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 formed by the points where two perpendicular tangent lines to the curve cross.

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 \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[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

\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[edit]

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

References[edit]