Several formulas related to the zonal calculation system

Abstract

In the article “The ultimate simplification of zonal calculations and crystallographic calculations in general,” I noted the extreme simplicity that calculations of spherical bipolar coordinates obtain if we take Miller’s formula as the basis. Now I will supplement the formulas derived then with those that relate to the calculation of the cotangents of angles formed with an arbitrarily chosen initial face of the zone (see the note). It can be can concluded that the system of zonal calculations with bipolar coordinates, in its particular simplicity, is applicable without the use of a sequential, recursive procedure of calculation, and directly to faces with arbitrary, complex indices.

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