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.
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)                                
[0]: cos(t), sin(t), 'mode=parametric'
>>> p = Plot()                              
>>> p[0] = c1                               
>>> radius = Segment(c1.center, c1.random_point())
>>> p[1] = radius                           
>>> p                                       
[0]: cos(t), sin(t), 'mode=parametric'
[1]: t*cos(1.546086215036205357975518382),
t*sin(1.546086215036205357975518382), 'mode=parametric'

Attributes

center  
hradius  
vradius  
area  
circumference  
eccentricity  
periapsis  
apoapsis  
focus_distance  
foci  
apoapsis

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
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.

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))
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

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)
circumference

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))
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

See also

sympy.geometry.point.Point

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’.
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
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)
foci

The foci of the ellipse.

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

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 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)
hradius

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
intersection(o)[source]

The intersection of this ellipse and another geometrical entity \(o\).

Parameters:o : GeometryEntity
Returns:intersection : list of GeometryEntity objects

See also

sympy.geometry.entity.GeometryEntity

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(-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)]
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.

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
major

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
minor

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
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 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.

periapsis

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
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]
random_point(seed=None)[source]

A random point on the ellipse.

Returns:point : Point

See also

sympy.geometry.point.Point

arbitrary_point
Returns parameterized point on ellipse

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.

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
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)
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.

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 
>>> p[1] = t
vradius

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
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
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))

Attributes

radius (synonymous with hradius, vradius, major and minor)  
circumference  
equation  
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))
[]
radius

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
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)
>>> c1.vradius
6