[Phys-L] Re: Hours of sunlight

[John Denker <jsd@AV8N.COM>]

Marc "Zeke" Kossover wrote:

> .... and a bunch of trig to
> figure out how much sun the area gets.
>
> Does anyone know of a resource to help me make these
> calculations?

That's a typical "compound rotation" problem.  The sun rotates (orbits)
in a plane that is rotated (inclined) relative to your horizontal.

If I were doing that calculation, I would do it using Clifford algebra.

For those who are not already up to speed on Clifford algebra, I sincerely
recommend looking into it.  If you know what a vector is, you can generalize
to multivectors more-or-less immediately.  Just accept one or two additional
axioms, such as the notion that the product of two vectors is a bivector,
and off you go.  The investment will be repaid almost immediately;  the
effort of learning Clifford algebra and using it to solve a compound
rotation problem the easy way is less than the effort of solving even one
compound rotation problem any other way that I know of.

Yeah, I know that sounds too good to be true.  That's what I said the first
time I heard of Clifford algebra ... but I found it to be even more useful
than the advertisments had claimed.

I'll pass up this opportunity to include the full advertisement, because
it would just scare people.  My point is that the easy parts of Clifford
algebra are very easy.  With some regularity I am called upon to explain
angular momentum and precession to people with only a high-school education,
or even less.  I explain it using bivectors.  It's waaay easier and yields
better results than any other approach I can imagine.

=============

FWIW, _quaternions_ are a special case of Clifford algebra.  I hate to
admit it, but for years I never really understood quaternions.  I was
exposed to them in school, but it didn't stick.  I couldn't visualize
them.  When I tried Clifford algebra for the first time, it was a
tremendous "AHA", a tremendous revelation.  I can visualize bivectors,
draw them, model them with my hands when talking to other people, et
cetera.

I've done enough compound rotation problems the dumb way (Euler angles
and/or rotation matrices) to really appreciate how much easier the
Clifford algebra approach is.

Everybody I know who does compound rotation problems (including for
autopilots and for 3D video games) does it using quaternions.  If you
learn Clifford algebra, you get quaternions for free ... plus a lot of
other good stuff.
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