# Ellipses¶

class sympy.geometry.ellipse.Ellipse[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. 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

apoapsis

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

arbitrary_point(parameter='t')[source]

A parameterized point on the ellipse.

Parameters: parameter : str, optional Default value is ‘t’. arbitrary_point : Point 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))

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

bounds

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

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)

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)

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 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')[source]

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’. equation : sympy expression

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

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’. 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)

foci

The foci of the ellipse.

Raises: ValueError When the major and minor axis cannot be determined.

sympy.geometry.point.Point

focus_distance
Returns the distance between focus and center

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))

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)

hradius

The horizontal radius of the ellipse.

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 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, 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)]

is_tangent(o)[source]

Is $$o$$ tangent to the ellipse?

Parameters: o : GeometryEntity An Ellipse, LinearEntity or Polygon is_tangent: boolean True if o is tangent to the ellipse, False otherwise. 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

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

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 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.

periapsis

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
-2*sqrt(2) + 3

plot_interval(parameter='t')[source]

The plot interval for the default geometric plot of the Ellipse.

Parameters: parameter : str, optional Default value is ‘t’. 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]

random_point(seed=None)[source]

A random point on the ellipse.

Returns: point : Point

sympy.geometry.point.Point

arbitrary_point
Returns parameterized point on ellipse

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


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)

reflect(line)[source]

Override GeometryEntity.reflect since the radius is not a GeometryEntity.

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.

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

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. 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.

References

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

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)

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

apoapsis
Returns greatest distance between focus and contour
periapsis
The shortest distance between the focus and the contour

References

Examples

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

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 tangent_lines : list with 1 or 2 Lines 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))]

vradius

The vertical radius of the ellipse.

Examples

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

class sympy.geometry.ellipse.Circle[source]

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 GeometryError When trying to construct circle from three collinear points. When trying to construct circle from incorrect parameters.

Examples

>>> from sympy.geometry import Point, Circle
>>> # 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))


Attributes

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’. 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 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))
[]

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)

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)