Page 212 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 18: Theory of Space Curves

Since the angle (b) at the end of the closed curve must coincide with the angle (a) at the Notes
beginning, up to an integer multiple of 2, it follows that the real numbers (x, z) and (x, z)
differ by an integer. Therefore, the fractional part of (x, z) depends only on the curve x and not
on the unit normal vector field used to define it. This common value (x) is called the total twist
of the curve x. It is a global invariant of the curve.
Proposition 3. If a closed curve lies on a sphere, then its total twist is zero.

Proof. If x lies on the surface of a sphere of radius r centered at the origin, then |x(t)| = x(t) . x(t)
2
x(t)
= r for all t. Thus, x’(t) . x(t) = 0 for all t, so x(t) is a normal vector at x(t). Therefore, z(t) = is
2
r
a unit normal vector field defined along x, and we may compute the total twist by evaluating
1
(x, z) =   [z'(t), z(t),T(t)]dt .
2
But

x'(t) x(t)
[z’(t), z(t),T(t)] = [ , ,T(t)] 0

r r
for all t since x’(t) is a multiple of T(t). In differential form notation, we get the same result:
1
[dz, z,T] = [x'(t),x(t),T(t)]dt  0. Therefore, (x, z) = 0, so the total twist of the curve x is zero.
r 2
Remark 1. W. Scherrer proved that this property characterized a sphere, i.e. if the total twist of
every curve on a closed surface is zero, then the surface is a sphere.
Remark 2. T. Banchoff and J. White proved that the total twist of a closed curve is invariant under
inversion with respect to a sphere with center not lying on the curve.
Remark 3. The total twist plays an important role in modern molecular biology, especially with
respect to the structure of DNA.

Exercise 10: Let x be the circle x(t) = (r cos(t), r sin(t), 0), where r is a constant > 1. Describe the
collection of points x(t) + z(t) where z(t) is a unit normal vector at x(t).
Exercise 11: Let  be the sphere of radius r > 0 about the origin. The inversion through the sphere

S maps a point x to the point x = r 2 x . Note that this mapping is not defined if x = 0, the center
|x| 2
of the sphere. Prove that the coordinates of the inversion of x = (x , x , x ) through S are given by
1
2
3
r x
2
x  i . Prove also that inversion preserves point that lie on the sphere S itself, and that
i
2
2
x  x  x 2 3
1
2
the image of a plane is a sphere through the origin, except for the origin itself.
Exercise 12: Prove that the total twist of a closed curve not passing through the origin is the same
as the total twist of its image by inversion through the sphere S of radius r centered at the origin.
18.4 Moving Frames
In the previous section, we introduced the notion of a frame in the unit normal bundle of a space
curve. We now consider a slightly more general notion. By a frame, or more precisely a right-
handed rectangular frame with origin, we mean a point x and a triple of mutually orthogonal
unit vectors E , E , E forming a right-handed system. The point x is called the origin of the frame.
1
3
2
Note that E . E = 1 if i = j and 0 if i  j.
i
j
LOVELY PROFESSIONAL UNIVERSITY 205