Director circle

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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 consisting of all points where two perpendicular tangent lines to the ellipse 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 a2 - b2, and so, may not exist in the Euclidean plane, but could be a circle with imaginary radius in the complex plane.

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 P1 is a fixed constant r, is another special case, with w1 = 1, w2 = −r2, 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]

  1. ^ Akopyan & Zaslavsky 2007, pp. 12–13
  2. ^ Faulkner 1952, p. 83

References[edit]

  • Akopyan, A. V.; Zaslavsky, A. A. (2007), Geometry of Conics, Mathematical World, 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, MR 1516260, doi:10.2307/2968867 .
  • 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 .