14.4 Arc Length and Curvature MAT 3238 Vector Calculus 14.4 Arc Length and Curvature

Homework Both written and WA HW due Next Tuesday You do not and should not wait until later to start your HW

Recall (Calculus III, 12.1) Arc length for a two dimensional curve

Arc Length

In Vector Function Form

Similarly, for a 3D curve...

Unify Formula in terms of 𝑟(𝑡)

Recall (Calculus III, 12.1) Representations of parametric curves are not unique

Example 1a

Example 1a

Example 1b

Example 1b

Example 1c

Example 1c

Q&A Q: Will I get a different arc length if a curve is represented by two different parametrizations?

Q&A A: No (Can you do the calculations in your head?)

Q&A Q: Can we somewhat “standardize” the parametrization process? That is, can we agree on a “standard” parameter?

Q&A A: Yes. Physicists and Engineers prefer a certain type of parametrization. We are going to describe “the” parameter below.

Arc Length Function The original parameter is 𝑡. 𝑠 is the “standard” parameter.

Parametrize a Curve with respect to Arc Length In this textbook, to simplify the calculations, it assumes 𝑎=0. Of course, this does not have to be the case.

Example 2

Curvature Curvature is a measure of how much a curve bends. It is used to study geometric properties of curves and motion along curves, and has applications in diverse areas.

Curvature

Curvature

Curvature – Use Std Paramter

Curvature: Second Formula The curvature is easier to compute if it is expressed in terms of the parameter 𝑡 instead of 𝑠. (So we do not need to switch to a new parameter.)

Example 3 Find the curvature of a circle with radius 𝑅.

Example 3 Find the curvature of a circle with radius 𝑅. Q1: What do you expect the curvature should be? Q2: What do you expect with the curvature when 𝑅 increases?

Example 3 Find the curvature of a circle with radius 𝑅.

Curvature: Third Formula Easy(?) to check. Use the fact that 𝑇 and 𝑇’ are orthogonal (compare: 𝑟(𝑡) and 𝑟’(𝑡) are orthogonal). Does not involve 𝑇.

Example 4

Normal and Binormal Vectors We want to establish a “coordinate frame” at each point of a curve. In addition to the tangent vector 𝑇(𝑡) ,we defined the following two unit vectors.

Normal and Binormal Vectors We want to establish a “coordinate frame” at each point of a curve. In addition to the tangent vector 𝑇(𝑡) ,we defined the following two unit vectors.

Example of Moving Frames

Normal and Binormal Vectors

Normal and Binormal Vectors

Normal and Osculating Planes at a Point 𝑃

Example 5