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

## TL;DR

This paper derives analytical equations for calculating the axis angle 2V from extinction measurements on a spindle stage, providing a closed form solution for error-free data and a maximum likelihood approach for noisy data, validated with real measurements.

## Contribution

It presents a purely analytical derivation of the Joel equations from wave equations, avoiding geometric stereographic arguments, and introduces a maximum likelihood method for data with measurement errors.

## Key findings

- Closed form solution for error-free data
- Maximum likelihood estimation for noisy data
- Validation with published experimental data

## Abstract

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.