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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.
For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle. A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces.
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 curvature lines are sections with one (blue) and two (purple) sheeted hyperboloids. The red points are umbilic points. Hyperboloid of one sheet (see diagram) Semi-axes: =, =, =. The curvature lines are intersections with ellipsoids (blue) and hyperboloids of two sheets (purple).
Hyperboloid of one sheet, such as cooling towers A hyperboloid of one sheet is a doubly ruled surface , and it may be generated by either of two families of straight lines. The hyperbolic paraboloid is a doubly ruled surface so it may be used to construct a saddle roof from straight beams.
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 ]
A two-sheet hyperboloid is an algebraic surface and the union of two non ... Implicit means that the equation defines implicitly one of the variables as a function of ...
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 +.