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Director circle

Circle formed by all 90° crossings of tangents of an ellipse or hyperbola

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated June 4, 2026
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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.

01Properties

The director circle of an ellipse circumscribes the minimum bounding box of the ellipse. It has the same center as the ellipse, with radius {\textstyle {\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.

The director circle of a hyperbola has radius {\textstyle {\sqrt {a^{2}-b^{2}}}, 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 {\textstyle {\sqrt {2}} times the radius of the original circle.

02Generalization

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 \sum _{i}w_{i}\,d(X,P_{i})^{2}=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 P2 is a fixed constant r, is another special case, with w1 = 1, w2 =, r 2, and C = 0.

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Sources and credits

This article is adapted from the Wikipedia article “Director circle”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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