If you rotate about X, then about Y, then undo the two rotations in the same order, you don't end up where you started. This can easily be observed by rotating a book in various ways, and observing that it doesn't end up the way it started.
But it gets really weird when you apply those rotations using very small angles. You turn X, turn Y, reverse X, reverse Y, and when all is said and done, what you've accomplished is a turn around Z.
Whoa.
As usual, I find that the demonstrations are lacking, in that they go the wrong way around, and give you the approximation at the wrong time.
[more ]One bit in the presentation of rotations is that they are often presented as exponentials. The common approach appears to be to first take approximate an infinitesmal rotation, then apply them in succession.
I'm not terribly keen on that approach, because it isn't clear how the approximation goes away and becomes an exact answer, and because it isn't necessary.
[more ]Of late, I've been distracted by the study of rotations - simple ordinary rotations in three space. It seems that I have had the results in my head for quite some time, but the derivations (and admittedly, therefore some degree of understanding) missing. So I recently went through the drill of working it out.
What I came away with was the notion that when this material was presented to me, it was backwards. Certainly that's the way it felt to me when I worked through the problems as they presented themselves.
So in this series, I reverse that order. We begin by re-examining the basics of rotations.