Analytical calculation of the axis angle 2V from extinction measurements on the spindle stage

A concise derivation of the "Joel equations", which allow for the determination of the axis angle 2V from measurements of extinction directions on a spindle stage, is provided starting from the wave-equation. Only analytic methods and no geometric arguments referring to stereographic projections are invoked. For error free data, the resulting equations allow for a closed form solution. If angle data with measurement error are to be used, a maximum likelihood estimation methodology is proposed, which may be solved using e.g. iterative reweighting. The method is tested with and compared to published data.