Hyperboloids


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Related to Hyperboloids: Hyperboloid of one sheet, Hyperboloid of two sheets, hyperbolic cylinder, Ellipsoids

Hyperboloids

 

open centric surfaces of the second degree. Two types of hyperboloids are distinguished: hyperboloid of one sheet (Figure 1) and hyperboloid of two sheets (Figure 2). They represent two of the five principal types of

Figure 1. Hyperboloid of one sheet

Figure 2. Hyperboloid of two sheets

second-degree surfaces and, on intersection with various planes, give all the conic sections—the ellipse, hyperbola, and parabola—as well as pairs of straight lines (in the case of a hyperboloid of one sheet). A hyperboloid comes infinitely close to a conic surface (the so-called asymptotic cone). The hyperboloid of one sheet is a ruled surface. In the corresponding coordinate system (see Figures 1 and 2), the equations of the hyperboloids have the form

References in periodicals archive ?
Furthermore, in order to resolve the scaling problems of Wyler's expression raised by Robertson, Gilmore showed why it is essential to use dimensionless volumes by setting the throat sizes of the Anti de Sitter hyperboloids to r = 1, because this is the only choice for r where all elements in the bounded domains are also coset representatives, and therefore, amount to honest group operations.
Since Gilmore, in response to Robertson's objections to Wyler's formula [2], has shown that one must set the scale r = 1 of the hyperboloids [H.
4] of 4 complex dimensions is an 8 real-dim Hyperboloid of constant negative scalar curvature that can be identified with the conformal relativistic curved phase space associated with the electron (a particle) moving in a 4D Anti de Sitter space Ad[S.
It is a striking fact that Genzken drew the ellipsoids and hyperboloids on a computer, an unusual way of working at the time.