X=Y=Z systems

The a, b, e, eoordinate system ean be formed by beginning with the original X, Y, Z system and sequentially ... 

Our convention is that a symmetry operation R changes the locations of points in space, while the coordinate axes remain fixed. In contrast in Section 1.2 we considered a change (proper or improper rotation) of coordinate axes, while, the points in space remained fixed Let x y z be a set of axes derived from the xyz axes by a proper or improper rotation. Consider a point fixed in space. We found that its coordinates in the x y z system are related to its coordinates in the xyz system by (1.120) or (2.29) ... 

Fig. 9. Geometrical arrangement of Trp-29 in erabutoxin b. Coordinate system of a spherical protein and tryptop-hanyl residue are shown by (x y z ) and (xyz), respectively. The origin of (x y z ) system is chosen at the CH2 group connecting the peptide chain and indole ring. Internal rotation of the tryptophan from the (x y z ) system to the (xyz) system is expressed with Euler s angles (a6y). The location of the quencher, -NHg of Lys-27, is also illustrated in the Fig. Polar coordinates of the N atom of the quencher in the system (x y z ) are indicated by a and 3q (33). ...

The general character of the motion can be summarised as follows The components of D are the co-ordinates (relative to the axes x, y, z) fixed in the top) of the point in which the invariable axis of the system (i.e. the axis of resultant angular momentum, which is fixed in space) penetrates the sphere (7). This point traverses the curve of intersection of the sphere with the ellipsoid (6), which is rigidly connected to the top. In the fixed co-ordinate system, therefore, the x, y, z system of axes, fixed in the top, executes a periodic nutation superposed on a precession about the axis of resultant angular momentum. In the case where the sphere touches the ellipsoid the motion becomes a rotation about a permanent axis. 

Cartesian. Robots belonging to this group can only perform translational movements, which are defined in a three-dimensional coordinate (x,y,z) system. 

Now, consider a three-dimensional system. We wish to describe a fixed vector A in the (,x, y, z) system, which will be thrice rotated to a (x, y, z) system. For ease of notation, we will label the (x, y, z) axes as (1, 2, 3) and the x,y,z) axes as (1,2,3). AU sets of axes will be assumed to represent orthogonal coordinate systems. 

The Maier-Saupe theory assmnes high symmetry for molecules forming liquid crystals. In reahty, this is usually not the case and the theory has been extended [3.18] to lath-like molecules. The order parameter tensor S is given by Eq. (3.8) for a biaxial molecule in a uniaxial phase. In the principal axis x y z) system of 5, only two order parameters, Szz and D = Sxx — Syy, are needed, which are related to the Wigner matrices according to Eq. (2.43) ... 

In the continuum theory of liquid crystals, the free energy density (per unit volume) is derived for infinitesimal elastic deformations of the continuum and characterized by changes in the director. To do this we introduce a local right-handed coordinate x, y, z) system with (z) at the origin parallel to the unit vector n (r) and x and y at right angles to each other in a plane perpendicular to z. We may then expand n (r) in a Taylor series in powers of x, y, z, such that the infinitesimal change in n (r) varies only slowly with position. In which case... 

As it is observed from Figure 4, the highest point on the suction cups (the point with maximum y coordinate in x y z system) has the most and earliest tendency to leave the surface, and the lowest point (the point with the least y coordinate in x y z system) has the least. Therefore, it is reasonable to assume that the reaction force acting on the suction cups has linear distribution along y axis shown in Figure 4. Based on this assumption, the reaction force density (f ) on every point of the suction cups can be expressed as a linear function of its y coordinate in x y z coordinate system, and the amount of the reaction force density at extreme points A, and B or ... 

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