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Circle-Ellipse Intersection

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CircleEllipseIntersection

An ellipse intersects a circle in 0, 1, 2, 3, or 4 points. The points of intersection of a circle of center (x_0,y_0) and radius r with an ellipse of semi-major and semi-minor axes a and b, respectively and center (x_e,y_e) can be determined by simultaneously solving

(x-x_0)^2+(y-y_0)^2=r^2
(1)
((x-x_e)^2)/(a^2)+((y-y_e)^2)/(b^2)=1.
(2)

If (x_0,y_0)=(x_e,y_e)=(0,0), then the solution takes on the particularly simple form

x=+/-asqrt((r^2-b^2)/(a^2-b^2))
(3)
y=+/-bsqrt((a^2-r^2)/(a^2-b^2)).
(4)

See also

Circle, Circle-Circle Intersection, Ellipse

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Cite this as:

Weisstein, Eric W. "Circle-Ellipse Intersection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Circle-EllipseIntersection.html

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