The lie group/lie algebra correspondence is one of the coolest things i wish i’d been taught in school. It’s the exponential and log map in the article, but in

The lie group/lie algebra correspondence is one of the coolest things i wish i’d been taught in school. It’s the exponential and log map in the article, but in a super reusable setting. You take some totally abstract thing you want to work with[0], say 3D rotations, but totally avoiding any details of specific coordinates that might get you into trouble. That’s the lie group. Then you can derive representations of it in coordinates that behave nicely. This is called the corresponding lie algebra. Then for free you get a way to go back and forth between the coordinates and the abstract thing[1], and ways to compose them and stuff. Turns out that this is the exponential map and the log map talked about in the article. What’s even cooler is that you can then compose them on either side, the abstract side or the coordinate side in a way that plays nicely. You get interpolation and averaging and all that in a fairly sensible way too in most cases you’ll deal with as an engineer. And best of all, when you can phrase your problem as a combination of lie groups, you can just google for what their algebras are and get lots of work for free that would be tons of time to do yourself. [0] the thing has to be something with a notion of smooth change, plus a bit more structure it probably has. [1] going back and forth has some ‘connected components’ issues sometimes, but that’s just another reason it’s great to piggyback on known results.