Procedurally Generating A Planet!!!

Perlin + fractal noise can give pretty nice results from my experience. If you want to generate it over a sphere, just displace each vertex on the sphere in the direction of the original vertex normal (calculating the original normal should be straightforward).

Also, if you own Photoshop, I think there's some filters to do something similar. I don't use Photoshop though so I'm not really an expert on this.

Beyond that, "Procedural Texturing and Modeling" has some nice algorithms with code (on the website) for terrain generation.

Now, if you're planning on zooming extremely far out and want something that looks like earth, well, I'm not sure if this technique will suffice. You can try it though, it might be acceptable for what you're doing.

EDIT:
I think I missed the point about computing the normal. I would just manually recompute all the normals in the mesh after generating the new verticies. That should work, right?

Last time I wrote a landscape generator I just used a sum of sine waves at different frequencies/amplitudes.

Your image wasn't a planet, it wasn't even height map of a planet surface. If you'd like to look at some images: This page has some starting points

Of course, you'd need something more, so after you'd guess some algorithms that could make similar results, you might come for more pointers. Note that above mentioned page shows only results of first three days of work, so you could try to be better.

can someone just show me a math example with a one dimensional random sin wave like the one above?

can someone just show me a math example with a one dimensional random sin wave like the one above?