Page 217 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes We have shown in the previous section that a moving frame is completely determined up to an
affine motion by the functions p (t) and q (t). In the case of the Frenet frame, this means that if
ij
i
two curves x and x have the same arclength s(t), the same curvature k(t), and the same torsion
w(t), then the curves are congruent, i.e. there is an affine motion of Euclidean three-space taking
x(t) to x(t) for all t. Another way of stating this result is:
Theorem 2. The Fundamental Theorem of Space Curves. Two curves parametrized by arclength
having the same curvature and torsion at corresponding points are congruent by an affine
motion.

Exercise 16: Compute the torsion of the circular helix. Show directly that the principal normals
of the helix are perpendicular to the vertical axis, and show that the binormal vectors make a
constant angle with this axis.
Exercise 17: Prove that if the curvature and torsion of a curve are both constant functions, then
the curve is a circular helix (i.e. a helix on a circular cylinder).

Exercise 18: Prove that a necessary and sufficient condition for a curve x to be a generalized helix
is that
.
x”(t) × x”’(t) xiv(t) = 0 .
.
Exercise 19: Let y(t) be a curve on the unit sphere, so that |y(t)| = 1 and y(t) y’(t) × y”(t)  0 for
1
all t. Show that the curve x(t) = c y(u)× y"(u)du with c  0 has constant torsion .

c
Exercise 20: (For students familiar with complex variables) If the coordinate functions of the
vectors in the Frenet frame are given by
T = (e , e , e ),
11
13
12
N = (e , e , e ),
22
23
21
b = (e , e , e ),
32
33
31
then we may form the three complex numbers
e + ie 1 e

1j
z = 1 – e 3j 2j  e  ie 3 j 2 j .
j
1j
Then the functions z satisfy the Riccati equation
j
i
2
z =-ik(s)zj+ w(s)( 1 z ).
'


j
j
2
This result is due to S. Lie and G. Darboux.
18.6 Local Equations of a Curve
We can “see” the shape of a curve more clearly in the neighborhood of a point x(t ) when we
0
consider its parametric equations with respect to the Frenet frame at the point. For simplicity,
we will assume that t = 0, and we may then write the curve as
0
x(t) = x(0) + x (t)T(0) + x (t)N(0) + x (t)b(0) .
3
1
2
On the other hand, using the Taylor series expansion of x(t) about the point t = 0, we obtain
t 2 t
x(t) = x(0) + tx'(0) + x"(0) + 3 x"'(0) + higher order terms .
2 6
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