mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a spherical conic or sphero-conic is a

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...

on the

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

, the intersection of the sphere with a concentric

elliptic cone In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the ''apex'' or ''vertex'' — and any point of some fixed space curve — the ''di ...

. It is the spherical analog of a

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...

(

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...

, or

hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) 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, cal ...

) in the plane, and as in the planar case, a spherical conic can be defined as the

locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...

of points the sum or difference of whose

great-circle distance The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle. It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a ...

s to two

foci Focus, or its plural form foci may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film *''Focus'', a 1962 TV film starring James Whitmore * ''Focus'' (2001 film), a 2001 film based ...

is constant. 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 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 ...

about the planar versions. Just as the arc length of an ellipse is given by an incomplete

elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...

of the second kind, the arc length of a spherical conic is given by an incomplete elliptic integral of the third kind. 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. The solution of the

Kepler problem In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be ei ...

in a space of uniform positive curvature is a spherical conic, with a potential proportional to the cotangent of geodesic distance. Because it preserves distances to a pair of specified points, the

two-point equidistant projection The two-point equidistant projection or doubly equidistant projection is a map projection first described by Hans Maurer in 1919 and Charles Close in 1921. It is a generalization of the much simpler azimuthal equidistant projection. In this two ...

maps the family of confocal conics on the sphere onto two families of confocal ellipses and hyperbolae in the plane. If a portion of the Earth is modeled as spherical, e.g. using the

osculating sphere Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. A sphere is a well-known historical approxima ...

at a point on an ellipsoid of revolution, the hyperbolae used in

hyperbolic navigation Hyperbolic navigation is a class of radio navigation systems in which a navigation receiver instrument is used to determine location based on the difference in timing (phase) of radio waves received from radio navigation beacon transmitters. Su ...

(which determines position based on the difference in received signal timing from fixed radio transmitters) are spherical conics.

Notes

References

* English edition:
*
Republished in
''Journal de mathématiques pures et appliquées 2e série''. 5: 425-454
PDF from mathdoc.fr
* * * * * * {{cite thesis , last=Tranacher , first=Harald , year=2006 , title=Sphärische Kegelschnitte – didaktisch aufbereitet , lang=de , trans-title=Spherical conics – didactically prepared , publisher= Technischen Universität Wien , url=https://www.geometrie.tuwien.ac.at/theses/pdf/diplomarbeit_tranacher.pdf Conic sections Spherical curves Spherical trigonometry Euclidean solid geometry Algebraic curves Analytic geometry