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 or hyperbola cross each other.

The director circle of an ellipse circumscribes the minimum bounding box 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

times the radius of the original circle.

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.

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