Triangle Spherical

Kugel-dioriti m. globular diorite. -dreieck, n. Geom.) spherical triangle, -druckprobe, /. ball pressure test (for hardness), Brinel)... 

Figure 9.5. 3D geometrical relations in the scattering pattern for the case of double fiber symmetry (F2). Dash-dotted are both the axis of the observed pattern, I, and one of its reflection circles. Drawn in solid line are both the axis of a tilted representative structural entity and a reflection circle of its fiber-symmetrical intensity, Iopt. Important for the simplification are the relations in the spherical triangle plotted in bold... 

In Fig. 9.5 two sides of the spherical triangle merge (

meridional reflection at constant s directly reflects the shape of the orientation distribution. 

Figure 12.7. E1 E2, and E3 mark out a spherical triangle which is the surface of one octant of the unit sphere. This figure shows that E3 + E2 + Ei = T0.

Figure 1.4 Equilateral division of a triangle. From left to right, divisions of order N = 2,3, 6. Each side of the original triangle is divided in N equal parts (in the case of spherical triangles the sides are circumference equatorial arcs). A segment (or an arc) is traced from each division point to the corresponding point on another side, so that the final result is a division of the original triangle in N2 triangles.

J. W. Harris and H. Stocker, General spherical triangle, 4.9.1 in Handbook of Mathematics and Computational Science, Springer-Verlag, New York 1998, pp 108-109. 

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. 

Note that one, two, or three right angles may coexist in the same right spherical triangle ... 

The celestial sphere, with local zenith Z, and a point B (a star). Note the spherical triangle BZPN. 

Fig. 1 The spherical triangle QAP formed by the direction OP of the magnetic field, the axis of sample rotation OQ and the interproton line OA. The proton pair is marked by two heavy dots (points O and A). By sample rotation the position A of one of the protons moves along the dashed circle, cp is the running angle of rotation...

This procedure is based on the use of unique sphere, with its tesserae defined in terms of a number of spherical triangles (from 60 to 940) derived by partition of the pentakisdodecahedron faces. The sphere is deformed to mimick the molecular shape, taking into account solvent excluded volume. The number of tesserae is automatically increased only in the portions of the surfaces where it is needed, keeping the shape of a curvilinear triangle. The gain in computational times reaches a 100 factor for the largest molecules. 

By solving the spherical triangle IPN, we can find the following general relation between the angles 0, and d ... 

Denoting the azimuth of the baseline by a and its arc length (approximately the Euclidean length) by s this relationsMp can be transformed. Considering the rectangular spherical triangle (on the unit sphere) shown in Fig. 2, we obtain the relations... 

To reach the abovementioned objective, we can easily determine from Figure 8.14(b) that the following two requirements must be satisfied (1) the arc TE should equal to the arc TA, and (2) the arc TC should equal to the arc EC. Besides, from Figure 8.14(b) we can also obtain Z.POB = zAOB = q>,TA=jt-2(p, and TC = cp. Based on spherical trigonometry, we can find the following relationships from the spherical triangles CTE and TEA ... 

Spherical Triangle Triangle whose three vertices are located on the surface of a sphere. 

---