Page 238 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 20: Curves

Notes

Notes The relation  is a reparametrization of  is an equivalence relation. A curve is
an equivalence class of parametrized curves. Furthermore, if  is regular then every
reparametrization of  is also regular, so we may speak of regular curves.

n
Definition 3. Let : Ig ®  be a regular curve. For any compact interval [a, b]  I, the arclength
of  over [a, b] is given by:
L    a,b   a b  ' dt.

Note that if  is a reparametrization of  then  and  have the same length. More specifically, if
    ,  then
L [ (c), (d)]    L [c,d] .
 
'
Definition 4. Let  be a regular curve. We say that  is parametrized by arc length if   1
Note that this is equivalent to the condition that for all t  I = [a, b] we have:

L  [a,t]   a.

t
Furthermore, any regular curve can be parametrized by arclength. Indeed, if  is a regular curve,
then the function

t
s(t)    ' ,
a
is strictly monotone increasing. Thus, s(t) has an inverse function (s) function, satisfying:

d  1 .
ds ' 

It is now straightforward to check that     is parametrized by arclength.

20.2 Local Theory for Curves in  3

3
3
We will assume throughout this section that : I   is a regular curve in  parametrized by


arclength and that "  0. Note that '  " 0.

3
'
3
Definition 5. Let : I   be a curve in  . The unit vector T   is called the unit tangent of
-1
. The curvature  is the scalar    " . The unit vector N = k T’ is called the principal normal.
The binormal is the unit vector B = T × N. The positively oriented orthonormal frame (T, N, B)
is called the Frenet frame of .
It is not difficult to see that N’ + T is perpendicular to both T and N, hence, we can define the
torsion  of  by: N’ + T = B. Note that the torsion, unlike the curvature, is signed. Finally, it is
easy to check that B’ = –N. Let X denote the 3 × 3 matrix whose columns are (T, N, B). We will call
X also the Frenet frame of . Define the rotation matrix of :

 0  0

 :  0    ...(1)

 0 0 
  

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