Curvilinear trajectories

Fig. 31.17. (a) In a classical PCA biplot, data values xy can be estimated by means of perpendicular projection of the ith row-point upon a unipolar axis which represents theyth column-item of the data table X. In this case the axis is a straight line through the origin (represented by a small cross), (b) In a non-linear PCA biplot, the jth column-item traces out a curvilinear trajectory. The data value is now estimated by defining the shortest distance between the ith row point and theyth trajectory. 

The same idea can be developed in the case of a non-Euclidean metric such as the city-block metric or L,-norm (Section 31.6.1). Here we find that the trajectories, traced out by the variable coefficient kj are curvilinear, rather than linear. Markers between equidistant values on the original scales of the columns of X are usually not equidistant on the corresponding curvilinear trajectories of the nonlinear biplot (Fig. 31.17b). Although the curvilinear trajectories intersect at the origin of space, the latter does not necessarily coincide with the centroid of the row-points of X. We briefly describe here the basic steps of the algorithm and we refer to the original work of Gower [53,54] for a formal proof. 

Figure 10. Modified Tera-Wasseibuig f° Pb/ ° Pb-2 U/2 Pb) plot with disequilibrium concordia (after Wendt and Carl 1985). The curvilinear trajectories (dotted lines), or concordia, show the change in 23 U/ 06p ) 207p y206p ggg incrcascs for different values of 8234u(0) and Pao = orh = 226Ra = q.

One can see from these expressions, that for identical time of driving (for example, for half of period) the paths of the particle on different trajectories are different. It is possible to tell, that the replacement of driving on curvilinear trajectories on uniform rectilinear driving always will require introducing some correction factor. 

Figure 8. Schematic presentation of rectilinear and curvilinear trajectories in particle collisions by differential sedimentation. (Adapted with permission from reference 24. Copyright 1991.)...

Figure 9. The result of KFpose estimative applied at irregular curvilinear trajectory...

At curvilinear movement the velocity vector is the product o = or, where T is a tangent ort. Because of the fact that the point is moving along a curvilinear trajectory and draws the unit vector t behind, its position is also dependent on time. In this case ... 

When air pollutants exit the smoke stack or exhaust pipe (called the sources), they are advected by winds and dispersed by turbulent diffusion. Winds blow from high pressure toward low pressure cells at speeds that depend on the pressure gradient. Because of the Coriolis force, wind trajectories are curvilinear in reference to fixed Earth coordinates, although within a relatively short (few to tens of km) distance, wind trajectories can be approximated as linear. Winds have a horizontal and vertical component. Over flat terrain the horizontal component predominates in mountainous and urban areas with tall buildings, the vertical component can be significant, as well at the land/sea interface. 

Equation 6.1 represents a system of three differential equations for the coordinates x, y, and z (or for some curvilinear coordinates qlt q2> q3) expressed as functions of time t. Solution of these equations defines a trajectory of the particle for certain initial conditions of position and velocity. Several examples will be examined. 

Interaction in the form of a perpendicular interlacing of two yarn systems confirms the presence of woven structures. No selvage was observed thus the warp system could not be distinguished from the weft system. Both curvilinear and zig-zag trajectories of certain yarns with respect to the interlacing systems were recorded. These may or may not indicate surface decoration in the form of embroidery. 

Hydrodynamic Models. The coagulation kernels are usually calculated for solid spheres with hydrodynamic models of different sophistication. The simplest calculation uses fluid flow in the absence of any effect of either particle on the flow. This flow level is known as rectilinear flow. The next level of sophistication involves calculating the flow around one particle, usually the larger of the two interacting particles. This level of calculation is known as curvilinear flow. Further levels of sophistication can be obtained by considering the particle trajectories as affected by the interacting flow fields of the particles, as well as any attractive or repulsive forces between them. 

Figure 3 illustrates several of the structural innovations that emerged from the discoveries made at LLNL in the first half of 2002. It remains to be seen which of these innovations are best and/or affordably testable in the near term, but all are likely to improve on conventional tanks. Several important holes in current theory and analyses emerged during the recent discovery process. These holes are big enough that several PhD theses may not suffice to fill each one. Fiber number flex conservation, departures from constant (assumed) fiber/matrix ratio, and actual curvilinear (non-axisymmetric, nongeodesic) fiber trajectories will contribute arduous but necessary improvements to current composite design methods. 

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