second_moment_of_area()[source]

Returns the polar second moment of area of an Ellipse

It is a constituent of the second moment of area, linked through the perpendicular axis theorem. While the planar second moment of area describes an object’s resistance to deflection (bending) when subjected to a force applied to a plane parallel to the central axis, the polar second moment of area describes an object’s resistance to deflection when subjected to a moment applied in a plane perpendicular to the object’s central axis (i.e. parallel to the cross-section)

Examples

>>> from sympy import symbols, Circle, Ellipse
>>> c = Circle((5, 5), 4)
>>> c.polar_second_moment_of_area()
128*pi
>>> a, b = symbols('a, b')
>>> e = Ellipse((0, 0), a, b)
>>> e.polar_second_moment_of_area()
pi*a**3*b/4 + pi*a*b**3/4


References

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

random_point(seed=None)[source]

A random point on the ellipse.

Returns

point : Point

Examples

>>> from sympy import Point, Ellipse
>>> 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

reflect(line)[source]

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.

rotate(angle=0, pt=None)[source]

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)

scale(x=1, y=1, pt=None)[source]

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)

second_moment_of_area(point=None)[source]

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

section_modulus(point=None)[source]

Returns a tuple with the section modulus of an ellipse

Section modulus is a geometric property of an ellipse defined as the ratio of second moment of area to the distance of the extreme end of the ellipse from the centroidal axis.

Parameters

point : Point, two-tuple of sympifyable objects, or None(default=None)

point is the point at which section modulus is to be found. If “point=None” section modulus will be calculated for the point farthest from the centroidal axis of the ellipse.

Returns

S_x, S_y: numbers or SymPy expressions

S_x is the section modulus with respect to the x-axis S_y is the section modulus with respect to the y-axis A negative sign indicates that the section modulus is determined for a point below the centroidal axis.

Examples

>>> from sympy import Symbol, Ellipse, Circle, Point2D
>>> d = Symbol('d', positive=True)
>>> c = Circle((0, 0), d/2)
>>> c.section_modulus()
(pi*d**3/32, pi*d**3/32)
>>> e = Ellipse(Point2D(0, 0), 2, 4)
>>> e.section_modulus()
(8*pi, 4*pi)
>>> e.section_modulus((2, 2))
(16*pi, 4*pi)


References

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

property semilatus_rectum

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

Examples

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


apoapsis

Returns greatest distance between focus and contour

periapsis

The shortest distance between the focus and the contour

References

tangent_lines(p)[source]

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.

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))]

property vradius

The vertical radius of the ellipse.

Returns

Examples

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


class sympy.geometry.ellipse.Circle(*args, **kwargs)[source]

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

Raises

GeometryError

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

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)


Attributes

property circumference

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

equation(x='x', y='y')[source]

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

intersection(o)[source]

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))
[]

property radius

Returns

radius : number or sympy expression

Examples

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

reflect(line)[source]

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)

scale(x=1, y=1, pt=None)[source]

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)

property vradius

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)