Polar coordinates, projection

Figure 9.2. Reflection 4m (s) smeared by misorientation of the ensemble of crystallites (structural entities). The center of gravity is found at s/jd- Definitions of polar coordinates Dashed lines indicate the radial direction of the integration that leads to a projection onto the orientation sphere resulting in the pole figure g kl ([Pg.208]

Spherical polar coordinates. The spherical polar coordinates r, 9,

radius vector r from the origin to point (x,y,z) 0 is the angle between r and the positive z axis r is the distance of (x,y,z) from the origin

relation between spherical polar and Cartesian coordinates is... 

The wave functions for the hydrogen atom are known exactly. They are functions of the three spatial coordinates of the electron and take their most simple form when we choose these coordinates to be the polar coordinates shown in Figure 8.1 in relation to a set of Cartesian axes. The point at jt y, and z in Cartesian coordinates is fixed by r, the radial distance, OP, from the origin of the coordinate system (always considered positive) 0, the angle between the z axis and the line OP and the angle between the x axis and the projection of OP on the xy plane. 

From a set of projections acquired under different angles (p in polar coordinates, a spin density map is reconstructed in back-projection imaging (cf. Section 6.1). For a 2D spin density Mq(jc, y) the projection onto an axis r which is at an angle (p with respect to the j -axis follows by integration over the space variable s orthogonal to r (Fig. 5.4.1),... 

If the magnetic-field gradient is applied in direction r in polar coordinates, which forms an angle

Cartesian reference frame, the projection P (r, spin density Mo(r, 5) along direction s, which is perpendicular to r (cf. Fig. 5.4.1),... 

Let us begin with the scheme for measuring differential photodissociation cross sections using a beam of unpolarized radiation propagating in the +Z direction as defined by Fig. 7.8. Photofragments are detected along k which has spherical polar coordinates 8, is the angle relative to X made by the projection of k into the XOY plane). The differential photodissociation cross section dcr/d l can be written (Beswick, 1993)... 

We now present several figures which illuminate these relations. The first figure is a projection of the magnetization in the x -y plane as a function of the offset for a 90 degree transmitter pulse at resonance. In terms of the equations in this section, we have plotted M and as a function of x with < >o constrained to tl/2 and Mq chosen to be 1.0 although an entirely equivalent interpretation is to consider the plot as a polar coordinate plot of and Cl as a function of x. The dots on the curve represent x in increments of 0.25. The... 

Figure 1.1 defines the spherical polar coordinate system. The radius vector, r, is the distance of any point from the origin 0 is the polar angle of the radius vector with respect to the z-axis and 4> is the azimuthal angle, the angle of rotation of the projection of the radius vector onto the xy plane, with respect to the x-axis. The Cartesian and polar coordinate systems relate to each other in the identities. 

Radial sampling was the first sparse sampling scheme introduced into NMR. Apart from FT [44, 55, 56], other data processing techniques were proposed. These include reduced dimensionality [57], projection-reconstruction [41], and multiway decomposition [58]. Radial sampling scheme consists of points placed on a set of lines in the time domain (see Fig. 6c, d). The coordinates of the th point, lying on the /th line, can be described in a polar coordinate system as... 

Figure 1. Transformations defining D-dimensional hyperspherical polar coordinates in terms of Cartesian coordinates, illustrated for D = 2,3,4,... On going from the D to D + 1 case, a further Cartesian axis xd+i is added the radius vector r is then projected on this axis via the cosine of the new polar angle 0d+i and projected on the D-dimensional subspace via the sine of that angle.

A formal description of the filtered back-projection process is best served by expressing the object function /(jc, y) defined in Eq. (26.64), in an alternative coordinate system. Here, die rectangular coordinate system in the frequency domain (u, v) is exchanged for the polar coordinate system iyv, 0). so that... 

This is called the projected surface free energy, because it is the surface free energy projected along the surface normal direction, z. One of the most important meanings of the projected surface free energy is that it is equivalent to the crystal shape through the Legendre transformation. The conversion of a polar coordinates (fr 9) into a... 

We resume our discussion with a molecular placed in a laser fields as in Eqs. (8.40) and (8.41). Suppose we have j (r, f R), as discussed in Sec. 7.1. Let Rc be the position of the molecular centroid. Make a sphere of a radius rs, whose center is located at Rc. A vector normal to the sphere is denoted as tb, and the polar coordinates are to be used to specify them as = What we really need is the flux that is normal to the sphere, which is readily projected as... 

For other coordinate systems, a factor analogous to the factor r sin(0) must be used. This factor is called a Jacobian. For example, for cylindrical polar coordinates, where the coordinates are z, (p (the same angle as in spherical polar coordinates, and p (the projection of r into the xy plane), the Jacobian is the factor p, so that the element of volume is pdpdzdp. We use the symbol d r for the three-dimensional volume element in any coordinate system, so that dxdydz, r sin 6)d[Pg.1241]

It is convenient to distinguish between rays which cross the fiber axis between reflections known as meridional rays-and rays which never cross the fiber axis-known as skew rays. We see from Fig. 2-2(a) that meridional rays lie in a plane of width 2p through the axis. Consequently, they have properties identical with rays of the corresponding planar waveguide, and Table 1-1, page 19, applies to meridional rays of fibers, if the cartesian coordinate x is replaced by the cylindrical polar coordinate r of Fig. 2-1. Skew rays, on the other hand, follow a helical path, whose projection onto the cross-section is a regular polygon-not necessarily closed-as shown in Fig. 2-2(b). The midpoints between successive reflections all touch a cylindrical surface of radius rj, known as the inner caustic. 

Fig. 4-4 Reflection and refraction of a source ray at Q on the fiber endface. Polar coordinates (r, ) define the position of Q, and the projection of the ray path onto the endface makes angle 0 with the azimuthal direction at Q.

In order to determine the strength of the dipole field at point A, the relationship between strength and potential can be used. We use a polar coordinate r and 0 with a polar axis coinciding with the dipole arm direction (Figure 4.15). Component (projection E on the radius vector r) is... 

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