Spherical conic

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.

[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.

When restricted to the surface of a sphere, the remaining coordinates are confocal spherical conics.

[4] 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.

[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.

Spherical conics drawn on a spherical chalkboard. Two confocal conics in blue and yellow share foci F 1 and F 2 . Angles formed with red great-circle arcs from the foci through one of the conics' intersections demonstrate the reflection property of spherical conics. Three mutually perpendicular conic centers and three lines of symmetry in green define a spherical octahedron aligned with the principal axes of the conic.
A grid on the square dihedron under inverse Peirce quincuncial projection is conformal except at four singularities around the equator, which become the foci of a grid of spherical conics.