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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 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. [ 5 ] Because it preserves distances to a pair of specified points, the two-point equidistant projection maps the family of confocal conics on the sphere onto two families of ...
If projected onto the xyz hyperplane, its image is a ball. If projected onto the xyw, xzw, or yzw hyperplanes, its image is a solid cone. If projected onto an oblique hyperplane, its image is either an ellipsoid or a solid cone with an ellipsoidal base (resembling an ice cream cone). These images are the analogues of the possible images of the ...
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]
Download as PDF; Printable version ... hide. A spherical cone may mean: a hypercone in 4D; a spherical sector in 3D; See also. Spherical conic This page was last ...
Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space. A ruled surface can be described as the set of points swept by a moving straight line.
It is readily seen that the image of the sphere under this transformation is null in the Minkowski space, and so it lies on the cone N +. Consequently, it determines a cross-section of the line bundle N + → S. Nevertheless, there was an arbitrary choice. If κ(x) is any positive function of x = (z, x 0, ..., x n), then the assignment
If the conic is non-degenerate, the conjugates of a point always form a line and the polarity defined by the conic is a bijection between the points and lines of the extended plane containing the conic (that is, the plane together with the points and line at infinity). If the point p lies on the conic Q, the polar line of p is the tangent line ...