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Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).
For example, Todhunter [1] gives two proofs of the cosine rule (Articles 37 and 60) and two proofs of the sine rule (Articles 40 and 42). The page on Spherical law of cosines gives four different proofs of the cosine rule. Text books on geodesy [2] and spherical astronomy [3] give different proofs and the online resources of MathWorld provide ...
The 3-sphere is the boundary of a -ball in four-dimensional space. The -sphere is the boundary of an -ball. Given a Cartesian coordinate system, the unit -sphere of radius can be defined as:
For fixed θ they describe circles on the 2-sphere which are perpendicular to the z-axis and these circles may be viewed as trajectories of a point on the sphere. A point {θ 0, φ 0} on the sphere, under a rotation about the z-axis, will follow a trajectory {θ 0, φ 0 + φ} as the angle φ varies.
Figure 1: Coordinate isosurfaces for a point P (shown as a black sphere) in oblate spheroidal coordinates (μ, ν, φ). The z-axis is vertical, and the foci are at ±2. The red oblate spheroid (flattened sphere) corresponds to μ = 1, whereas the blue half-hyperboloid corresponds to ν = 45°.
2 FeSO 4 Fe 2 O 3 + SO 2 + SO 3. Like other iron(II) salts, iron(II) sulfate is a reducing agent. For example, it reduces nitric acid to nitrogen monoxide and chlorine to chloride: 6 FeSO 4 + 3 H 2 SO 4 + 2 HNO 3 → 3 Fe 2 (SO 4) 3 + 4 H 2 O + 2 NO 6 FeSO 4 + 3 Cl 2 → 2 Fe 2 (SO 4) 3 + 2 FeCl 3. Its mild reducing power is of value in organic ...
0, 4, 8, 16, 32, 48, 72, 88, 120, 152, 192 … (sequence A175341 in the OEIS ). Using the same ideas as the usual Gauss circle problem and the fact that the probability that two integers are coprime is 6 / π 2 {\displaystyle 6/\pi ^{2}} , it is relatively straightforward to show that
Given a unit sphere, a "triangle" on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere. If the lengths of these three sides are a (from u to v ), b (from u to w ), and c (from v to w ), and the angle of the corner opposite c is C , then the law of haversines states: [ 10 ]