MA 242.003

• Day 14- January 25, 2013• Chapter 10, sections 10.3 and 10.4

Section 10.3Arc Length and Curvature

• To describe the acceleration r’’(t) it turns out that the crucial idea is CURVATURE of the curve.

Section 10.3Arc 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

•Curvature of a straight line is then ZERO

•Curvature of a non-straight line is then NON-ZERO

•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.

The Solution

• Everyone must use the same parameter for the curve

The Solution

• Everyone must use the same parameter for the curve

• The only unique parameter along the curve is the arc length parameter s

Solution:

(Continuation of problem)

Solution:

(continuation of problem)

Solution:

(continuation of problem)

(continuation of problem)

So now we have a good definition of curvature.

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

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

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

New calculation of the curvature of a helix:

The Unit Normal Vector N(t)