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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 ...
The elliptic cones intersect the sphere in spherical conics. Conical coordinates , sometimes called sphero-conal or sphero-conical coordinates, are a three-dimensional orthogonal coordinate system consisting of concentric spheres (described by their radius r ) and by two families of perpendicular elliptic cones, aligned along the z - and x ...
The equation for a conic section with apex at the origin and tangent to the y axis is + (+) = alternately = + (+) where R is the radius of curvature at x = 0. This formulation is used in geometric optics to specify oblate elliptical ( K > 0 ), spherical ( K = 0 ), prolate elliptical ( 0 > K > −1 ), parabolic ( K = −1 ), and hyperbolic ( K ...
In geometry, a hypercone (or spherical cone) is the figure in the 4-dimensional Euclidean space represented by the equation x 2 + y 2 + z 2 − w 2 = 0. {\displaystyle x^{2}+y^{2}+z^{2}-w^{2}=0.} It is a quadric surface, and is one of the possible 3- manifolds which are 4-dimensional equivalents of the conical surface in 3 dimensions.
More generally, when the directrix is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of , one obtains an elliptic cone [4] (also called a conical quadric or quadratic cone), [5] which is a special case of a quadric surface. [4] [5]
The analog of a conic section on the sphere is a spherical conic, a quartic curve which can be defined in several equivalent ways. The intersection of a sphere with a quadratic cone whose vertex is the sphere center; The intersection of a sphere with an elliptic or hyperbolic cylinder whose axis passes through the sphere center
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requiring a conic to pass through a point imposes a linear condition on the coordinates: for a fixed (,), the equation + + + + + = is a linear equation in (,,,,,); by dimension counting , five constraints (that the curve passes through five points) are necessary to specify a conic, as each constraint cuts the dimension of possibilities by 1 ...