Page 225 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes  If x is an immersed curve, with x’(t)  0 for all t in the domain, then we may define the unit
x'(t)
tangent vector T(t) to be . If the parameter is arclength, then the unit tangent vector
|x'(t)|
T(s) is given simply by x’(s). The line through x(t ) in the direction of T(t ) is called the
0
0
tangent line at x(t ). We can write this line as y(u) = x(t ) + uT(t ), where u is a parameter
0
0
0
that can take on all real values.
Since T(t) · T(t) = 1 for all t, we can differentiate both sides of this expression, and we obtain
2T’(t)· T(t) = 0. Therefore T’(t) is orthogonal to T(t). The curvature of the space curve x(t) is
T'(t)
defined by the condition (t) = , so = (t)s’(t) = |T’(t)|. If the parameter is arclength,
|x'(t)|
then x’(s) = T(s) and (s) = |T’(s)| = |x”(s)|.
The unit tangent vectors emanating from the origin form a curve T(t) on the unit sphere

called the tangential indicatrix of the curve x.
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.
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.
The total twist plays an important role in modern molecular biology, especially with

respect to the structure of DNA.
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).
18.11 Keywords

Curvature is one of the simplest and at the same time one of the most important properties of a
curve.
Fenchel’s Theorem: The total curvature of a closed space curve x is greater than or equal to 2.

(s)ds  2
Non-inflectional: A curve x is called non-inflectional if the curvature (t) is never zero. By our
earlier calculations, this condition is equivalent to the requirement that x’(t) and x”(t) are linearly
independent at every point x(t), i.e. x’(t) × x”(t)  0 for all t.

18.12 Self Assessment

1. If k(t) = 0 for all t, then the curve lies along a ....................
2. The unit tangent vectors emanating from the origin form a curve T(t) on the unit sphere
called the ................... of the curve x.
3. ................... The total curvature of a closed space curve x is greater than or equal
to 2.
(s)ds  2
4. A curve x is called ................... if the curvature k(t) is never zero. By our earlier calculations,
this condition is equivalent to the requirement that x’(t) and x”(t) are linearly independent
at every point x(t), i.e. x’(t) × x”(t)  0 for all t.

218 LOVELY PROFESSIONAL UNIVERSITY