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The paraboloid is hyperbolic if every other plane section is either a hyperbola, or two crossing lines (in the case of a section by a tangent plane). The paraboloid is elliptic if every other nonempty plane section is either an ellipse, or a single point (in the case of a section by a tangent plane). A paraboloid is either elliptic or hyperbolic.
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.
Hence, the coordinates can be used to solve these equations in geometries with paraboloidal symmetry, i.e. with boundary conditions specified on sections of paraboloids. The Helmholtz equation is ( ∇ 2 + k 2 ) ψ = 0 {\displaystyle (\nabla ^{2}+k^{2})\psi =0} .
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).
It will only work in space; but in orbit, gravity will not distort the mirror's shape into a paraboloid. The design features a liquid stored in a flat-bottomed ring-shaped container with raised interior edges. The central focal area would be rectangular, but a secondary rectangular-parabolic mirror would gather the light to a focal point.
Join the paraboloids y = xz and x = yz. The result is shown in Figure 1. Figure 1. The paraboloid y = x z is shown in blue and orange. The paraboloid x = y z is shown in cyan and purple. In the image the paraboloids are seen to intersect along the z = 0 axis. If the paraboloids are extended, they should also be seen to intersect along the lines ...
If both curves are contained in a common plane, the translation surface is planar (part of a plane). This case is generally ignored. ellipt. paraboloid, parabol. cylinder, hyperbol. paraboloid as translation surface translation surface: the generating curves are a sine arc and a parabola arc Shifting a horizontal circle along a helix. Simple ...
The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines through each of its points (Fuchs & Tabachnikov 2007). The properties of being ruled or doubly ruled are preserved by projective maps, and therefore are concepts of projective geometry.