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# Solid Angles - what are they, really?

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#1
Crossbones+ - Reputation: **8187**

Posted 27 November 2012 - 07:18 AM

The slowsort algorithm is a perfect illustration of the multiply and surrender paradigm, which is perhaps the single most important paradigm in the development of reluctant algorithms. The basic multiply and surrender strategy consists in replacing the problem at hand by two or more subproblems, each slightly simpler than the original, and continue multiplying subproblems and subsubproblems recursively in this fashion as long as possible. At some point the subproblems will all become so simple that their solution can no longer be postponed, and we will have to surrender. Experience shows that, in most cases, by the time this point is reached the total work will be substantially higher than what could have been wasted by a more direct approach.

- *Pessimal Algorithms and Simplexity Analysis*

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#2
Moderators - Reputation: **27879**

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: **11951**

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: **8187**

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?

The slowsort algorithm is a perfect illustration of the multiply and surrender paradigm, which is perhaps the single most important paradigm in the development of reluctant algorithms. The basic multiply and surrender strategy consists in replacing the problem at hand by two or more subproblems, each slightly simpler than the original, and continue multiplying subproblems and subsubproblems recursively in this fashion as long as possible. At some point the subproblems will all become so simple that their solution can no longer be postponed, and we will have to surrender. Experience shows that, in most cases, by the time this point is reached the total work will be substantially higher than what could have been wasted by a more direct approach.

- *Pessimal Algorithms and Simplexity Analysis*

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#5
Members - Reputation: **544**

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.**