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) >>> 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)
See also
Attributes
center
hradius
vradius
area
circumference
eccentricity
periapsis
apoapsis
focus_distance
foci
- 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
See also
periapsisReturns 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))
See also
- 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)
See also
- 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
- 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
See also
- 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
See also
arbitrary_pointReturns parameterized point on ellipse
References
- 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))
- 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)
See also
- property hradius¶
The horizontal radius of the ellipse.
- Returns
hradius : number
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
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)]
See also
- 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
See also
- 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)
See also
apoapsisReturns 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
- 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
- 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
angleradians counterclockwise about Pointpt.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
- 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
See also
References
[1] http://mathworld.wolfram.com/SemilatusRectum.html [2] https://en.wikipedia.org/wiki/Ellipse#Semi-latus_rectum
- 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
vradius : number
Examples
>>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.vradius 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) >>> c1.hradius, c1.vradius, c1.radius (5, 5, 5)
A circle constructed 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))
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)
See also
Attributes
radius (synonymous with hradius, vradius, major and minor)
circumference
equation
- 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¶
The radius of the circle.
- Returns
radius : number or sympy expression
Examples
>>> from sympy import Point, Circle >>> c1 = Circle(Point(3, 4), 6) >>> c1.radius 6
See also
Ellipse.major,Ellipse.minor,Ellipse.hradius,Ellipse.vradius
- 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) >>> c1.vradius 6
