Projection, stereographic

Vertical planes related by a /(-fold rotation axis. 

Stereographic projection of crystal class (point group) 4. 

All thirty-two crystal classes in stereographic projection are tabulated in Fig. 9.1.8. For each crystal class, two diagrams showing the equipoints and symmetry elements are displayed side by side. Note that the thicker lines indicate mirror planes, and the meaning of the graphic symbols of the symmetry elements are given in Table 9.3.2. 

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In the remainder of this section, we focus on the two lowest doublet states of Li3. Figures 3 and 4 show relaxed triangular plots [68] of the lower and upper sheets of the 03 DMBE III [69,70] potential energy surface using hyper-spherical coordinates. Each plot corresponds to a stereographic projection of the... 

A final example is the concept of QM state. It is often stated that the wave function must be square integrable because the modulus square of the wave function is a probability distribution. States in QM are rays in Hilbert space, which are equivalence classes of wave functions. The equivalence relation between two wave functions is that one wave function is equal to the other multiplied by a complex number. The space of QM states is then a projective space, which by an infinite stereographic projection is isomorphic to a sphere in Hilbert space with any radius, conventionally chosen as one. Hence states can be identified with normalized wave functions as representatives from each equivalence class. This fact is important for the probability interpretation, but it is not a consequence of the probability interpretation. 

Fig 1. Stereographic projection of the crystal structure of the 2 1 inclusion compound between CS2 and cavitand 1. One CS2 ( guest ) molecule is encapsulated within the host cavity, the second CS2 ( solvent ) being located between the complexed entities (taken from Ref.27>)... 

Figure 5. Stereographic projection of data for figure 4a. The point x=° is at the center of the figure. The meridians...

A theoretical analysis is presented for the binding of the four dia-stereoisomers of benzo[a]pyrene diol epoxides (BPDEs) to N2(g), N6(a), 06(G) and NU(c). Molecular models for binding and stereoselectivity involving intercalation, intercalative covalently and externally bound forms are presented. Molecular mechanics calculations provide the energetics which suggest possible structures for the formation of each of the principal DNA-BPDE complexes. Stereographic projections are used to illustrate the molecular structures and steric fits. The results of previous calculations on intercalation and adduct formation of BPDE l(+) in kinked DNA (37) are summarized and extended to include the four diastereoisomers l( ) and II( ). The theoretical model is consistent with the observed experimental data. 

The N2(G) adducts are more stable than the N6(a) and 06(G) and NU(c) adducts. Because cis addition products are present, minor amounts of the other adducts are found. If only cis addition occurred, then the i(-) and Il(-) isomers would be stereoselected by N2(G), and the l(+) and Il(+) isomers would be stereoselected by N6(A) and 06(G). Although we did not perform calculations on the cis adducts, it can be seen from the stereographic projections that the change accompanied by a reflection of only the BPDE atoms through the plane of the pyrene changes the (+) into (-) isomers. Thus, the rules of stereoselectivity are reversed. However, the small amount of cis adduct yields these minor components the l(-)-N2(G) adduct is most prevalent (38) for reactions of BPDE i(-) with DNA and we assume that this arises from the cis addition. If both trans and cis addition occurred equally, we predict that stereoselectivity would not be observed. 

Fig. 1 Stereographic projection of brucite. Reprinted with permission from L. Eriksson, U. Palmqvist, Studsvik Neutron Research Laboratory Report 482. Copyright University of Stockholm...

This transformation is analogous to a stereographic projection from a hyperplane onto a hypersphere in a four dimensional space ... 

Problem 4-4. Construct stereographic projection diagrams for the symmetry groups of HOCl, cyclopropane, allene, and benzene. 

In drawing stereographic projection diagrams, it is conventional to indicate improper axes, by an open polygon at the center, and a proper axis, C n, by a filled polygon (see Figure 4.1c). 

A.5 Stereographic projections of the S3mmetry involved in the thirty-two crystal classes (point groups)... 

Fig. 3.6 (a) Stereographic projection map of a (001) oriented cubic crystal. The facet sizes reflect approximately those of the field ion image of a fee crystal. 

In his work on the wave equation of the Kepler problem in momentum space (ZS. f. Phys. 74, 216, 1932), E. Hellras has derived a differential equation [Equations (9g) and (10b) in his article] which—after a simple transformation — can be understood as the differential equation of the four-dimensional spherical harmonics in stereographic projection. [With the gracious approval of E. Helleras, we correct the following misprints in his article the number E that appears in the last term of his equations (9f) and (9g) should be multiplied by 4.]... 

More precisely, we can use stereographic projection to find an injective, surjective function from the projective space P(C2) to the sphere 5, via the plane of ratios. Stereographic projection is a function F from the xy-planc in R- into the unit sphere in R. We define... 

See Figure 10.2. Some properties stereographic projection are given in Ex-... 

Because stereographic projection is an injective function, the point [co ci]... 

Exercise 10.6 (Used in Proposition 10.2) Show that stereographic projection (the correspondence between PCC ) and the unit sphere in defined in Section 10.1) is given by... 

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