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Paraboloid of revolution. In geometry, ... (measured along the axis of symmetry from the vertex to the plane of the rim), and R is the radius of the rim.
r = the radius of the cone's base h = the distance is from base to the apex ... Solid paraboloid of revolution around z-axis: a = the radius of the base circle
If the dish is symmetrical and made of uniform material of constant thickness, and if F represents the focal length of the paraboloid, this "focus-balanced" condition occurs if the depth of the dish, measured along the axis of the paraboloid from the vertex to the plane of the rim of the dish, is 1.8478 times F. The radius of the rim is 2.7187 F.
Parabolic antennas are based on the geometrical property of the paraboloid that the paths FP 1 Q 1, FP 2 Q 2, FP 3 Q 3 are all the same length. Thus, a spherical wavefront emitted by a feed antenna at the dish's focus F will be reflected into an outgoing plane wave L travelling parallel to the dish's axis VF.
The radius r has a simple formula as well r = x 2 + y 2 = 1 2 ( σ 2 + τ 2 ) {\displaystyle r={\sqrt {x^{2}+y^{2}}}={\frac {1}{2}}\left(\sigma ^{2}+\tau ^{2}\right)} that proves useful in solving the Hamilton–Jacobi equation in parabolic coordinates for the inverse-square central force problem of mechanics ; for further details, see the ...
Paraboloidal coordinates are three-dimensional orthogonal coordinates (,,) that generalize two-dimensional parabolic coordinates.They possess elliptic paraboloids as one-coordinate surfaces.
The radius of curvature at the origin, which is the vertex of the parabola, is twice the focal length. Corollary A concave mirror that is a small segment of a sphere behaves approximately like a parabolic mirror, focusing parallel light to a point midway between the centre and the surface of the sphere.
The red paraboloid corresponds to τ=2, the blue paraboloid corresponds to σ=1, and the yellow half-plane corresponds to φ=-60°. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (1.0, -1.732, 1.5).