Approximation tangent plane

Figure 13.1 Reciprocal space (2D) representation of the diffraction condition Ewald sphere (radius 1/2), limiting sphere (radius 2/2) and PD sphere (double line, radius d ). Left Enlargement of the intersection between PD sphere and reciprocal space point, with approximating tangent plane (dash). The arrow shows the direction of expansion of the diffraction sphere during a PD measurement.

The exact values of E and 5E / 5n are in general unknown and the Kirchhoff or physical optics method consists in approximating the values of these two quantities on the surface and then evaluating the Helmholtz integral. We shall approximate the field at any point of the surface by the field that would be present on a tangent plane at the point. With this approximation, the field on the surface and its normal derivative are... 

The tangent plane approximation is valid the curvature of the Ewald sphere is negligible at small scattering angles. 

WAXS and MAXS. Fiber symmetry means that, even in WAXS and MAXS, the scattering pattern is completely described by a slice in reciprocal space that contains the fiber axis. Nevertheless, for 20 > 9° the tangent plane approximation is no longer valid and the detector plane is mapped on a spherical surface in reciprocal space. 

If, in the detector plane, the effective slit is wider than the region of the pattern in which significant intensity is observed, the approximation of an infinite slit is valid. Let the slit be infinitively long in ft -direction but very narrow in 53-direction then in the tangent plane approximation the recorded scattering curve... 

For USAXS and SAXS data the tangent-plane approximation is valid and the relation between scattering angle and the units of reciprocal space are given by Eq. (2.7). If the scattering pattern is properly aligned with the vertical direction identical to a fiber axis or the polymer chain direction, then sy = 53. In similar manner the. vx-axis of the detector is related to the actual orientation of the sample with respect to the beam. 

If the tangent planes in M are more or less parallel then P, and P, are approximately equally affected by a variation in the mixture composition and the correlation of the response surfaces in M is high and the variance in the selectivity is small. If the tangent planes in M have opposite slopes then P, and Pj are affected in an opposite way by a variation in the mixture composition and the correlation of the response surfaces in Mwill be low and the variance in the selectivity high. The calculation of the correlation r (i.e. the parallelism of the response surfaces) is outlined below. 

When the diffusion time is short enough, the translation on the spherical surface is approximately identical with that on the tangent plane to the spherical surface. The latter is the two-dimensional diffusion process treated by the Green function method in Appendix C, and we can use Eq. (C17) again. Since the rotational diffusion coefficient Dr is related to the translational diffusion coefficient D<2) in Eq. (Cl7) by Dr = D(2)/(Le/2)2, we have... 

If the above rule is applied to the entire sequence of envelope surfaces, then one obtains a collection of at most (n x m x N) line segments generating a set of piecewise linear broken lines of approximately uniform distribution and approximately perpendicular to the tangent plane of each local spherical domain where the lines cross the envelope surfaces. Some of these lines may merge and possibly separate again as some of the selected points get buried and possibly reappear in the enlargement process of the spheres. 

The tangent plane approximation is valid for sufficiently small points with equiaxial shapes. It cannot be used, for example, when faulting is present and some reciprocal space points are rods extended along the stacking direction (Section 13.2.3). In this case, integration over a spherical (or at least cylindrical) cross section is necessary. ... 

One alternative apparently uncorrelated to the previous ones is to approximate the function using a tangent plane and use a method valid for linear functions with... 

Consider a particular point on such a surface and its tangent plane. In a vicinity of the point, the surface can be approximated by a paraboloid. For a convenient description, use local Cartesian coordinates tangent plane. The paraboloid is the second-order expansion of the surface function C = )- The first derivatives vanish by... 

Tangent-plane approximation, 12 Tensile testing, 23 Tie-molecule, 3 TPU, 24, 100... 

A completely different case is presented by solid elastic interfaces (e.g., rubber membranes). Another interesting case is that of a rod-shaped micelle, which is usually considered to have at least approximately a cylindrical shape. Presumably, the entanglement of the hydrocarbon chains endows the interface (which comprises practically all of a micellar aggregate) with sufficient rigidity to support anisotropic tensions in the tangent plane of the dividing surface so that the condition = 2[Pg.587]

The curvature at a given point on a surface is characterized by the maximum and minimum radii, R and R2, of the circles in mutually perpendicular planes perpendicular to the plane tangent to the surface at that point, best approximating the curves formed by the intersection of the surface with these planes [221], (Fig. 1). The principal curvatures, k and k2 are defined as... 

What happens if the meadow is not planar At a given point, we can approximate a nonplanar meadow with the plane that is tangent to the meadow at... 

There are several known models of plain knitted loop geometry as follows Peirce s model (1947) is a purely geometrical model that approximates the loop shape on its projection to the fabric plane by two circular arcs of radius r and a straight line tangent to both arcs ... 

If discs are employed, and one places them at a distance equal to approximately a third of their diameter, and one withdraws sufficient liquid, the angle between the end tangents of the surface of the mass and the plane of each disc decreases until being cancelled completely, so that this surface is then tangent to the disc planes (fig. 36), and that thus the end tangents of the meridian curve are parallel to the axis of symmetry. It is extremely difficult to consider... 

There was formed, at all pressures, at the point of meeting of the two jets, i.e. in the middle of the interval which separated the openings, a circular plane sheet, thicker in its central part than at its contour, whose plane was normal to the tangent of the two jets, and which ended in an annular zone, which was disturbed, agitated and noisy, when the pressure exceeded 120 centimefres, but which became perfectly smooth and flat in all its extent when the pressure dropped below this point. For the strongest pressure, which was 488 centimefres, the diameter of this sheet was initially 24 to 25 centimefres then it increased little by little as the pressure became lower, and when it was only from 115 to 120 centimetres, it was then approximately 38 centimefres then it decreased again. ... 

---