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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.
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 ).
Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet. Often these are tall structures, such as towers, where the hyperboloid geometry's structural strength is used to support an object high above the ground. Hyperboloid geometry is often used for decorative effect as well as structural economy.
The third case generates the hyperbolic paraboloid or the hyperboloid of one sheet, depending on whether the plane at infinity cuts it in two lines, or in a nondegenerate conic respectively. These are doubly ruled surfaces of negative Gaussian curvature. The degenerate form =
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 ...
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 ...
A hyperboloid of two sheets. Sphere; Spheroid. Oblate spheroid; Prolate spheroid; Ellipsoid; Cone (geometry) Hyperboloid of one sheet; Hyperboloid of two sheets; Hyperbolic paraboloid (a ruled surface) Paraboloid
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 +.