Today we solved many problems by using the Frenet-Serret apparatus and equations. The fun part is to see how quickly we can get real geometry theorems using this tool.
From Shifrin, Corbyn did 1.2.7. Jesse did 1.2.9. From Struik, TJ did 1-6#2 and Mark did 1-6#4.
I handed out project description sheets, too!
We will have the pen & paper competency exam for the basics of curves on Friday in class.
Our next topic of study will be “osculation.” I want to get into the details of understanding the osculating plane, circle, sphere, and helix. The main tool here will be the Mean Value Theorem and a particular kind of limiting argument.
There is exposition of this material in Struik \(\S\S\)1-3, 1-4, and 1-7. We will complete exercises $\S$1-6 #3,8,10,11,15 and \(\S\)1-11 #5. Also, there is a bit of overlap with Shifrin exercise 1.2.24.