1. 14.4 Arc Length and Curvature
MAT 3238 Vector Calculus
14.4 Arc Length and Curvature

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

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

4. Arc Length

5. In Vector Function Form

6. Similarly, for a 3D curve...

7. Unify Formula in terms of 𝑟(𝑡)

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

9. Example 1a

10. Example 1a

11. Example 1b

12. Example 1b

13. Example 1c

14. Example 1c

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

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

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

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

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

20. 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.

21. Example 2

22. 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.

23. Curvature

24. Curvature

25. Curvature – Use Std Paramter

26. 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.)

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

28. 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?

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

30. 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.

31. Example 4

32. 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.

33. 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.

34. Example of Moving Frames

35. Normal and Binormal Vectors

36. Normal and Binormal Vectors

37. Normal and Osculating Planes at a Point 𝑃

38. Example 5