Rendering an atmosphere around a planet with shading

Question

I have a made a planet and wanted to make an atmosphere around it. So I was referring to this site:

Click to visit site

I don't understand this:

As with the lookup table proposed in Nishita et al. 1993, we can get the optical depth for the ray to the sun from any sample point in the atmosphere. All we need is the height of the sample point (x) and the angle from vertical to the sun (y), and we look up (x, y) in the table. This eliminates the need to calculate one of the out-scattering integrals. In addition, the optical depth for the ray to the camera can be figured out in the same way, right? Well, almost. It works the same way when the camera is in space, but not when the camera is in the atmosphere. That's because the sample rays used in the lookup table go from some point at height x all the way to the top of the atmosphere. They don't stop at some point in the middle of the atmosphere, as they would need to when the camera is inside the atmosphere.

Fortunately, the solution to this is very simple. First we do a lookup from sample point P to the camera to get the optical depth of the ray passing through the camera to the top of the atmosphere. Then we do a second lookup for the same ray, but starting at the camera instead of starting at P. This will give us the optical depth for the part of the ray that we don't want, and we can subtract it from the result of the first lookup. Examine the rays starting from the ground vertex (B 1) in Figure 16-3 for a graphical representation of this.

First Question - isn't optical depth dependent on how you see that is, on the viewing angle? If yes, the table just gives me the optical depth of the rays going from land to the top of the atmosphere in a straight line. So what about the case where the rays pierce the atmosphere to reach the camera? How do I get the optical depth in this case?

Second Question - What is the vertical angle it is talking about...like, is it the same as the angle with the z-axis as we use in polar coordinates?

Third Question - The article talks about scattering of the rays going to the sun..shouldn't it be the other way around? like coming from the sun to a point?

Any explanation on the article or on my questions will help a lot.

Thanks in advance!

Answer 1

I am no expert in the matter but have played with Atmospheric scattering and various physical and optical simulations. I strongly recommend to look at this:

• my VEEERRRYYY Simplified version of atmospheric scattering in GLSL

It odes not do the full volume intergration but just linear path integration along the ray and does only the Rayleight scatering with isotropic coefficients. As you can see its still good enough.

In real scattering the viewing angle is impacting the real scattering equation as the scattering coefficients are different in different angles (against main light source and viewer) So answer to your first question is Yes it does.

Not sure what you are refer to in your second question. The scattering itself is dependent on angle between light source, particle and camera. That lies on arbitrary plane. However if the Earth surface is accounted to the equation too then its dependent on the horizontal and vertical angles (against terrain) so azimuth,elevation as usually more light is reflected when camera is facing sun (azimuth) and the reflected rays are closer to your elevation. So my guess is that's what the horizontal angle is about accounting for reflected light from the surface.

To answer your 3th question is called back ray tracing. You can cast rays both ways (from camera or from sun) however if you start from light source you do not know which way to go to hit a pixel on camera screen so you need to cast a lot of rays to increase the probability of hit enough to fill the screen which is too slow and inaccurate (produce holes). If you start from screen pixel then you cast just single or per wavelength ray instead which is much much faster. The resulting color is the same.

[Edit1] vertical angle

OK I read the linked topic a bit and this is How I understand it:

So its just angle between surface normal and the casted ray. Its scaled so vert.angle=0 means that ray and normal are the same and vert.angle=1 means the are opposite directions.