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As can be seen from Fig. 1, these problems involve solving the triangle NAB given one angle, α 1 for the direct problem and λ 12 = λ 2 − λ 1 for the inverse problem, and its two adjacent sides. For a sphere the solutions to these problems are simple exercises in spherical trigonometry , whose solution is given by formulas for solving a ...
The solutions to both problems in plane geometry reduce to simple trigonometry and are valid for small areas on Earth's surface; on a sphere, solutions become significantly more complex as, for example, in the inverse problem, the azimuths differ going between the two end points along the arc of the connecting great circle.
The sphere's radius is taken as unity. For specific practical problems on a sphere of radius R the measured lengths of the sides must be divided by R before using the identities given below. Likewise, after a calculation on the unit sphere the sides a, b, and c must be multiplied by R.
Visual calculus, an intuitive way to solve this type of problem, originally applied to finding the area of an annulus, given only its chord length; Napkin ring problem, another problem where the radius of a sphere is counter-intuitively irrelevant
For this reason, the expression for m in terms of β and its inverse given above play a key role in the solution of the geodesic problem with m replaced by s, the distance along the geodesic, and β replaced by σ, the arc length on the auxiliary sphere. [22] [30] The requisite series extended to sixth order are given by Charles Karney, [31] Eqs.
The center of the osculating sphere is offset from the center of the ellipsoid, but is at the center of curvature for the given point on the ellipsoid surface. This concept aids the interpretation of terrestrial and planetary radio occultation refraction measurements and in some navigation and surveillance applications. [15] [16]
In geometry, the Tammes problem is a problem in packing a given number of points on the surface of a sphere such that the minimum distance between points is maximized. It is named after the Dutch botanist Pieter Merkus Lambertus Tammes (the nephew of pioneering botanist Jantina Tammes ) who posed the problem in his 1930 doctoral dissertation on ...
Many problems in the chemical and physical sciences can be related to packing problems where more than one size of sphere is available. Here there is a choice between separating the spheres into regions of close-packed equal spheres, or combining the multiple sizes of spheres into a compound or interstitial packing.