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 dθ/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–
, 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.
756
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