Definition of focal conics A,C: vertices of the ellipse and foci of the hyperbola E,F: foci of the ellipse and vertices of the hyperbolaFocal conics: two parabolas A: vertex of the red parabola and focus of the blue parabola F: focus of the red parabola and vertex of the blue parabola
In geometry, focal conics are a pair of curves consisting of[1][2]
either
an ellipse and a hyperbola, where the hyperbola is contained in a plane, which is orthogonal to the plane containing the ellipse. The vertices of the hyperbola are the foci of the ellipse and its foci are the vertices of the ellipse (see diagram).
or
two parabolas, which are contained in two orthogonal planes and the vertex of one parabola is the focus of the other and vice versa.
Focal conics play an essential role answering the question: "Which right circular cones contain a given ellipse or hyperbola or parabola (see below)".
Focal conics can be seen as degenerate focal surfaces: Dupin cyclides are the only surfaces, where focal surfaces collapse to a pair of curves, namely focal conics.[5]
Right circular cone (green) through an ellipse (blue)
Right circular cones through an ellipse
The apices of the right circular cones through a given ellipse lie on the focal hyperbola belonging to the ellipse.
Right circular cones through an ellipse
Proof
Given: Ellipse with vertices and foci and a right circular cone with apex containing the ellipse (see diagram).
Because of symmetry the axis of the cone has to be contained in the plane through the foci, which is orthogonal to the ellipse's plane. There exists a Dandelin sphere, which touches the ellipse's plane at the focus and the cone at a circle. From the diagram and the fact that all tangential distances of a point to a sphere are equal one gets:
Hence:
const.
and the set of all possible apices lie on the hyperbola with the vertices and the foci .
Analogously one proves the cases, where the cones contain a hyperbola or a parabola.[7]