Time of Day systems simulate the movement of the sun across the sky. There are several challenges to implement such a system, but this post deals with the most fundamental of these - how to generate the sun position in the sky. While an accurate position is not required, it is fairly simple to generate and very satisfying. I came across this in “A Practical Analytic Model for Daylight” by Preetham et. al. and wanted to extract the relevant portion here for ease of discoverability.

The three parameters that determine the position of the sun in the sky are:

  • Solar time \(t\) in decimal hours
  • Julian date \(J\) which is the day of the year in [1,365] range
  • Latitute \(l\) of the site in radians \([-\frac{\pi}{2}, \frac{\pi}{2}]\)

Solar time is the calculation of time based on the position of the sun, and it deviates from standard time by a few seconds based on the Julian date and site location. If you want to work with standard time, the paper provides a conversion but the difference is not significant unless you want an exact reproduction.

The first step is to calculate the solar declination angle (\(\delta\)). This is the angle between the Earth-Sun line and the equatorial plane. It reaches a max value of \(23.44^\circ\) on the June solstice and a minimum angle of \(-23.44^\circ\) on the December solstice. It is approximated by the following formula:

\[\delta = 0.4093 sin(\frac{2\pi(J - 81)}{368})\]

The sun elevation (\(\theta\)) and azimuth (\(\phi\)) can then be computed as follows:

\[\theta_s = \frac{\pi}{2} - \arcsin(\sin l\sin\delta - \cos l\cos\delta\cos\frac{\pi t}{12})\] \[\phi = \arctan(\frac{-\cos\delta\sin\frac{\pi t}{12}}{\cos l\sin\delta - \sin l\cos\delta\cos\frac{\pi t}{12}})\]

Code snippet for the above:

Vector3 GetSunDirection(float t, int J, float l)
	// Solar declination
	float delta = 0.4093f * sin(PI * (J - 81.f) / 368.f);

	float sin_l = sin(l);
	float cos_l = cos(l);
	float sin_delta = sin(delta);
	float cos_delta = cos(delta);
	float t = PI * t / 12.f;
	float sin_t = sin(t);
	float cos_t = cos(t);

	// Elevation
	float theta = 0.5f * PI - asin(sin_l * sin_delta - cos_l * cos_delta * cos_t);

	// Azimuth
	float phi = atan(-cos_delta * sin_t / (cos_l * sin_delta - sin_l * cos_delta * cos_t));

	// Sun dir based on Time of Day
	Vector3 sunDir;
	sunDir.x = sin(theta) * cos(phi);
	sunDir.z = sin(theta) * sin(phi);
	sunDir.y = cos(theta);

	return sunDir;

This is how it looks in action.