ma 242.003 day 14- january 25, 2013 chapter 10, sections 10.3 and 10.4
TRANSCRIPT
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MA 242.003
• Day 14- January 25, 2013• Chapter 10, sections 10.3 and 10.4
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Section 10.3Arc Length and Curvature
• To describe the acceleration r’’(t) it turns out that the crucial idea is CURVATURE of the curve.
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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
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•Curvature of a straight line is then ZERO
•Curvature of a non-straight line is then NON-ZERO
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•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.
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The Solution
• Everyone must use the same parameter for the curve
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The Solution
• Everyone must use the same parameter for the curve
• The only unique parameter along the curve is the arc length parameter s
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Solution:
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(Continuation of problem)
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Solution:
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(continuation of problem)
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Solution:
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(continuation of problem)
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(continuation of problem)
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So now we have a good definition of curvature.
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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
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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
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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
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New calculation of the curvature of a helix:
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The Unit Normal Vector N(t)
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