Celestial Programming : Vertical Sundial

The equations for a vertical sundial are quite similar to those for a horizontal sundial. And the results might look like the dial is just rotated, but the angles are different.

Computation of the angles for the hour lines is fairly simple, I give two methods below. They both produce the same results, but the first method allows the use of the ATAN2 function to ensure $\theta$ is in the correct quadrant. The second method is just shorter.

Hour Lines

$$ \begin{align*} x &=\sin H \\ y &=\frac{\cos H}{\cos \phi} \\ \tan \theta &= \frac{x}{y} \\ \\ Alternate: \\ \tan \theta &= \tan H \cos \phi \end{align*} $$
$\theta $ = Line's angle for given hour
$H $ = Hour angle of the Sun
$\phi $ = Latitude

Latitude (e.g. 37.25)