A trick to eliminate 2π (sometimes)

Re:A trick to eliminate 2π (sometimes) by Misdicorl on Wednesday June 28, 2023 @09:08PM (#36515054)

I think this will devolve at this point. But you should consider that you are perhaps far too confident in your position here. My last attempt is due you to consider what happens when you

A) multiply two degrees together

B) multiply two dBm values together

The output "units" in A change but do not in B. dB and angles are very different.

Edit: the units in B do change, but the dB part doesn't. Tired.

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Re:A trick to eliminate 2π (sometimes) by jacobolus on Wednesday June 28, 2023 @10:23PM (#36515592)

Multiplying degrees together is not a meaningful concept.

But you can compose rotations.

Multiplying decibels together is also not meaningful, but you can compose scaling.

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Re:A trick to eliminate 2π (sometimes) by Misdicorl on Thursday June 29, 2023 @10:26AM (#36520796)

> Multiplying degrees together is not a meaningful concept.

Curious what you think of the d_theta * d_phi term in a spherical coordinates integral...

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Re:A trick to eliminate 2π (sometimes) by jacobolus on Friday June 30, 2023 @11:48AM (#36537112)

You can compute solid angle (a.k.a. spherical excess, normalized spherical surface area) by taking a surface integral, but the units are not "square degrees" or "square radians", but instead an entirely new type, usually just measured in radians ("steradians"). Some people have defined https://en.wikipedia.org/wiki/Square_degree but that is a stupid unit.

While rotation is naturally oriented like a bivector (plane), solid angle is naturally oriented like a trivector (3-space).

The natural representation is as a kind of (unitless) ratio formed from 3 vectors, not the product of two vector–vector ratios.

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