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)

12	0.0°
1	11.9°
2	24.4°
3	38.1°
4	53.7°
5	71.1°
6	90.0°
7	108.9°
8	126.3°
9	141.9°
10	155.6°
11	168.1°


//Greg Miller (gmiller@gregmiller.net) https://www.celestialprogramming.com/ 2024
//Released as public domain

function getAngles(lat){
    const angles=new Array();
    for(let i=0;i<12;i++){
        const H=i*15*rad; //convert hour from hours to radians
        
        const x=Math.sin(H);
        const y=Math.cos(H)/Math.cos(lat);

        const θ = Math.atan2(x,y);

        angles[i]=θ;
    }
    return angles;
}