Director Circle

In geometry, the director circle of an ellipse or hyperbola (also called the Isoptic, 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

The director circle of an ellipse Circumscribed circle, circumscribes the minimum bounding box of the ellipse. It has the same center as the ellipse, with radius $\sqrt$, 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 , and so, may not exist in the Euclidean plane, but could be a circle with imaginary radius in the complex plane.

Generalization

More generally, for any collection of points , weights , and constant , one can define a circle as the locus of points 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 and at the foci of the ellipse, weights , and equal to the square of the major axis of the ellipse. The Circle#Apollonius circle, Apollonius circle, the locus of points such that the ratio of distances of to two foci and is a fixed constant , is another special case, with , , and .

Related constructions

In the case of a parabola the director circle degenerates to a straight line, the Directrix (conic section), directrix of the parabola.

Notes

References

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*{{citation, first=George Albert, last=Wentworth, title=Elements of Analytic Geometry, publisher=Ginn & Company, year=1886, page=150.

Conic sections
Circles