Congruence transformation

Because of the statement made in the beginning of this section as to the Gaussian nature of the subchains, the matrix for transformation into the normal coordinates is the same for the x- and the -directions. This means that matrix (-4ir) is transformed in the same way as matrix aik), i.e. by a congruence transformation [see eq. (3.12)]. One obtains instead of eq. (3.33) ... 

Fourier transform, the crystallographer moves back and forth between real and reciprocal space to nurse the model into congruence with the data. 

In a next step the result obtained with the help of the first estimate on E/R is optimized by trial and error. The assessment of the degree of congruence achieved can either be performed with optical control or with the help of statistical methods vriiich are used to characterize variability. For this example, the highest degree of congruence of the curves obtained fi om the transformation to equivalent isothermal experiments was reached with a value of E/R = 7300 K (c.f. Figure 4-74). 

It can easily be shown that K,+i has the same mathematical fonn as the spatial inertia of a single rigid body. Thus, the congruence transformation of this matrix also requires (49 multiplications, 49 additions) as described above. The addition of I, requires only an extra 10 additions, if we consido the symmetry and form of Ki and I<. Note that the bottom right submatrix of is simply the diagonal matrix of the composite mass of links t through N. Thus, since this composite... 

It may be shown that this procedure for ccmiputing the spatial composite-rigid-body inertia is exactly equivalent to the procedure used by Walker and Orin in [42]. That is, if the composite mass, composite center of mass, and composite moment of inertia matrix are computed for links i through iV, they may be combined to obtain the spatial matrix K,-. After studying this equivalent approach, howev, it appears that the congruence transformation method given hoe is more efficioit... 

A set of points on the sphere may be transformed by isometries or congruence mappings that preserve the distances between all pairs of points. All isometries, in turn, can be built up from three basic types of transformations. (i) rotations about an axis, (ii) mirror reflections in a plane, and (iii) parallel displacements of all points. If the mappings are restricted to a fixed sphere, parallel displacements play no role, and the congruence transformations reduce to proper isometries or (rigid body)... 

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