Ellipses¶
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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
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
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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
Examples
>>> from sympy import Point, Ellipse >>> e1 = Ellipse(Point(0, 0), 3, 2) >>> e1.arbitrary_point() Point2D(3*cos(t), 2*sin(t))
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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
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bounds
¶ Return a tuple (xmin, ymin, xmax, ymax) representing the bounding rectangle for the geometric figure.
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center
¶ The center of the ellipse.
Returns: center : number See also
Examples
>>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.center Point2D(0, 0)
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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))
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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
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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
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
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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
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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)
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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))
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focus_distance
¶ The focale distance of the ellipse.
The distance between the center and one focus.
Returns: focus_distance : number See also
Examples
>>> from sympy import Point, Ellipse >>> p1 = Point(0, 0) >>> e1 = Ellipse(p1, 3, 1) >>> e1.focus_distance 2*sqrt(2)
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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
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intersection
(o)[source]¶ The intersection of this ellipse and another geometrical entity \(o\).
Parameters: o : GeometryEntity Returns: intersection : list of GeometryEntity objects See also
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)]
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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
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
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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
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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
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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.
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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
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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]
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random_point
(seed=None)[source]¶ A random point on the ellipse.
Returns: point : Point 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
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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
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rotate
(angle=0, pt=None)[source]¶ Rotate
angle
radians 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)
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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)
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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)) [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
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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
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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
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
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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
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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)) []
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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
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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)
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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)
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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
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