geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, 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 ...

hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...

(also called the orthoptic circle or Fermat–Apollonius circle) is a

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

consisting of all points where two

perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...

tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

s to the ellipse or hyperbola cross each other.

Properties

The director circle of an ellipse circumscribes the

minimum bounding box In geometry, the minimum bounding box or smallest bounding box (also known as the minimum enclosing box or smallest enclosing box) for a point set in dimensions is the box with the smallest measure (area, volume, or hypervolume in higher dime ...

of the ellipse. It has the same center as the ellipse, with radius

\sqrt

, where

a

and

b

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 longe ...

and

semi-minor 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 longe ...

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

\sqrt

, and so, may not exist in the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

, but could be a circle with imaginary radius in the

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

. The director circle of a circle is a concentric circle having radius

\sqrt

times the radius of the original circle.

Generalization

More generally, for any collection of points , weights , and constant , one can define a circle as the locus of points 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 and at the foci of the ellipse, weights , and equal to the square of the major axis of the ellipse. The 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 In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...

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

Notes

References

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