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

Chapter 13, Problem 82

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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 d/ds > 0 (see Exercise 69). 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 1(s) = r (s) + k(s). (b) As P moves around the oval counterclockwise, increases by 2 [Figure 22(A)]. Use this and a change of variables to prove that L 0 (s) ds = 2. (c) Show that C1 has length L + 2k. (A) An oval T N C1 is the expansion of C in the normal direction. (B) C1 C x y = 0 P P C FIGURE 22 As P moves around the oval, increases by 2.