Curve on the sphere analogous to an ellipse or hyperbola
Not to be confused with Spherical cone.
Spherical conics drawn on a spherical chalkboard. Two confocal conics in blue and yellow share foci F1 and F2. 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.
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]
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a sphericalconic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog...
planar conic sections also extend to sphericalconics. If a sphere is intersected by another surface, there may be more complicated spherical curves....
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola...
The intersection of an elliptic cone with a concentric sphere is a sphericalconic. In projective geometry, a cylinder is simply a cone whose apex is...
distances along all other parallels are stretched. Conic projections that are commonly used are: Equidistant conic, which keeps parallels evenly spaced along...
can be treated as a spherical surface with the same radius. Some[which?] non-optical design references use the letter p as the conic constant. In these...
A Lambert conformal conic projection (LCC) is a conic map projection used for aeronautical charts, portions of the State Plane Coordinate System, and...
positive value of f, this exceeds 180°. Spherical astronomy SphericalconicSpherical distance Spherical polyhedron Spherics Half-side formula Lénárt sphere Versor...
projection's characteristics. Coordinates from a spherical datum can be transformed into Albers equal-area conic projection coordinates with the following formulas...
The intersection between one of the cones and the sphere forms a sphericalconic. The conical coordinates ( r , μ , ν ) {\displaystyle (r,\mu ,\nu )}...
the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface...
throughout the region of interest. Coordinates from a spherical datum can be transformed to an equidistant conic projection with rectangular coordinates by using...
A spherical cone may mean: a hypercone in 4D a spherical sector in 3D Sphericalconic This disambiguation page lists articles associated with the title...
(described below) are often much more difficult to create. The sides of a conic profile are straight lines, so the diameter equation is simply: y = x R...
such systems. The conformal model discussed below is homogeneous, as is "Conic Geometric Algebra", and see Plane-based geometric algebra for discussion...
points on a line that were in a ratio to the others. Apollonius in the Conics further developed a method that is so similar to analytic geometry that...
Second Degree and on the SphericalConics, adding a significant amount of original material. Jakob Steiner had proposed Steiner's conic problem of enumerating...
1925. The projection has seen use in digital photography for portraying spherical panoramas. The maturation of complex analysis led to general techniques...
scale. An example of the use of the rectifying latitude is the equidistant conic projection. (Snyder, Section 16). The rectifying latitude is also of great...
from a small set of axioms. He also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour. In addition...
the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin spheres...
straight from pole to equator), regularly spaced along parallels. Conic In normal aspect, conic (or conical) projections map meridians as straight lines, and...
setting for spherical analytic geometry instead of spherical polar coordinates or three-dimensional cartesian coordinates. This is the spherical analog of...
reduced so that the cylinder slices through the model globe. Both exist in spherical and ellipsoidal versions. Both projections are conformal, so that the...
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