1. Chapter 13 – Vector Functions
13.3 Arc Length and Curvature
Objectives:
Find vector, parametric, and general forms of equations of lines and planes.
Find distances and angles between lines and planes
13.3 Arc Length and Curvature

2. Arc Length
Assuming that the space curve is traversed exactly once on [a,b], and the component functions are differentiable on [a,b], then arc length is given by the integrals below.
13.3 Arc Length and Curvature

3. Note
Plane curves are described in 2 while space curves are defined in 3.
13.3 Arc Length and Curvature

4. Example 1 – pg. 860 #4 Find the length of the curve.
13.3 Arc Length and Curvature

5. Parameterization in Terms of Arc Length
The relation below allows us to use distance along the curve as the parameter.
This parameterization does not depend on coordinate system.
Replace r(t) with r(t(s)) .
13.3 Arc Length and Curvature

6. Example 2 – pg.860 #14
Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t.
13.3 Arc Length and Curvature

7. Recall
If C is a smooth curve defined by the vector r, recall that the unit tangent is given by
and indicates the direction of the curve.
13.3 Arc Length and Curvature

8. Visualization The Unit Tangent Vector
T(t) changes direction very slowly when C is fairly straight, but it changes direction more quickly when C bends or twists more sharply.
13.3 Arc Length and Curvature

9. Definition - Curvature
13.3 Arc Length and Curvature

10. Curvature and the Chain Rule
If we use the Chain rule on curvature, we will have:
So we have:
13.3 Arc Length and Curvature

11. Note: Small circles have large curvature.
Large circles have small curvature.
Curvature of a straight line is always 0 because the tangent vector is constant.
13.3 Arc Length and Curvature

12. Theorem - Curvature
We can always use equation 9 to compute curvature, but the below theorem is easier to apply.
13.3 Arc Length and Curvature

13. Example 3 Use Theorem 10 to find the curvature.
13.3 Arc Length and Curvature

14. Definition – Unit Normal
As you can see, N is
perpendicular to T(t).
13.3 Arc Length and Curvature

15. Definition – Binormal Vector
The Binormal Vector is perpendicular to both T and N. It is also a unit vector and is defined as:
13.3 Arc Length and Curvature

16. Visualization
The TNB Frame
13.3 Arc Length and Curvature

17. Example 4 – pg.861 # 48
Find the vectors T, N, and B at the given point.
13.3 Arc Length and Curvature

18. Other Definitions
The normal plane is determined by the vectors N and B at a point P on the curve C. It consists of all lines that are orthogonal to the tangent vector.
The osculating plane of C and P is determined by the vectors T and N.
An osculating circle is a circle that lies in the oculating place of C at P, has the same tangent as C at P, lies on the concave side of C (towards N), and has radius =1/.
13.3 Arc Length and Curvature

19. Visualization
Osculating Circle
13.3 Arc Length and Curvature

20. Summary of Formulas
13.3 Arc Length and Curvature

21. In groups, work on the following problems
Problem 1 – pg. 860 #6
Find the arc length of the curve.
13.3 Arc Length and Curvature

22. In groups, work on the following problems
Problem 2 – page 860 #16
Reparametrize the curve below with respect to arc length measured from the point (1,0) in the direction of increasing t. Express the reparametrization in its simplest form. What can you conclude about the curve?
13.3 Arc Length and Curvature

23. In groups, work on the following problems
a) Find the unit tangent and unit normal vectors.
b) Use formula 9 to find curvature.
13.3 Arc Length and Curvature

24. In groups, work on the following problems
Problem 4 – pg. 860 #31
At what point does the curve have a maximum curvature? What happens to the curvature as x.
13.3 Arc Length and Curvature

25. In groups, work on the following problems
Problem 5 – pg. 861 #50
Find equations of the normal plane and osculating plane of the curve at the given point.
13.3 Arc Length and Curvature

26. More Examples
The video examples below are from section 13.3 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length.
Example 1
Example 3
Example 7
13.3 Arc Length and Curvature

27. Demonstrations Feel free to explore these demonstrations below.
TBN Frame
Circle of Curvature
13.3 Arc Length and Curvature