In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other.
Properties[edit]
The director circle of an ellipse circumscribes the minimum bounding box of the ellipse. It has the same center as the ellipse, with radius , where and 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.[1]
The director circle of a hyperbola has radius , and so, may not exist in the Euclidean plane, but could be a circle with imaginary radius in the complex plane.
The director circle of a circle is a concentric circle having radius times the radius of the original circle.
Generalization[edit]
More generally, for any collection of points Pi, weights wi, and constant C, one can define a circle as the locus of points X such that
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 P2 is a fixed constant r, is another special case, with w1 = 1, w2 = –r 2, and C = 0.
Related constructions[edit]
In the case of a parabola the director circle degenerates to a straight line, the directrix of the parabola.[2]
Notes[edit]
- ^ Akopyan & Zaslavsky 2007, pp. 12–13
- ^ Faulkner 1952, p. 83
References[edit]
- Akopyan, A. V.; Zaslavsky, A. A. (2007), Geometry of Conics, Mathematical World, vol. 26, American Mathematical Society, ISBN 978-0-8218-4323-9
- Cremona, Luigi (1885), Elements of Projective Geometry, Oxford: Clarendon Press, p. 369
- Faulkner, T. Ewan (1952), Projective Geometry, Edinburgh and London: Oliver and Boyd
- Hawkesworth, Alan S. (1905), "Some new ratios of conic curves", The American Mathematical Monthly, 12 (1): 1–8, doi:10.2307/2968867, JSTOR 2968867, MR 1516260
- Loney, Sidney Luxton (1897), The Elements of Coordinate Geometry, London: Macmillan and Company, Limited, p. 365
- Wentworth, George Albert (1886), Elements of Analytic Geometry, Ginn & Company, p. 150