Math 227 Problem Set III
Due Wednesday, January 27
1. Let Y (x) be an infinitely differentiable function on −1 < x < 1. Parametrize the curve y = Y (x),
−1 < x < 1, by x. That is, set r(t) = tı̂ı + Y (t)̂.
(a) Develop formulae for κ(t), T̂(t), N̂(t), B̂(t) and the centre of curvature c(t). (View the xy–plane as
the plane z = 0 in IR3 .)
(b) Give an example of an infinitely differentiable function Y (x) on −1 < x < 1 for which N̂(t), B̂(t)
and c(t) are infinitely differentiable except at t = 0, where they are not even continuous. Sketch the
curve, a circle of curvature for one x < 0 and a circle of curvature for one x > 0.
2. In this problem, we define what we mean by “the circle that fits the parametrized curve C best near r0 ”.
◦ For convenience, let r(s) be a parametrization of C by arc length with r(0) = r0 . Define T̂, N̂ and
d2 r d2 r
κ by T̂ = dr
ds (0), κ = ds2 (0) and κN̂ = ds2 (0). In this problem, we only consider the case that
κ > 0, so that N̂ is a well defined unit vector that is perpendicular to T̂.
◦ Pick any c ∈ IR3 , any ρ′ > 0 and any two mutually perpendicular unit vectors T̂′ and N̂′ . Then
R(s) = c − ρ′ cos ρs′ N̂′ + ρ′ sin ρs′ T̂′
is a circle, parametrized by arc length. We may parametrize any circle by choosing c, ρ′ , T̂′ and
N̂′ appropriately. (See the notes “Parametrizing Circles”.)
◦ Set D(s) = |R(s) − r(s)|2 . It is, of course, the square of the distance from the point R(s) on the
circle to the point r(s) on C.
◦ We’ll say that the circle above fits C best near r0 if D(0) = D′ (0) = D′′ (0) = D(3) (0) = D(4) (0) = 0.
p
(These conditions imply that the distance from r(s) to R(s), i.e. D(s), goes to zero faster than
order s2 as s tends to zero.)
Prove that D(0) = D′ (0) = D′′ (0) = D(3) (0) = D(4) (0) = 0 if and only if T̂′ = T̂, N̂′ = N̂, ρ′ = κ1 and
c = r0 + κ1 N̂.
3. The Frenet–Serret formulae may be written
 

T̂(s)
0
κ(s)
d 
0
N̂(s)  =  −κ(s)
ds
0
−τ (s)
B̂(s)


T̂(s)
0
τ (s)   N̂(s) 
0
B̂(s)
This is a system of first order linear ordinary differential equations. In general, a system of first order
linear ordinary differential equations is one of the form dx
ds (s) = M (s) x(s), where for each s ≥ 0,
x(s) ∈ IRn is an n–component vector and M (s) is an n × n matrix. Let
◦ x0 ∈ IRn
◦ for each s ≥ 0, M (s) be an n × n matrix
Assume that each matrix element of M (s) is continuous in s. In the case of the Frenet–Serret formulae,
we also have that
◦ M (s) is antisymmetric, meaning that M (s)ij = −M (s)ji for all 1 ≤ i, j ≤ n and s ≥ 0
(a) Assume that M (s) is antisymmetric. Prove that if x(s) obeys dx
ds (s) = M (s) x(s) for all s > 0, then
|x(s)| is a constant, independent of s.
(b) Again assume that M (s) is antisymmetric. Prove that if both xa (s) and xb (s) solve the initial value
problem dx
ds (s) = M (s) x(s), x(0) = x0 , then xa (s) = xb (s) for all s ≥ 0.
(c) Let κ(s) and τ (s) be continuous functions. Assume that T(s), N(s), B(s) obeys the Frenet–Serret
formulae and that T(0), N(0) and B(0) are mutually perpendicular unit vectors. Prove that, for each
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s > 0, T(s), N(s) and B(s) are also mutually perpendicular unit vectors. Hint: It is true, but beyond
the scope of this course to prove, that there is a unique solution to the initial value problem x(0) = x0 ,
dx
ds (s) = M (s) x(s). You may use this result.
4. Do either part (a) or part (b) of the following question.
Let the curve r(s) be parametrized by arc length and have κ(s) > 0 and τ (s) 6= 0.
(a) Suppose that the curve lies on the sphere with centre c and radius R. Prove that
r(s) − c = −ρ(s)N̂(s) − ρ′ (s)σ(s) B̂(s)
1
1
where ρ(s) = κ(s)
and σ(s) = τ (s)
. In particular R2 = ρ(s)2 + ρ′ (s)2 σ 2 .
(b) Prove that, conversely, if ρ(s)2 + ρ′ (s)2 σ(s)2 is a constant and ρ′ (s) 6= 0, then r(s) lies on a sphere.
Hint: Prove that r(s) + ρ(s) N̂(s) + ρ′ (s)σ(s) B̂(s) is a constant.
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