Plane triangle

Left) Spherical triangle with sides a, b, cand opposite angles A, B, and C. (Right) Computation of orthodrome between San Francisco (SFO) and Tokyo (TYO), and plane triangle ABD. 

There are only three regular tesselations of the plane triangles, squares, and hexagons. 

Plane Triangle Triangle whose three vertices lie in a plane. 

Plane Trigonometry. A plane triangle is two-dimensional. Its three vertices are connected by line... 

The presented method allows to avoid application of any interface element, the shape of the contact zone is consistent with the used finite element. The method of setting the local coordinate systems is compatible with the one used for shells consisted of plane triangles (linear tetrahedral case) and quadratic triangles for the second order tetrahedrzds. The way of introducing the method for 3D case is compatible with previously developed algorithm for 2D case which has been succesfuliy applied to the solution of bulk material flow problems. The most important feature of the method is the use of skew and conical... 

Fig. 15. - Symbols traditional ly used tor the depiction of the four elements, upper line shows tlie composition of the plane triangles w hile the lower line is made up of the explicit geometrical bodies. On the right, the geometrical model of an aggregated universe is revealed with the outermost sphere belonging to Saturn, and with Mercury and the Sun in die centCT. (The illustration is the imitation drawn after the Keppler s picture in his Mysterium Cosmographictiin , printed in about 17th Century). ...

For a three-axis system, three triangles in mutually perpendicular planes may be used. For navigational purposes, the output of the laser gyroscope may be integrated to determine the heading of an aircraft. Laser-based navigation systems have been in use on commercial and military aircraft since the early 1980s. 

FIG. 2 Phase diagram in the M-z plane for a square lattice (MC) and for a Bethe lattice q = A). Dashed lines Exact results for the Bethe lattice for the transition lines from the gas phase to the crystal phase, from the gas to the demixed phase and from the crystal to the demixed phase full lines asymptotic expansions. Symbols for MC transition points from the gas phase to the crystal phase (circles), from the gas to the demixed phase (triangles) and from the crystal to the demixed phase (squares). (Reprinted with permission from Ref. 190, Fig. 7. 1995, American Physical Society.)... 

N occupy an sp lone-pair in the plane of the ring (or the plane of the local PNP triangle) as in Fig. 12.26a. The situation at P is less clear mainly because of uncertainties concerning the d-orbital energies and the radial extent (size) of these orbitals in the bonding situation (as distinct from the free atom). In so far as symmetry is concerned, the sp lone-pair on each N can be involved in coordinate bonding in the jcy plane... 

Reductions of [PtClg] " in an atmosphere of CO provide a series of clusters, [Pt3(CO)6] ( = 1-6,10) consisting of stacks of Pt3 triangles in slightly twisted columns Pt-Pt = 266 pm in triangles, 303-309 pm between triangular planes (Fig. 27.12). A feature of these and other Pt clusters is that they mostly have electron counts lower than predicted by the usual electron counting rules. In the series just mentioned for instance, = 1 and n — 2 have electron counts of 44 and 86 whereas 48 and 90 would... 

A great deal of evidence has shown that carbocations are planar. The divalent carbon is 5p2-hybridized, and the three substituents are oriented to the corners of an equilateral triangle, as indicated in Figure 6.9. Because there are only six valence electrons on carbon and all six are used in the three a bonds, the p orbital extending above and below the plane is unoccupied. 

These identical orbitals all lie in the same plane and point toward the corners of an equilateral triangle. 

In this case, there are three equivalent hybrid orbitals, each called sp (trigonal hybridization). This method of designating hybrid orbitals is perhaps unfortunate since nonhybrid orbitals are designated by single letters, but it must be kept in mind that each of the three orbitals is called sp. These orbitals are shown in Figure 1.4. The three axes are all in one plane and point to the comers of an equilateral triangle. This accords with the known structure of boron trifluoride (BF3), a planar molecule with angles of 120°. 

Unlike the geometries for other steric numbers, the five positions in a trigonal bipyramld are not all equivalent, as shown in Figure 9-21a. Three positions lie at the comers of an equilateral triangle around the phosphorus atom, separated by 120° bond angles. Atoms in the trigonal plane are In equatorial positions. The other two positions... 

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