In
geometry, the director circle of an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
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.
Properties
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
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the long ...
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
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
.
The director circle of a hyperbola has radius , and so, may not exist in the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
, but could be a circle with imaginary radius in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
.
Generalization
More generally, for any collection of points , weights , and constant , one can define a circle as the locus of points such that
:
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
Apollonius circle
The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for exampl ...
, 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 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