Celestial Programming : Shadow Path of Tip of Vertical Object (Nodus)

This shows the x,y point of a shadow or beam for a given height. Usually the point is either the tip of a sundial gnonmon, or an obstruction with a hole to allow a small beam of light to shine on a sundial (often called a nodus). This is primarily used to mark the lines of the equinoxes, solstices, and possibly dates on a sundial. Hour lines can be added using the usual equations.

Generally the equation is used by setting a particular declination value for the Sun (e.g. 23.4), then looping through all X values for however wide the sundial is to produce a list of x,y points.

$$ y = \frac{-a \sin \phi \cos \phi + \sin \delta \sqrt{x^2(\cos^2 \phi - \sin^2 \delta) +a^2\cos^2 \delta} } {\sin^2 \delta - cos^2 \phi} $$
a = height of tip
\(\phi \) = Latitude
\(\delta \) = Declination of the Sun

The Gnomon Height can be in any units (feet, meters, AU), the resulting x,y values will be in the same units.