MA3D9 Example Sheet 2
1. Let α(s), s ∈ [0, l] be a closed convex plane curve positively oriented (i.e. κs > 0). The curve
β(s) = α(s) − rn(s), where r is a positive constant, is called a parallel curve of α. Show that
a. The length l(β) = l(α) + 2πr.
b. The area A(β) = A(α) + rl + πr2 .
c. κβ =
κα
.
1+rκα (s)
Solution:
a.) Assume α(s) is unit-speed. We have
β 0 (s) = α0 (s) − rn0 = (1 + rκα (s))α0 (s).
So
Z
l(α)
Z
κα ds = l(α) + 2πr
(1 + rκα (s))ds = l(α) + r
l(β) =
l(α)
0
0
by Hopf’s Umlaufsatz.
R
R
b.) A(β) = 21 ||β × β 0 ||ds = 12 ||(α − rn) × (t − rn0 )||ds. It is
1
2
Z
l
0
l
Z
||α × α ||ds + r
0
||b||ds + r
0
2
Z
l
κα ds = A(α) + rl + πr2 .
0
An easier way is to integrate the formula of (a) from 0 to r, which gives
Z r
A(β) − A(α) =
(l(α + 2πρ)dρ = A(α) + rl + πr2 .
0
c.) β 0 (s) = (1 + rκα (s))α0 (s), so ||β 0 (s)|| = 1 + rκα (s). Since α and β have the same normal
n(s),
d
d
ds
κα
κβ (s) = ||
n(s)|| = || n(s)
|| =
.
dsβ
ds
dsβ
1 + rκα (s)
2. Let α(s) be a regular plane curve with unit-speed parametrization. Assume that κs is nonzero
everywhere. The curve
1
β(s) = α(s) + ns
κs
is called the evolute of α.
a. Show that the tangent at s0 of the evolute of α is the normal to α at s0 .
b. Show that a plane curve α has a vertex at s0 if and only if the evolute β of α has a singular
point at s0 .
Solution:
a.) β 0 (s) = α0 (s) +
1
(−κs t)
κs
+ ( κ1s )0 ns = ( κ1s )0 ns , which is normal to α.
b.) β 0 (s0 ) = 0 if and only if ( κ1s )0 (s0 ) = 0, if and only if κ0s (s0 ) = 0.
3. Let γ(s), s ∈ [0, l], be a plane simple closed curve. Assume that the curvature κ(s) ≤ c, where
c is a constant. Prove that
2π
.
l(γ) ≥
c
Solution:
Assume γ(s) is unit-speed. By Hopf’s Umlaufsatz,
Rl
0
κ(s)ds = 2π. Since κ(s) ≤ c, 2π ≤ cl(γ).
4. Find all the vertices for the ellipse γ(t) = (p cos t, q sin t) when p 6= q.
Solution:
κs =
y 00 x0 − x00 y 0
(x02 + y 02 )
3
2
=
pq
3
(p2 sin2 t + q 2 cos2 t) 2
3pq(p2 − q 2 ) sin t cos t
dκs
=−
5
dt
(p2 sin2 t + q 2 cos2 t) 2
So
dκs
dt
= 0 if and only if sin t cos t = 0 if and only if t = 0, π2 , π, 3π
, i.e. at (p, 0), (0, q), (−p, 0), (0, −q).
2
2