Hyperboloid, equations

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. 

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... 

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. 

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. 

The curve whose equation in polar coordinates is r theta=a. hyperboloid... 

This is the title of Chapter 3 in Ref. [9], Advances in Quantum Chemistry, Vol. 57, dedicated to confined quantum systems. The conoidal boundaries involve spheres, circular cones, dihedral angles, confocal paraboloids, con-focal prolate spheroids, and confocal hyperboloids as natural boundaries of confinement for the hydrogen atom. In fact, such boundaries are associated with the respective coordinates in which the Schrodinger equation is separable and the boundary conditions for confinement are easily implemented. While spheres and spheroids model the confinement in finite volumes, the other surfaces correspond to the confinement in semi-infite spaces. 

The ionization limit of the Schrodinger equation and its eigenfunctions for the free hydrogen atom, at a vanishing energy value, corresponds to Bessel functions in the radial coordinate as known in the literature and illustrated in 2.1. The counterparts for paraboloidal [21], hyperboloidal [9], and polar angle [22] coordinates have also been shown to involve Bessel functions. These limits and their counterparts for the other coordinates are reviewed successively in this section. 

Indeed, the quadratic term in / representing the energy vanishes, the separation constant A becomes infinitely large, the confining spheroids become very eccentric, so that u becomes almost one. This suggests using the transformation complementary to the one introduced in equation (11) of [9] for the hyperboloidal boundary ... 

By recalling that Equations (81) and (82) are the same, but in different domains, it should not be surprising that the change of variable in Equation (120) transforms Equation (82) into Equation (114). Consequently, the boundary condition on the corresponding ordinary Bessel function leads to the positions of the hyperboloids for which the energy is zero ... 

The free electron confined by a hyperboloid has the same type of wave function described in the paragraph of Equation (116). The difference consists in the boundary condition applied to the angular spheroidal functions, which are expressed as infinite series of associated Legendre polynomials of order m + r, degree m, and argument v ... 

The family of confocal ellipsoids and hyberboloids represented by the prolate spheroidal coordinates allows us now to treat the case of a many-electron atom spatially limited by an open surface in half-space. A special case of the family of hyperboloids corresponds to an infinite plane defined by jj = 0 according to Equations (35) and (36). We now treat the specific case of an atom whose nuclear position is located at the focus a distance D from the plane as shown in Figure 4. 

It has the dimension of nr2 and is positive for spheres, negative for one sheet hyperboloids and zero for planes. The Gaussian curvature, K, and mean curvature, H, are related to each other by a quadratic equation [ K2 - 2Hk+ K= 0], which has solutions of kx=H- H2 -K and Ki=H- H2 —K. We may write the mean curvature in terms of the Gaussian curvature... 

The range of the variable is from 1 to oo, and of ij from — 1 to +1. The surfaces = constant are confocal ellipsoids of revolution, with the nuclei at the foci, and the surfaces ij = constant are confocal hyperboloids. The parameters m, X, and n must assume characteristic values in order that the equations possess acceptable solutions. The familiar

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The hyperboloid curve implied in Equation (7.5) is shown in Figure 7.3 together with the experimental data. 

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... 

Prominent surface of draw plate 2 is profiled in its central part as an ellipsoid 7 passing to its periphery as a gently sloping curve on the form of one cavity hyperboloid satisfying the following equation ... 

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