Ray-tracing a sphere with displacement mapping

Question

I'm interested in building a simple "Google Earth" type app (for overlaying my own information, not the huge quantity of data that Google has). I'd like it to just be a simple X11 app that ray-traces a sphere with displacement (topographic) information. Ray-sphere intersection is pretty simple, but when the displayment mapping is throw in there it starts to get muddy in my head.

I was wondering if there's a simple technique to extend basic ray-sphere intersection to include displacement data...

Answer 1

I know its been 12 years since first post, but I've uploaded a complete solution to this problem here: https://github.com/ramakarl/just_math

See the sample for math_displace_sphere. This raytraces a displaced sphere terrain at arbitrary detail w/o any mesh. Pure CPU code, could be ported to shader/GPU. MIT licensed.

The solution is to march along a ray, while projecting the ray sample points back down to lat-long, and sampling the terrain texture to determine the displaced height at that sample. Heights are specified relative to center of sphere.

Pseudo-code:

for ( ; ray_hgt >= terrain_hgt && ray_hgt <= shell_hgt; ) { 

        sample += ray_dir * dt; 

        ray_hgt = (sample - sphere_center).Length();

        // given sample, compute surface_pnt and lat, long
        ComputeSphereUV ( sample, surface_pnt, lat, long ); 
                
        pixel_val = terrain_map->GetPixelUV ( lat, long ).x;                                    

        terrain_hgt = sphere_radius + pixel_val * terrain_depth;    
}