Today was a bit slow. So, I talked a little bit about the Fundamental Theorem of Curve theory, which is really a corollary to the Existence and Uniqueness theorem for ordinary differential equations. The idea is that the curvature and torsion functions are all the information we need to write down the Frenet-Serret equations. And those are a coupled system of ordinary differential equations! In the end, that means that knowledge of \(\kappa\) and \(\tau\) is enough to completely determine the curve, up to some rigid motion of Euclidean space.
I briefly introduced the ideas of involutes and evolutes. I want to talk about those some more this week. I wrote about those a few years ago. Here are some links:
http://theronhitchman.blogspot.com/2013/02/involutes.html http://theronhitchman.blogspot.com/2013/03/evolutes.html
It looks like the 3d plotting from those Sage cells is off. I will think about that.
At this point, we still have the following uncompleted exercises from Struik: \(\S\)1-6 # 8,10,11,15.
Before we move on to surfaces, I want to talk about the following things: