Focal conics
In geometry, focal conics are a pair of curves consisting of[1][2] either or 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 are used as directrices for generating Dupin cyclides as canal surfaces in two ways.
[3][4] 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] In Physical chemistry focal conics are used for describing geometrical properties of liquid crystals.
If one describes the ellipse in the x-y-plane in the common way by the equation then the corresponding focal hyperbola in the x-z-plane has equation where
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.
, which touches the ellipse's plane at the focus
From the diagram and the fact that all tangential distances of a point to a sphere are equal one gets: Hence: and the set of all possible apices lie on the hyperbola with the vertices
Analogously one proves the cases, where the cones contain a hyperbola or a parabola.
A,C: vertices of the ellipse and foci of the hyperbola
E,F: foci of the ellipse and vertices of the hyperbola
A: vertex of the red parabola and focus of the blue parabola
F: focus of the red parabola and vertex of the blue parabola
