Hyperboloid

Symmetry 50. Intercepts 50. Asymptotes 50. Equations of Slope 51. Tangents 51. Equations of a Straight Line 52. Equations of a Circle 53. Equations of a Parabola 53. Equations of an Ellipse of Eccentricity e 54. Equations of a Hyperbola 55. Equations of Three-Dimensional Coordinate Systems 56. Equations of a Plane 56. Equations of a Line 57. Equations of Angles 57. Equation of a Sphere 57. Equation of an Ellipsoid 57. Equations of Hyperboloids and Paraboloids 58. Equation of an Elliptic Cone 59. Equation of an Elliptic Cylinder 59. 

Chimney-assisted natural draft towers are of hyperboloidal shapes because they have greater strength for a given thickness a tower 250 ft high has concrete walls 5-6 in. thick. The enlarged cross section at the top aids in dispersion of. exit humid air into the atmosphere. 

The secret is to build a primary that is aspheric to get the desirable optical properties of these designs while keeping the segments as close as possible to a shape that is spherical or can be polished almost as easily as a sphere such as a toroid, a surface with two constant radii. To study this idea further, consider a mirror that is a parabola of revolution. We use a parabola because the more realistic hyperboloid is only a few percent different from the parabola but the equations are simpler and thus give more insight into the real issues of fabrication. The sagitta, or sag, of a parabola is its depth measured along a diameter with respect to its vertex, or... 

Equation (2.110) describes hyperboloids of one sheet. Finally, within the interval —P KBK—b this equation characterizes hyperboloids of two sheets. Thus, one surface of each family passes through any point of space. It is a simple matter to find equations for the normal to these surfaces and demonstrate that they are orthogonal to each other. Now we show that these surfaces can be eqiupotential surfaces. Let us introduce the functions... 

Now we demonstrate the system of coordinates, where the ellipsoids of rotation and hyperboloids of one sheet form two mutually orthogonal coordinate families of surfaces. First, we introduce the Cartesian system at the center of the mass and suppose that semi-axes of the ellipsoid of rotation obey the condition brelation between coordinates of the Cartesian and cylindrical... 

Another curvilinear coordinate system of importance in two-centre problems, such as the diatomic molecule, derives from the more general system of confo-cal elliptical coordinates. The general discussion as represented, for instance by Margenau and Murphy [5], will not be repeated here. Of special interest is the case of prolate spheroidal coordinates. In this system each point lies at the intersection of an ellipsoid, a hyperboloid and and a cylinder, such that... 

The other four integrals are also equal to one another, and this is a function of the distance, R, between the two atoms called the overlap integral, S R). The overlap integral is an elementary integral in the appropriate coordinate system, confocal ellipsoidal-hyperboloidal coordinates [27]. In terms of the function of Eq. (2.12) it has the form... 

While a proper aiming of the atom-probe can be experimentally determined, information on field lines and on equipotential lines is difficult to derive with an experimental method because of the small size of the tip. Yet this information is needed for interpreting quantitatively many experiments in field emission and in field ion emission. We describe here a highly idealized tip-counter electrode configuration which may be useful for describing field lines at a short distance away from the tip surface but far enough removed from the lattice steps of the surface. The electrode is assumed to consist of a hyperboloidal tip and a planar counter-electrode.30 In the prolate spheroidal coordinates, the boundary surfaces correspond to coordinate surfaces and Laplace s equation is separable, so that the boundary conditions can be easily satisfied. 

Fig. 3.9 Field distribution and equipotential surfaces for a hyperboloidal tip with a planar counter-electrode. The tip radius is 420 A, and the tip to counterelectrode distance is 1.2 mm. The vertical line represents a position 5r, away from the tip. The field lines are drawn so that their density is proportional to the field...

The value of k depends on the exact geometry of the tip. If the latter were a free sphere, k would be unity. For a hyperboloid of revolution of radius of curvature r Muller (1) uses... 

As for the electron Fermi surfaces, the e, X, p and v branches should be noted for consideration. The e, p and v branches are due to electrons since these branches constitute an electron Fermi surface as explained below. As for the Fermi surface shape, it should be noted that the 1/F2 vs. cos2 8 relation (Figure 6.7) shows that the e and X branches are ellipsoidal and the v branch is hyperboloidal as long as observed frequencies are concerned. With the frequency value and the Fermi surface... 

Stability of Hyperboloidal Shells Veronda, Daniel R. Weingarten, Victor I. 

The equilibrium equations of a hyperboloid of revolution used for cooling towers derived by using membrane theory under an arbitrary static normal load are reduced to a single partial differential equation with constant coefficients. The problem of finding displacements is reduced to a similar type of equation so that the solution for this problem becomes straightforward. 11 refs, cited. 

A dynamic statistical approach is used to predict dynamic stresses in a hyperboloidal cooling tower due to earthquakes. It is shown that the configuration associated with one circumferential wave is the only one which is excitable by earthquake force and that the first mode of such configuration is dominant. An equivalent static load is calculated on this basis. Numerical data presented give coefficients for equivalent static loads, natural frequencies of cooling towers, and static stresses for a seismic load. 21 refs, cited. 

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