Paraboloid

Since 1963 spectra of many molecules have been detected, mainly in emission but some in absorption. Telescopes have been constructed with more accurately engineered paraboloids in order to extend observations into the microwave and millimetre wave regions. 

For this reason these vibrations influence tunneling in an entirely different way. For a model in which the reactant and product valleys are represented by paraboloids with frequencies coq and transition probability has been found to be [Benderskii et al. 1991a, b]... 

Here we shall describe how the periodic-orbit theory of section 3.4, relating the energy levels with the poles of the spectral function g E), can be extended to two dimensions. For simplicity we shall exemplify this extension by the simplest model in which the total PES is constructed of two paraboloids crossing at some dividing line. Each paraboloid is characterized by two eigenfrequen-cies, o + and [Pg.72]

The first observation is that the cured shape of an unsymmetric cross-ply laminate is often cylindrical, whereas we would predict it to be a saddle shape (hyperbolic paraboloid) from classical lamination theory (the curvatures can be shown to be = - Ky or - = Ky). A thick lami-... 

Figure 5-25. Sections through a hyperbolic paraboloid energy surface constructed over an RIP diagram. The intrinsic barrier AG of the main reaction and the intrinsic well AC of the disparity reaction are shown.

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. 

M = area of a plane section midway between the bases Paraboloid of revolution (Figure 1-25)... 

Figure 1-48. Elliptic paraboloid. Figure 1-49. Hyperbolic paraboloid.

For any constant pressure P, equation 2.78 is the equation of a parabola, and therefore all surfaces of constant pressure are paraboloids of revolution. The free surface of the liquid is everywhere at the pressure P0 of the surrounding atmosphere and therefore is itself a paraboloid of revolution. Putting P = Pa in equation 2.78 for the free surface... 

Stressed Mirror Polishing Fabrication of an Off-Axis Section of a Paraboloid. Nelson, J., Gabor, G., Hunt, L., Lubliner, J., Mast, T., 1980, Appl. Optics 19, 2341... 

Following on the above, we consider the departure of off-axis paraboloidal segments from a sphere of the same vertex radius. Figure 1 shows the geometry of an arbitrary segment, taken to be circular for ease of argument. 

Asag for the S5mimetric paraboloid is known from Eq. 2. To find Agag over the off-axis aperture a coordinate transformation is performed to shift the center of the coordinate system to the center of the off-axis segment and to scale the off-axis aperture radius to a. Then we have... 

The term is piston and is included only for completeness since it has no effect on the measurement. The b term is the tilt of the off-axis paraboloidal... 

The geometric interpretation for the preceding problem requires visualizing the objective function as the surface of a paraboloid in three-dimensional space, as shown in Figure 8.1. The projection of the intersection of the paraboloid and the plane representing the constraint onto the/(x2) = x2 plane is a parabola. We then find the minimum of the resulting parabola. The elimination procedure described earlier is tantamount to projecting the intersection locus onto the x2 axis. The intersection locus could also be projected onto the xx axis (by elimination of x2). Would you obtain the same result for x as before ... 

Equations 9.1 and 9.2 correspond with equations 2.80 and 2.79 in Volume 1, Chapter 2. Taking the base of the bowl as the origin for the measurement of zo, positive values of za correspond to conditions where the whole of the bottom of the bowl is covered by liquid. Negative values of za imply that the paraboloid of revolution describing the free surface would cut the axis of rotation below the bottom, and therefore the central portion of the bowl will be dry. 

The reconstruction of the exit wave occurs in four steps. It starts with an alignment step where the images of the focal-series are aligned with respect to each other. In the second step, an analytical inversion of the linear imaging problem is achieved by using the paraboloid method (PAM) to generate a first approximation to the exit wave function. This approximated exit wave function is then refined in the third step by a maximum likelihood (MAE) approach that accounts for the non-linear image contributions. Finally, the exit wave is corrected for residual aberrations of the microscope. 

Figure 7. The linear image contributions of a focal-series are located on the surface of two paraboloids obtained by 3D Fourier transformation of the focal series. The two paraboloids correspond to the electron wave function and its complex conjugate.

The result of the PAM reconstruction is, in general, only an approximation of the exit wave function. Some non-linear terms may be present exactly on the paraboloid surfaces, and, thus result in artifacts for the PAM reconstruction. However, the PAM result is a good approximation to the exit wave function, which, in the present implementation, is used as a starting point for a maximum likelihood (MAL) reconstruction that takes the non-linear image contributions fully into account (Coene et al. 1996, Thust etal. 1996a). 

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