Page 207 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes differentiation in differential notation as we go along. For example, the statement

'
2
s'(t)   3 i 1 x (t) can be rewritten as
t

2 3 2
 ds   dx i  ,

    
 dt  i 1  dt 

and this may be expressed in the form
3
2
2
ds   dx ,
i
i 1

which has the advantage that it is independent of the parameter used to describe the curve. ds is
called the element of arc. It can be visualized as the distance between two neighboring points.
One of the most useful ways to parametrize a curve is by the arc length s itself. If we let s = s(t),
then we have
s’(t) = |x’(t)| = |x’(s)|s’(t),
from which it follows that |x’(s)| = 1 for all s. So the derivative of x with respect to arc length is
always a unit vector.
This parameter s is defined up to the transformation s  ±s + c, where c is a constant.
Geometrically, this means the freedom in the choice of initial point and direction in which to
traverse the curve in measuring the arc length.
Exercise 1: One of the most important space curves is the circular helix
x(t) = (a cos t, a sin t, bt),

where a  0 and b are constants. Find the length of this curve over the interval [0, 2].
Exercise 2: Find a constant c such that the helix
x(t) = (a cos(ct), a sin(ct), bt)
is parametrized by arclength, so that |x’(t)| = 1 for all t.

Exercise 3: The astroid is the curve defined by
x(t) = (a cos t, a sin t, 0),
3
3
on the domain [0, 2]. Find the points at which x(t) does not define an immersion, i.e., the points
for which x’(t) = 0.
Exercise 4: The trefoil curve is defined by
x(t) = ((a + b cos(3t)) cos(2t), (a + b cos(3t)) sin(2t), b sin(3t)),
where a and b are constants with a > b > 0 and 0  t  2. Sketch this curve, and give an argument
to show why it is knotted, i.e. why it cannot be deformed into a circle without intersecting itself
in the process.
Exercise 5: (For the serious mathematician) Two parametrized curves x(t) and y(u) are said to be
equivalent if there is a function u(t) such that u’(t) > 0 for all a < t < b and such that y(u(t)) = x(t).
Show that relation satisfies the following three properties:
1. Every curve x is equivalent to itself
2. If x is equivalent to y, then y is equivalent to x

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