Arc length and curvature
Math 21a
Fall 2015
Arc length
1 Elsa lives on the third floor landing of her frozen castle. There is a helical staircase
from the first floor foyer to the third floor, which is approximated by the parametrized curve
~r(t) = h5 cos t, 5 sin t, ti, 0 ≤ t ≤ 4π. One morning, she slipped at the top of the stairway,
and slid down the stairs (ice is very slippery), only managing to stop herself on the first
floor. How far did she travel on the icy slide?
Arc length
If r(t) = hx(t), y(t), z(t)i, a ≤ t ≤ b, is a curve, then its arc length is given by
Z
L=
b
Z bp
|r (t)| dt =
x0 (t)2 + y 0 (t)2 + z 0 (t)2 dt.
0
a
a
This quantity does not depend on the chosen parametrization of the curve.
2 Compute the arc lengths of the following curves:
√
(a) ~r(t) = he2t , e−2t , 2 2ti, 0 ≤ t ≤ 1
(b) ~r(t) = hcos t, sin t, ln cos ti, 0 ≤ t ≤
π
4
Curvature
The unit tangent vector
If r(t) is a curve which has nonzero speed at t, the unit tangent vector of the curve at
0 (t)
. It does not depend on the parametrization.
t is defined to be T(t) = |rr0 (t)|
3 Consider the circle of radius a centered at the origin in the plane R2 . For c 6= 0, we
obtain parametrizations ~rc (t) = ha cos(ct), a sin(ct)i.
(a) Compute the velocity of the curve for each parametrization ~rc .
(b) Compute the unit tangent vector T~c (t) for each parametrization ~rc .
~0
(t)|
(c) Show that the quantity κ(t) = ||~Tr0c(t)|
does not depend on c. What is relationship between
c
κ(t) and a, the radius of the circle?
Curvature
For a curve with parametrization r(t), the quantity κ(t) defined above is called its
curvature. The curvature of a curve does not depend on the chosen parametrization –
it is an intrinsic property of the curve. One can also show the following useful formula
for curvature:
|r0 (t) × r00 (t)|
|T0 (t)|
=
.
κ(t) = 0
|r (t)|
|r0 (t)|3
4
(a) Let ~r(t) = ht + 7, −2t, 3ti be a line. Compute its curvature. Does your answer make
sense? In general, what is the curvature for any line?
(b) Let ~r(t) = ht2 , t, −4i be a parabola contained in the plane z = −4. Compute its
curvature. At which point on the parabola is the curvature largest? Does this make
sense?
5 The circle involute is the plane curve with parametric equation
~r(t) = ht sin t + cos t, sin t − t cos ti,
t ≥ 0.
Let s(t) be the length of this curve from ~r(0) to ~r(t), and let κ(t) be the curvature of this
curve at ~r(t). Show that 2s(t)κ(t)2 = 1 for all t.
Normal and binormal vectors
Normal and binormal vectors
Let C be a space curve with parametrization r(t) and let P be a point on C.
• The (principal) unit normal vector is N(t) =
T0 (t)
.
|T0 (t)|
• The binormal vector is B(t) = T(t) × N(t).
• The normal plane of C at P is the plane perpendicular to T at the point P .
• The osculating plane of C at P is the plane spanned by T and N at P .
• The osculating circle of C at P is the circle in the osculating plane tangent to
the curve at P lying on the concave side of C with radius κ1 .
The word “osculate” comes from the Latin osculatum, meaning “kiss”. The osculating
plane is the plane that comes closest to containing the curve C near P , and the
osculating circle is the best approximation of the curve C near P by a circle: it shares
the same tangent, normal, and curvature at P .
~ , and B
~ at the given point.
6 Find the vectors T~ , N
(i) ~r(t) = ht2 , 23 t3 , ti at the point (1, 23 , 1).
(ii) ~r(t) = hcos t, sin t, ln cos ti at the point (1, 0, 0).
~
~.
7 For any curve, show that ddsT , where s is arc length, is equal to κN
8 Suppose the osculating plane at every point on a space curve is the same plane. What
can you conclude about the curve?