Further Insights and Challenges

Question 13.249

Show that the curvature of Viviani’s curve, given by r(t) = 〈1 + cos t, sin t, 2 sin(t/2)〉, is

Question 13.250

Let r(s) be an arc length parametrization of a closed curve of length L. We call an oval if /ds > 0 (see Exercise 61). Observe that −N points to the outside of . For k > 0, the curve 1 defined by r1(s) = r(s) − kN is called the expansion of c(s) in the normal direction.

  • Show that .

  • As P moves around the oval counterclockwise, θ increases by 2π [Figure 21(A)]. Use this and a change of variables to prove that .

    As P moves around the oval, θ increases by 2π.
  • Show that 1 has length L + 2πk.

In Exercises 75–82, let B denote the binormal vector at a point on a space curve , defined by B = T × N.

Question 13.251

Show that B is a unit vector.

Question 13.252

Follow steps (a)–(c) to prove that there is a number τ (lowercase Greek “tau”) called the torsion such that

  • Show that and conclude that dB/ds is orthogonal to T.

  • Differentiate B · B = 1 with respect to s to show that dB/ds is orthogonal to B.

  • Conclude that dB/ds is a multiple of N.

Question 13.253

Show that if is contained in a plane , then B is a unit vector normal to . Conclude that τ = 0 for a plane curve.

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Question 13.254

Torsion means “twisting.” Is this an appropriate name for τ? Explain by interpreting τ geometrically.

Question 13.255

Use the identity

a × (b × c) = (a · c)b − (a · b)c

to prove

Question 13.256

Follow steps (a)–(b) to prove

  • Show that dN/ds is orthogonal to N. Conclude that dN/ds lies in the plane spanned by T and B, and hence, dN/ds = aT + bB for some scalars a, b.

  • Use N · T = 0 to show that and compute a. Compute b similarly. Equations (14) and (16) together with dT/dt = κN are called the Frenet formulas and were discovered by the French geometer Jean Frenet (1816–1900).

Question 13.257

Show that r′ × r″is a multiple of B. Conclude that

Question 13.258

The vector N can be computed using N = B × T [Eq. (15)] with B, as in Eq. (17). Use this method to find N in the following cases:

  • r(t) = 〈cos t, t, t2〉 at t = 0

  • r(t) = 〈t2, t−1, t〉 at t = 1