Fermat–Apollonius circle

Fermat–Apollonius circle

In geometry, the Fermat–Apollonius circle of an ellipse is a circle that circumscribes the minimum bounding box of the ellipse. It has the same center as the ellipse, with radius √("a"2 + "b"2), where "a" and "b" are the semi-major axis and semi-minor axis of the ellipse. Additionally, it has the property that, when viewed from any point on the circle, the ellipse spans a right angle.

More generally, for any collection of points "Pi", weights "wi", and constant "C", one can define a Fermat–Apollonius circle, the locus of points "X" such that

:sum w_i , d^2(X,P_i) = C.

The Fermat–Apollonius circle of an ellipse is a special case of this more general construction with two points "P"1 and "P"2 at the foci of the ellipse, weights "w"1 = "w"2 = 1, and "C" equal to the square of the major axis of the ellipse. The Apollonius circle, the locus of points "X" such that the ratio of distances of "X" to two foci "P"1 and "P"1 is a fixed constant "r", is another special case, with "w"1 = 1, "w"2 = −"r"2, and "C" = 0.

References

*citation
last1 = Akopyan | first1 = A. V.
last2 = Zaslavsky | first2 = A. A.
title = Geometry of Conics
publisher = American Mathematical Society
series = Mathematical World | volume = 26
year = 2007 | isbn = 978-08218-4323-9
pages = 12–13.


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