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# Solid Angles - what are they, really?

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#1
Crossbones+ - Reputation: **13064**

Posted 27 November 2012 - 07:18 AM

*“If I understand the standard right it is legal and safe to do this but the resulting value could be anything.”*

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#2
Moderators - Reputation: **49118**

Posted 27 November 2012 - 08:02 AM

[edit]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, 28 November 2012 - 12:29 AM.**

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#3
Crossbones+ - Reputation: **19952**

Posted 27 November 2012 - 08:24 AM

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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#4
Crossbones+ - Reputation: **13064**

Posted 27 November 2012 - 08:53 AM

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?)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....?

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 and angles "? And they use the solid angle notation for conciseness instead of messing around with double integrals ... right, I think I understand. Thanks!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?

*“If I understand the standard right it is legal and safe to do this but the resulting value could be anything.”*

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#5
Members - Reputation: **921**

Posted 28 November 2012 - 07:47 AM

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, 28 November 2012 - 07:52 AM.**