Projective plane

Pore size The distance between two adjacent warp or weft threads, measured in the projected plane. Only applies to fabrics above 10 microns. 

Plan, tn, plane, plain plan (Painting) ground. planen, v.t. plan, design, project plane. Planenstoff, tn. awning cloth tarpaulin, planetarisch, a. planetary. 

Cable trays Not less than 0.3 gpm/ft2 of projected plane area (horizontal or vertical). 

Figure 7.36. Projections on characteristic planes of the unit cells of AlB2-derivative structures (binary deformation variants). Open circles represent atoms on the projection plane, dashed circles atoms on other parallel planes. For A1B2 compare with Fig. 7.5 and, for KHg2, with 7.33. (Adapted from Gladyshevskii et al. 1992)...

W. Barth, Moduli of bundles on the projective plane. Invent. Math. 42 (1977), 63-91. 

Fig. 10.2 Surface hydroxyl configuration on the goethite 001, 101, 100 and 210 faces. Distances of O and Fe ions to the projection plane are indicated next to the corresponding row of ions. Rows of singly, doubly, and triply coordinated O ions are indicated as S, D, and T, respectively. Solid line rectangles represent the two-dimensional (surface) unit cell. Dotted-line rectangles show contiguous singly coordinated hydroxyls (Barron and Torrent, 1996, with permission).

Since the fuzzy cylinders act as obstacles when entering the laminar region in Fig. 15a, the probability of their appearance (and disappearance) on the projection plane should be governed essentially by the longitudinal diffusion of the hindering or test fuzzy cylinder, and the lifetime t should obey the relation... 

Fig. 2.38 Structure of j8-BaFe2S4. (a) Projection on (001). Tetrahedral FeS is outlined. An arrow is normal to the projection plane of (b). (b) Projection on (110). FCS4- and Ba-chain subcells along [001] are seen.

If the problem is restricted to two dimensions (for a projection of the structure), calculations are practicable, or, better still, can be avoided altogether by using the optical analogue method (Lipson and Taylor, 1951, 1958) moreover, for a flat molecule the transform need only be obtained for the plane of the molecule. The effect of tilting the molecule with respect to the projection plane so that it is foreshortened in a particular direction is simply to lengthen the transform in that direction. 

In the ac projection, the origin and axes are again chosen to conform with a right-handed axis set. The screw axes are now represented by the symbol that shows them to be perpendicular to the projection plane. The glide planes are now shown by the symbol at the upper left, which gives both the elevation (y = i and, by implication, 5) and glide direction (c). 

For tori, we take their universal covers on the plane and use the primal-dual representation obtained from the program TorusDraw ([Dut04b]). For the projective plane F2, we take its universal cover, which is the sphere, and draw a circular frame,... 

The projective plane arises as a quotient space of the sphere, the required group being C,-. It is obtained by identifying antipodal points of the spherical surface in other words, it is the antipodal quotient of the sphere (see Section 1.2.2). P2 is the simplest compact non-orientable surface in the sense that it can be obtained from the sphere by adding just one cross-cap. 

Prisntm, m > 2 (on S2, with two m-gons seen as holes) and their non-orientable quotients, for m> 2 even (on projective plane, with one hole),... 

Stereographic projection provides a convenient way of displaying the angular relations between planes and directions in a crystal in two dimensions. The system involves first projecting planes and directions of interest onto a spherical surface and then mapping the spherical surface. Figure 4.1 illustrates how planes and directions are projected onto a sphere. If an infinitesimal crystal were placed at the center of a sphere and its planes extended, they would intersect the sphere as great circles and their directions would intersect the sphere as points. 

Mapping of planes and directions by placing an infinitesimal crystal at the center of a sphere and projecting planes onto the sphere to form great circles and lines to form points. 

Take the sphere and attach q crosscaps to form the non-orientable surface Nq which has the Euler characteristic x(Nq) = 2 — q. For example the projective plane is homeomorphic to the sphere with one crosscap and has... 

Fig. 2. The embedding of K6 in the projective plane. The Euler equation reads V — E + F = 6 — 15 + 10 = 1. The surface is obtained by gluing together opposite edges so that arrows match.

In this paper, we have introduced the polyhedral representation of reaction surfaces for chemical interconversion processes, and applied it to the interconversion of JT distortions of icosahedral molecules. In this case, the minimal hypersurface is 5D. Two types of distortions are investigated pentagonal and trigonal. Interconversions between pentagonal distortions can simply be represented by a triangulation of the projective plane. This is the prototype of a JT surface in a... 

Figure 5.5.11-1 Absorption characteristics of a complete disk showing the variation in absorption properties as a function of spatial angle and absorbing species. Top 3-D isometric view. Bottom 2-D projection, plane contains vertical axis perpendicular to disk surface. The shared quantum-mechanical structure of the liquid crystalline chromophore(s) creates a highly focused (anisotropic) absorption profile. This structure is in quantum-mechanical contact with the microtubules surrounding the disk. The retinoids within the opsin proteins are not in quantum-mechanical contact with each other or the microtubules.

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