Let r(s) be an arc length parametrization of a closed curve C of length L. We call

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Let r(s) be an arc length parametrization of a closed curve C of length L. We call C an oval if dθ/ds > 0 (see Exercise 71). Observe that −N points to the outside of C. For k > 0, the curve C1 defined by r1(s) = r(s) − kN is called the expansion of c(s) in the normal direction.(a) Show that ||r(s)|| = ||r'(s)|| + kk(s). (b) As P moves around the oval counterclockwise, increases by 2


Data From Exercise 71

The angle of inclination at a point P on a plane curve is the angle θ between the unit tangent vector T and the x-axis (Figure 21). Assume that r(s) is a arc length parametrization, and let θ = θ(s) be the angle of inclination at r(s). Prove that

k(s) = de ds

Observe that T(s) = (cos θ(s), sin θ(s)).

y P 0 T = (cos 0, sin 0) -X

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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