We started working through computations of the Frenet framing of a curve. This is a set of vectors \(T, N, B\) defined at each point of the curve. The best way to think of them is as a moving frame. The basic geometry of a curve will be encoded in the way this frame moves around.
Today’s progress: We worked exclusively from Shifrin. Sladana did 1.2.1a; Jesse M did 1.2.1b; Sladana did 1.2.1c. Then Mark did 1.2.2a and Corbyn did 1.2.2b.
Note:in 1.2.2, both of our problems were solved by first reparameterizing the curve to be unit speed. I challenge you to figure out how to compute the moving frame without that! Sometimes that reparameterization is completely impractical, and you still want to move on.
We did not finish discussing the tasks I assigned for today, so we’ll pick up where we left off next time. Jesse T is up with 1.2.2c to get us started.
Do Shifrin section 1.2 #4-6, and Struik section 1.7 #2. This starts to get toward proving geometry theorems!