Minard’s chart

Let's assume \( \epsilon\) the difference between \(l \) the spherical distance, and \( x\) the direct distance between the two points. The inequality to use to determine when the difference between these two distances becomes significant is as follows:

\[ \begin{array}{rrl} & &\displaystyle\frac{l - x}{l} > \epsilon \\ & \Leftrightarrow & \displaystyle\frac{r \alpha - 2 r \sin \left(\frac{\alpha}{2}\right)}{r \alpha} > \epsilon \\ & \Leftrightarrow & \displaystyle\frac{\alpha - 2 \sin \left(\frac{\alpha}{2}\right)}{\alpha} > \epsilon \\ & \Leftrightarrow & 1 - \displaystyle\frac{2}{\alpha} \sin \left(\displaystyle\frac{\alpha}{2}\right) > \epsilon \end{array} \]

From there, I chose to solve the inequality analytically. The plot below illustrates the results with a threshold set at 1%. Since different thresholds could be selected, the upper portion of the plot displays the full range of possible values. Because this range is too broad to clearly see the 1% threshold, the lower part zooms in on that specific region to highlight the result for 1%.