page_title: Hyperbola
section_title: Hyperbola as plane section of a cone
hierarchical_section_title: Hyperbola / Hyperbola as plane section of a cone
context_page_description: In mathematics, a hyperbola is a type of smooth 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 components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.
  Hyperbolas arise in many ways:
  as the curve representing the function in the Cartesian plane,
  as the path followed by the shadow of the tip of a sundial,
  as the shape of an open orbit, such as the orbit of a spacecraft during a gravity assisted swing-by of a planet or, more generally, any spacecraft exceeding the escape velocity of the nearest planet,
  as the path of a single-apparition comet,
  as the scattering trajectory of a subatomic particle,
  in radio navigation, when the difference between distances to two points, but not the distances themselves, can be determined,
  and so on.
context_section_description: The intersection of an upright double cone by a plane not through the vertex with slope greater than the slope of the lines on the cone is a hyperbola (see diagram: red curve). In order to prove the defining property of a hyperbola (see above) one uses two Dandelin spheres , which are spheres that touch the cone along circles  ,  and the intersecting (hyperbola) plane at points  and . It turns out:  are the foci of the hyperbola.
  Let   be an arbitrary point of the intersection curve .
  The generatrix of the cone containing  intersects circle  at point  and circle  at a point .
  The line segments  and  are tangential to the sphere  and, hence, are of equal length.
  The line segments  and  are tangential to the sphere  and, hence, are of equal length.
  The result is:  is independent of the hyperbola point , because no matter where point  is,  have to be on circles  , , and line segment  has to cross the apex. Therefore, as point  moves along the red curve (hyperbola), line segment  simply rotates about apex without changing its length.
caption: Hyperbola (red): two views of a cone and two Dandelin spheres d1, d2