Reference hyperboloid

The moire of two cross sections of the hyperboloids represents the difference of two interference patterns. If the two cross sections are identical but one is displaced, the moire fringes also represent the loci of constant resolution in the displacement direction. If one focal point is fixed, and the other one is displaced, the resulting moire pattern forms a new set of hyperboloids whose foci are the two positions of the displaced focal point. This new pattern is independent of the position of the fixed focal point (a rotation of one of the original spherical wavefronts of Fig. 5 produces no moire effect). Therefore hologram interference fringes are independent of the position of the point source of the fixed reference beam. 

If all three principal values are positive, the quadric surface is an ellipsoid with semiaxes a, = TfiVz, but if one or two of the principal values are negative the quadric surface is a hyperboloid. For example, the (relative) impermeability tensor 3 is defined by nfn, where k is the permittivity and n0 is the permittivity of free space. As for any symmetric 7(2) the components of 3 define the representation quadric I3ijxixj= 1, which here is called the indicatrix or optical index ellipsoid. Referred to principal axes the indicatrix is... 

This contribution as a chapter in the special volume of ADVANCES IN QUANTUM CHEMISTRY on Confined Quantum Systems is focussed on (i) the hydrogen atom, (ii) confinement by conoidal boundaries, and (iii) semi-infinite spaces however, some of its discussions may extend their validity to other physical systems and to confinement in closed volumes. The limitations in the title are given as a point of reference, and also take into account that several of the other chapters deal with confinement in finite volumes. A semantic parenthesis is also appropriate and self-explanatory Compare conical curves (circles, ellipses, parabolas, hyperbolas and their radial asymptotes) with conoidal surfaces (spheres, spheroids, paraboloids, hyperboloids and their radial asymptotic cones). 

Since the hyperboloid is a surface of revolution, the tangent plane to any point of the waist circle will give a similar pair of straight lines on the hyperboloid so that there exist two families of straight lines each of which covers the completely. Referring to Figures AIF.3 and the following equations may be establis... 

The U.S. Space Agency, NASA, is also building a new X-ray satellite telescope. Advanced X-Ray Astrophysics Facilities (AXAF), for the spectral region of 0.1 to 10 KeV for an Earth orbit at 600 km altitude (Fig. 4.67). The launch is planned for 1998. This Wolter telescope, with six paraboloids and hyperboloids each nested into one another, consists of Zerodur cylinders with diameters between 600 and 1274 mm, whose fabrication required the development of many new techniques (refer to Sect. 4.3.3). 

Figure 27. Characteristic surfaces of second-rank polar tensors (a) ellipsoid, (b) hyperboloid of one sheet, (c) hyperboloid of two sheets. (After reference 36.)...

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