Solid Angles - what are they, really?

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I've been looking at improving my understanding of solid angles, and the Wikipedia page left me confused. They describe solid angles as being the projected area of an object as perceived by an observer at some distance, which kind of makes sense. However, when I look at solid angles in the context of computer graphics radiometry, they are being treated as differential oriented angles. as an azimuth/elevation pair with respect to the surface normal! Is it really the same thing?

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I've always thought of them as the same thing as angles, but with regards to the sphere in 3d, instead of the circle in 2D. Instead of a wedge/slice of a circle, they're a conical slice of a sphere.
and just as a 2d wedge of a circle can go past 180º and look like pac-man, these 3d cones can do the same, where they'd actually look like a sphere with a cone missing from it.[/edit]

I believe the solid angle that's common in CG is the same as this one, yes, but there's also the "differential solid angle", which is a different thing.

Isn't an azimuth/elevation pair just another way of representing a direction, which is the same thing as a surface normal?
Given two directions, you could define a cone, which could describe some slice of a sphere, which is a solid angle....? Edited by Hodgman

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It's not only for cones. I think of it as the measure of a subset of the sphere (the whole sphere being 4*pi steradians). In the Wikipedia page this subset is taken to be the projection of some object onto the sphere.

If you want to numerically integrate some quantity over the sphere (often just a half sphere when you are doing graphics), you can quantize the azimuth and elevation and you'll get a partition of the sphere into cells. When you want to sum up you'll multiply the value of the function in a cell times the cell's solid angle. Perhaps that's the kind of usage of solid angles that you have seen?

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[quote name='Hodgman' timestamp='1354024961' post='5004514']
Isn't an azimuth/elevation pair just another way of representing a direction, which is the same thing as a surface normal?
Given two directions, you could define a cone, which could describe some slice of a sphere, which is a solid angle....?
[/quote]
That's what I thought at first, but I was confused with the vector/cone thing. I wasn't sure how the cone would be defined from the vector (how wide should the cone be?)

[quote name='Álvaro' timestamp='1354026295' post='5004518']
If you want to numerically integrate some quantity over the sphere (often just a half sphere when you are doing graphics), you can quantize the azimuth and elevation and you'll get a partition of the sphere into cells. When you want to sum up you'll multiply the value of the function in a cell times the cell's solid angle. Perhaps that's the kind of usage of solid angles that you have seen?
[/quote]
This actually makes a lot of sense - so when they say "integrate over a hemisphere of inward directions" it's just a shortcut to say "divide the hemisphere's surface into lots of infinitesimally small, same-area cells with solid angle [eqn]\mathrm{d} \omega[/eqn] and angles [eqn](\phi, \theta)[/eqn]"? And they use the solid angle notation for conciseness instead of messing around with double integrals [eqn]\mathrm{d} \theta ~ \mathrm{d} \phi[/eqn]... right, I think I understand. Thanks!

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Isn't the notion of solid angle in this context an abstraction of what we're trying to accomplish? What I mean is, when you're trying to numerically integrate a light source over a hemisphere, a differential solid angle is just basically saying, "perform a one dimentional integral over all of the surface area pieces of the sphere". Well, to do that, we ultimately need spherical coordinates to compute dA via dtheta and dphi, and it turns into a two dimentional integral that is more in line with our actual implementation.

I guess I think of the solid angle representation as a concise theoretical format that can be converted to spherical when I need to actually integrate the thing. Is that an accurate way to think about it?

EDIT: That appears to be what Bacterius observed as well in the above post. I didn't get my coffee yet. Edited by ZBethel

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