Director circle

An ellipse, its minimum bounding box, and its director circle.

In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle formed by the points where two perpendicular tangent lines to the curve cross.

Properties

The director circle of an ellipse circumscribes the minimum bounding box of the ellipse. It has the same center as the ellipse, with radius $\sqrt{a^2+b^2}$ , 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.

Generalization

More generally, for any collection of points P_i, weights w_i, and constant C, one can define a circle as the locus of points X such that

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

The director circle of an ellipse is a special case of this more general construction with two points P₁ and P₂ at the foci of the ellipse, weights w₁ = w₂ = 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 P₁ and P₁ is a fixed constant r, is another special case, with w₁ = 1, w₂ = −r², and C = 0.

Related constructions

In the case of a parabola the director circle degenerates to a straight line, the directrix of the parabola.

References

Akopyan, A. V.; Zaslavsky, A. A. (2007), Geometry of Conics, Mathematical World 26, American Mathematical Society, pp. 12–13, ISBN 978-0-8218-4323-9
Luigi Cremona, Elements of Projective Geometry, Oxford, Clarendon Press, 1885. See page 369
Alan S. Hawkesworth, "Some New Ratios of Conic Curves", American Mathematical Monthly, January 1950, page 1.
Sidney Luxton Loney, The Elements of Coordinate Geometry, Macmillan and Company, Limited, London 1897. See page 365.
George Albert Wentworth, Elements of Analytic Geometry, Ginn & Company, 1886. See page 150.