[whycombinater]

Regarding the "pedagogy", my favorite method of learning is straightforward problem solving: http://blog.sigfpe.com/2006/08/you-could-have-invented-monad...

If you wanted to have state with pure functions, you would, according to DRY, be compelled to write the bind operator. You wouldn't have known to call it a monad or why some basement theoretician likens it to a burrtiofunctor, but you nevertheless would have made the obvious coding solution.

Is this possible for rotation without a PhD in geometry or algebra?

[ajkjk]

Well! I don't have a PhD at all, but I think the exponential map is a surprisingly intuitive operation once you get used to it. It works on all kinds of operators. I'd argue for teaching it much earlier, in intro calculus, to introduce the idea that e^(a d_x) f(x) = f(x + a). But it still feels like there is some magic in the fact that works that I don't have a really satisfying explanation for (although it is easy to see by expanding the Taylor series).

[alecst]

The magic comes from the fact that you can decompose a translation (as in your example) into a bunch of little ones. So you want an operator that has the property that F(a)g(x) = g(x+a) = F(a/N)^N g(x). Equating F(a) to F(a/N)^N (for any N) reveals the exponential structure. I’m sure there are other ways but this is the first that comes to mind. You can also try using a very small translation F(da) and that will give you some insight too.

[fault1]

Another way to see the it is to explore it from the matrix exponential structure and the link with trig (esp odd/even functions) or example this video: https://www.youtube.com/watch?v=UWrt9Fj80Kc&list=PLlXfTHzgMR...

so much structure even in 2x2 rotations

[ajkjk]

Yeah, I know how to derive it, but it still feels very unsatisfying to say: voila, you can put derivatives inside functions. It would be a hard sell to an intro calculus student, even though the concept would be very useful at that level.

[fault1]

You certainly do not need to have a phd in geometry and algebra for this. For example, the exp map is used all the time in robotics particularly in rigid body dynamics and kinematics.

Here is a free book made for undergraduates that teaches in that manner that just assumes some elementary linear algebra; http://hades.mech.northwestern.edu/index.php/Modern_Robotics

IMHO, geometric algebra makes certain things clear, but it's also oversold as something new or novel. It can be recast in the language of differential forms (covariant multivectors) which is very often used in physics.

[ajkjk]

For what it's worth, differential forms have plenty of their own pedagogical problems. IMO most of the value in them comes from the concept of multivectors, and very little from the fact that they're 'dual' to regular vectors (which is a finicky detail necessary to do covariant geometry correctly, but not useful for general intuition).