Spherical conic

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a spherical conic or sphero-conic is a <a href="/facts/Curve/xBQ0pTBW">curve</a> on the <a href="/facts/Sphere/jywYLNM2">sphere</a>, the intersection of the sphere with a concentric <a href="/facts/Elliptic_cone/n6h4xmLu">elliptic cone</a>. It is the spherical analog of a <a href="/facts/Conic_section/teqgLptX">conic section</a> (<a href="/facts/Ellipse/1wUDnGxY">ellipse</a>, <a href="/facts/Parabola/Cweo3wgw">parabola</a>, or <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a>) in the plane, and as in the planar case, a spherical conic can be defined as the <a href="/facts/Locus_(mathematics)/t4swdGJq">locus</a> of points the sum or difference of whose <a href="/facts/Great-circle_distance/s6HNWn8e">great-circle distances</a> to two <a href="/facts/Focus_(geometry)/WFtFnqoa">foci</a> 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 <a href="/facts/Quartic_function/lHuBhMk7">quartic</a>, though its orthogonal projections in three <a href="/facts/Principal_axis_theorem/cr8VMued">principal axes</a> 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 <a href="/facts/Confocal_conic_sections/sNANGYBF">confocal conic sections</a> about the planar versions.
Just as the <a href="/facts/Ellipse/1wUDnGxY">arc length of an ellipse</a> is given by an incomplete <a href="/facts/Elliptic_integral/q8EGYzQ3">elliptic integral</a> of the second kind, the arc length of a spherical conic is given by an incomplete elliptic integral of the third kind.
An <a href="/facts/Orthogonal_coordinates/LLoa2lAJ">orthogonal coordinate system</a> in Euclidean space based on concentric spheres and quadratic cones is called a <a href="/facts/Conical_coordinates/0eYMnnMx">conical</a> 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 <a href="/facts/Elliptic_coordinate_system/aGfvboJP">elliptic coordinate system</a>. Such coordinates can be used in the computation of conformal maps from the sphere to the plane.

Spherical conic open-in-new

Spherical conic