Cartesian diagram

The interpretation of biplots is made easier by the construction of axes in it. These axes are used in the same way as in a bivariate Cartesian diagram. Perpendicular projection of the points in the diagrams upon a coordinate axis allows us to determine (or reconstruct) the values in the table. 

Figure 31.4 shows the biplot of the trace elements and wind directions for the case when a = p = 0.5. Since here we have that a + P equals 1, we can reconstruct the values in the columns of the data table X by means of perpendicular projections upon unipolar axes. In Fig. 31.4a we have drawn a unipolar axis through Cl. Perpendicular projection of the four wind directions upon this axis reconstructs the order of the concentrations of Cl at the four wind directions as listed in Table 31.1. Now we have established a way which leads back from the graphic display to the tabulated data. This interpretation of the biplot emphasizes the one-to-one relationship between the data and the plot. Such a relationship is also inherent in the ordinary bivariate (or Cartesian) diagram. 

Proof For an abelian surface this has already been shown in [Beauville (1)]. We briefly repeat the argument let (n) A —> A be the multiplication by n. Then we have the cartesian diagram... 

Whenever it is possible to analyze in a given rock at least two minerals that crystallized at the same initial time t = 0, equation 11.86 can be solved in t. On a Cartesian diagram with coordinates Sr/ Sr and Rb/ Sr, equation 11.86 appears as a straight line ( isochron ) with slope exp(At) — 1 and intercept ( Sr/ Sr)o. As shown in figure 11.14A, all minerals crystallized at the same t from the same initial system of composition ( Sr/ Sr)o rest on the same isochron, whose slope exp(At) — 1 increases progressively with t. 

Proof Again we can assume S= Spec A, S Spf A and put S Spec . Consider the cartesian diagram (with s tS1 over the point seS)... 

Introducing the complex notation enables the impedance relationships to be presented as Argand diagrams in both Cartesian and polar co-ordinates (r,rp). The fomier leads to the Nyquist impedance spectrum, where the real impedance is plotted against the imaginary and the latter to the Bode spectrum, where both the modulus of impedance, r, and the phase angle are plotted as a fiinction of the frequency. In AC impedance tire cell is essentially replaced by a suitable model system in which the properties of the interface and the electrolyte are represented by appropriate electrical analogues and the impedance of the cell is then measured over a wide... 

Thus, solving a problem in particle statics reduces to finding the unknown force or forces such that the resultant force will be zero. To facilitate this process it is useful to draw a diagram showing the particle of interest and all the forces acting upon it. This is called a free-body diagram. Next a coordinate system (usually Cartesian) is superimposed on the free-body diagram, and tbe force.s are decomposed into their... 

In this equation, V2 = d2/dx2 + d2/dy2 + d2/dz2 denotes the Laplacian operator of cartesian second derivatives, p(r) is the charge density in a spherical shell of radius r and infinitesimal thickness dr centered at the particle of interest (see diagram), k is the effective dielectric constant, and e0 is the permittivity of free space (8.854 x 10 12 in SI units). The energy of interaction / , of ions of charge z,c with their surroundings,... 

Figure 8.1 Diagram showing the relation of polar coordinates, r, 0, (j>, to Cartesian coordinates for the point P.

Figure 10.5 Diagrams showing the effect of a threefold rotation on the set of Cartesian displacement vectors.

Figure 13. Cartesian [center-of-mass (CM)] contour diagrams for NH+ produced from reaction of N+ with H2. Numbers indicate relative product intensity corresponding to each contour. Direction of N+ reactant beam is 0° in center-of-mass system. For clarity, beam profiles have been displaced from their true positions (located by dots and 0°). Tip of velocity vector of center of mass with respect to laboratory system is located at origin of coordinate system (+). Scale for production velocities in center-of-mass system is shown at bottom left of each diagram (a) reactant N+ ions formed by impact of 160-eV electrons on N2 two components can be discerned, one approximately symmetric about the center of mass and the other ascribed to N+(IZ3), forward scattered with its maximum intensity near spectator stripping velocity (b) ground-state N+(3/>) reactant ions formed in a microwave discharge in N2. Only one feature is apparent—contours are nearly symmetric about center-of-mass velocity.12 ...

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