page_title: Confocal conic sections
section_title: Focal curves
hierarchical_section_title: Confocal conic sections / Confocal quadrics / Focal curves
context_page_description: In geometry, two conic sections are called confocal, if they have the same foci. Because ellipses and hyperbolas possess two foci, there are confocal ellipses, confocal hyperbolas and confocal mixtures of ellipses and hyperbolas. In the mixture of confocal ellipses and hyperbolas, any ellipse intersects any hyperbola orthogonally. Parabolas possess only one focus, so, by convention, confocal parabolas have the same focus and the same axis of symmetry. Consequently, any point not on the axis of symmetry lies on two confocal parabolas which intersect orthogonally.
  The formal extension of the concept of confocal conics to surfaces leads to confocal quadrics.
context_section_description: Limit surfaces for :
  Varying the ellipsoids by increasing parameter  such that it approaches the value  from below one gets an infinite flat ellipsoid. More precise: the area of the x-y-plane, that consists of the ellipse  with equation  and its doubly covered interior (in the diagram: below, on the left, red).
  Varying the 1-sheeted hyperboloids by decreasing parameter  such that it approaches the value  from above one gets an infinite flat hyperboloid. More precise: the area of the x-y-plane, that consists of the same ellipse  and its doubly covered exterior (in the diagram: bottom, on the left, blue).
  
  That means: The two limit surfaces have the points of ellipse 
  in common.
  Limit surfaces for :
  Analogous considerations at the position  yields:
  The two limit surfaces (in diagram: bottom, right, blue and purple) at position  have the hyperbola
  in common.
  Focal curves:
  One easily checks, that the foci of the ellipse are the vertices of the hyperbola and vice versa. That means: Ellipse  and hyperbola  are a pair of focal conics.
  Reverse: Because any quadric of the pencil of confocal quadrics determined by  can be constructed by a pins-and-string method (see ellipsoid) the focal conics  play the role of infinite many foci and are called focal curves of the pencil of confocal quadrics.
caption: Focal conics (ellipse, hyperbola, black)