# Ellipses¶

class diofant.geometry.ellipse.Ellipse[source]

An elliptical GeometryEntity.

Parameters: center (Point, optional) – Default value is Point(0, 0) hradius (number or Diofant expression, optional) vradius (number or Diofant expression, optional) eccentricity (number or Diofant expression, optional) – Two of $$hradius$$, $$vradius$$ and $$eccentricity$$ must be supplied to create an Ellipse. The third is derived from the two supplied.
center
hradius
vradius
area
circumference
eccentricity
periapsis
apoapsis
focus_distance
foci
Raises: diofant.geometry.exceptions.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

>>> 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)

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

>>> 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

>>> 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

>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.area
3*pi

center

The center of the ellipse.

Returns: center (number)

Examples

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

circumference

The circumference of the ellipse.

Examples

>>> 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

>>> 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.

Notes

Being on the border of self is considered False.

Parameters: p (Point) encloses_point (True, False or None)

Examples

>>> 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 (diofant expression)

See also

arbitrary_point()
Returns parameterized point on ellipse

Examples

>>> 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 (diofant expression)

Examples

>>> 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.

Notes

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

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

See also

diofant.geometry.point.Point

focus_distance
Returns the distance between focus and center

Examples

>>> 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

>>> 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

Examples

>>> 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) intersection (list of GeometryEntity objects)

Notes

Currently supports intersections with Point, Line, Segment, Ray, Circle and Ellipse types.

Examples

>>> 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 NotImplementedError – When the wrong type of argument is supplied. is_tangent (boolean) – True if o is tangent to the ellipse, False otherwise.

Examples

>>> 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

>>> 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

>>> 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

>>> 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

>>> 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

>>> 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

diofant.geometry.point.Point()

arbitrary_point()
Returns parameterized point on ellipse

Examples

>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.random_point() # gives some random point
Point2D(...)
>>> p1 = e1.random_point(seed=0); p1.evalf(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


Notes

A random point may not appear to be on the ellipse, ie, $$p in e$$ may return False. This is because the coordinates of the point will be floating point values, and when these values are substituted into the equation for the ellipse the result may not be zero because of floating point rounding error.

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:

>>> 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.

reflect(line)[source]

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

Examples

>>> Circle((0, 1), 1).reflect(Line((0, 0), (1, 1)))
Circle(Point2D(1, 0), -1)
>>> 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

>>> 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

>>> 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) tangent_lines (list with 1 or 2 Lines) NotImplementedError – Can only find tangent lines for a point, $$p$$, on the ellipse.

Examples

>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.tangent_lines(Point(3, 0))
[Line(Point2D(3, 0), Point2D(3, -12))]

vradius

The vertical radius of the ellipse.

Returns: vradius (number)

See also

Examples

>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.vradius
1

class diofant.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 diofant expression) points (sequence of three Points)
radius(synonymous with hradius, vradius, major and minor)
circumference
equation[source]
Raises: diofant.geometry.exceptions.GeometryError – When trying to construct circle from three collinear points. When trying to construct circle from incorrect parameters.

Examples

>>> # 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))

circumference

The circumference of the circle.

Returns: circumference (number or Diofant expression)

Examples

>>> 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 (Diofant expression)

Examples

>>> 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

>>> 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 diofant expression)

Examples

>>> c1 = Circle(Point(3, 4), 6)
>>> c1.radius
6

reflect(line)[source]

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

Examples

>>> 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

>>> 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

>>> c1 = Circle(Point(3, 4), 6)
>>> c1.vradius
6