In
mathematics, a spherical conic or sphero-conic is a
curve on the
sphere, the intersection of the sphere with a concentric
elliptic cone. It is the spherical analog of a
conic section (
ellipse,
parabola, or
hyperbola) in the plane, and as in the planar case, a spherical conic can be defined as the
locus of points the sum or difference of whose
great-circle distances to two
foci is constant.^{
[1]} By taking the antipodal point to one focus, every spherical ellipse is also a spherical hyperbola, and vice versa. As a space curve, a spherical conic is a
quartic, though its orthogonal projections in three
principal axes are planar conics. Like planar conics, spherical conics also satisfy a "reflection property": the great-circle arcs from the two foci to any point on the conic have the tangent and normal to the conic at that point as their angle bisectors.
Many theorems about conics in the plane extend to spherical conics. For example, Graves's theorem and Ivory's theorem about confocal conics can also be proven on the sphere; see
confocal conic sections about the planar versions.^{
[2]}
Just as the
arc length of an ellipse is given by an incomplete
elliptic integral of the second kind, the arc length of a spherical conic is given by an incomplete elliptic integral of the third kind.^{
[3]}
An
orthogonal coordinate system in Euclidean space based on concentric spheres and quadratic cones is called a
conical or sphero-conical coordinate system. When restricted to the surface of a sphere, the remaining coordinates are confocal spherical conics. Sometimes this is called an elliptic coordinate system on the sphere, by analogy to a planar
elliptic coordinate system. Such coordinates can be used in the computation of conformal maps from the sphere to the plane.^{
[4]}
Applications
The solution of the
Kepler problem in a space of uniform positive curvature is a spherical conic, with a potential proportional to the cotangent of geodesic distance.^{
[5]}
Because it preserves distances to a pair of specified points, the
two-point equidistant projection maps the family of confocal conics on the sphere onto two families of confocal ellipses and hyperbolae in the plane.^{
[6]}
If a portion of the Earth is modeled as spherical, e.g. using the
osculating sphere at a point on an ellipsoid of revolution, the hyperbolae used in
hyperbolic navigation (which determines position based on the difference in received signal timing from fixed radio transmitters) are spherical conics.^{
[7]}
Glaeser, Georg;
Stachel, Hellmuth; Odehnal, Boris (2016). "10.1 Spherical conics". The Universe of Conics: From the ancient Greeks to 21st century developments. Springer. pp. 436–467.
doi:
10.1007/978-3-662-45450-3_10.