Today we took time to talk a bit more about the detail for curves. Emily finished Struik \(S\)1-6 #11 and we learned the new vocabulary word vertex. A vertex is a point where \(\kappa’=0\), and this connects with the task Emily did by seeing the radius of the osculating circle seems to be not changing at that point.
Then we discussed \(S\) 1-6 #8, with Corbyn taking the lead. His argument was inspired by the mean value theorem arguments we have been making, but viewed in reverse. I gave an alternate that used just the Frenet-Serret equations. These amount to the same thing, though, and I see no reason to prefer one over the other.
Then I filled time by discussing the “normal form” for a curve. The idea is to use Taylor’s theorem to third order, but then plug in what you know from our later study. This allows us to build pictures of projections of a curve on its three standard framing planes: the osculating plane, the normal plane, and the rectifying plane. They gives us alternate ways of thinking about curvature and torsion.
I posted broken links last time, so I’ll clean those up. For friday, I want to talk about involutes and evolutes. I’ll post an update with some tasks to do later this afternoon.