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One has a hyperboloid of revolution if and only if =. Otherwise, the axes are uniquely defined (up to the exchange of the x-axis and the y-axis). There are two kinds of hyperboloids. In the first case (+1 in the right-hand side of the equation): a one-sheet hyperboloid, also called a hyperbolic hyperboloid.
A parabola has only one focus, and can be considered as a limit curve of a set of ellipses (or a set of hyperbolas), where one focus and one vertex are kept fixed, while the second focus is moved to infinity. If this transformation is performed on each conic in an orthogonal net of confocal ellipses and hyperbolas, the limit is an orthogonal ...
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 vectors v ∈ R n+1 such that Q(v) = -1 form an n-dimensional hyperboloid S consisting of two connected components, or sheets: the forward, or future, sheet S +, where x 0 >0 and the backward, or past, sheet S −, where x 0 <0. The points of the n-dimensional hyperboloid model are the points on the forward sheet S +.
The hyperboloid model or Lorentz model employs a 2-dimensional hyperboloid of revolution (of two sheets, but using one) embedded in 3-dimensional Minkowski space. This model is generally credited to Poincaré, but Reynolds [33] says that Wilhelm Killing used this model in 1885
However, the surface of constant τ is a full one-sheet hyperboloid, shown in blue. This produces a two-fold degeneracy, shown by the two black spheres located at ( x , y , ± z ). An alternative and geometrically intuitive set of oblate spheroidal coordinates (σ, τ, φ) are sometimes used, where σ = cosh μ and τ = cos ν. [ 1 ]
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From the point of view of projective geometry, a hyperbolic paraboloid is one-sheet hyperboloid that is tangent to the plane at infinity. A hyperbolic paraboloid of equation z = a x y {\displaystyle z=axy} or z = a 2 ( x 2 − y 2 ) {\displaystyle z={\tfrac {a}{2}}(x^{2}-y^{2})} (this is the same up to a rotation of axes ) may be called a ...