1. MA 242.003 Day 14- January 25, 2013 Chapter 10, sections 10.3 and 10.4

5. Section 10.3 Arc Length and Curvature To describe the acceleration r’’(t) it turns out that the crucial idea is CURVATURE of the curve.

6. Section 10.3 Arc Length and Curvature To describe the acceleration r’’(t) it turns out that the crucial idea is CURVATURE of the curve. Compare the unit tangents for – 1. a straight line – 2. a curved line

8. Curvature of a straight line is then ZERO Curvature of a non-straight line is then NON-ZERO

9. Curvature of a straight line is then ZERO Curvature of a non-straight line is then NON-ZERO Problem: The number for the curvature depends on choice of parameter.

14. The Solution Everyone must use the same parameter for the curve

15. The Solution Everyone must use the same parameter for the curve The only unique parameter along the curve is the arc length parameter s

19. Solution:

20. (Continuation of problem)

23. Solution:

24. (continuation of problem)

25. Solution:

26. (continuation of problem)

32. So now we have a good definition of curvature.

33. The problem now is that we CANNOT do the calculation for general curves because we cannot compute

34. So now we have a good definition of curvature. The problem now is that we CANNOT do the calculation for general curves because we cannot compute

35. So now we have a good definition of curvature. The problem now is that we CANNOT do the calculation for general curves because we cannot compute

54. New calculation of the curvature of a helix:

55. The Unit Normal Vector N(t)