 Based on the definition of coordinate system of ray marching point in atmosphere in a previous article, this article talks about how to transform the space from one point to another during ray marching.

For a given point p, whose coordinate in its own space is (0, r + h). Starting from p, if we move p to p_prime along a direction with vertical angle theta (in p’s space), we get p_prime in p’s space. Then how to get p_prime’s coordinate and the direction in p_prime’s space?

• 1) Compute p_prime’s coordinate in p’s space. Given p (x, y), the direction (x_d, y_d) = (cos(theta), sin(theta)), and the distance to move to p_prime: d, we can get p_prime’s coordinate in p’s space by: (x_prime, y_prime) = p + dist * direction = (x, y) + d * (x_d, y_d) = (x + d * x_d, y + d * y_d).
• 2) Transform p_prime’s coordinate from p’s space to p_prime’s space. Get the distance from p_prime in p’s space to the origin: distance_to_origin = sqrt(x_prime^2 + y_prime^2). Then p_prime’s coordinate in its own space is: (x_prime_own, y_prime_own) = (0, distance_to_origin).
• 3) Compute the direction in p_prime’s space. To get the direction in p_prime’s space, we just need to get theta_prime as shown in the figure. theta_prime in p’s space can be obtained by theta_prime = acos(dot_product((x_d, y_d), (x_prime, y_prime))).

So that’s the basic idea. In practical (i.e. in glsl shader), the input is p represented by a normalized altitude and a normalized vertical angle. The output should be p_prime with its normalized altitude and normalized vertical angle in its own space. To do this, we need to:

• 4) Compute p’s coordinate and the direction in p’s space, then
• 5) Follow the above three steps to get p_prime’s coordinate and the direction in p_prime’s space.
• 6) Compute p_prime’s normalized altitude and the normalized vertical angle with the result from the previous step.

Now let’s do 4) and 6).

• 4) Compute p’s coordinate and the direction in p’s space.
• Compute earth’s relative radius based on the ratio of earth/outter-atmospheric-depth and the normalized outter-atmospheric-depth: ratio * 1.0 = 106. (=6360/(6420-6360). Simulating the Colors of Sky)
• Compute p’s coordinate (x, y) = (0, 1280 + normalized-altitude).
• Compute theta (domain is [0, PI]) with the normalized vertical angle (domain is [0, 1]): theta = normalized-vertical-angle * PI.
• 5) Compute p_prime’s normalized altitude and the normalized vertical angle.
• p_prime’s normalized altitude = y_prime_own (i.e. in 2)) – earth’s relative radius = y_prime_own – 1280.
• normalized vertical angle = theta_prime / PI (NOTE: theta_prime is calculated from 3)).

In conclusion, the steps for computing p_prime’s normalized altitude and normalized vertical angle in its own space with p’s normalized altitude and normalized vertical angle are:

4), 1), 2) 3), 5).