Jan 30, 2015: Frenet-Serret Framing 2

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We only had two presentations today. Jesse T did 1.2.2c from Shifrin and Corbyn did 1.2.3a.

Then I took a long detour to talk about the geometric interpretation of the Frenet-Serret framing of a space curve, \({T, N, B}\). This is often called the “moving frame.” The important parts are these:

• \(T\) measures the direction of travel of the curve
• \(N\) measures the direction in which the curve bends (to second order)
• \(\kappa\) measures the speed of turning of the tangent vector, and hence the amount of bending of the curve.

Another way to look at this uses the osculating circle and osculating plane, which we will discuss more later. The osculating circle is the “circle of best fit”, and the osculating plane is the plane which contains that circle. If there were a plane that could contain the curve, the osculating plane is the one that is closest to doing the job. Note that the osculating plane has \( T, N\) as a basis.

• \(B = T \times N\) measures the orientation of the osculating plane in space
• \(\tau \) measures how fast the osculating plane moves (via the rate of turning of \(B\).)

What will turn out to be awesome is that we can understand how a curve behaves by understanding this framing, and curvature and torsion are exactly the things that describe how the framing moves!

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For Monday:

We will spend some time discussing in groups. Be sure you are done with Shifrin 1.2 #1-6, and Struik 1.6 #1,2.