Page 230 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 19: Serret-Frenet Formulae

Now, if (e (t), . . ., e (t)) is the distinguished Frenet frame, by construction, e (t) is a linear Notes
1 n i
'
combination of f (t), . . ., f (t), and thus e (t) is a linear combination of f (t), . . ., f (i+1) (t), hence, of
(2)
(i)
(1)
i
(e (t), . . ., e (t)).
1
i+1
In matrix form, when (e (t), . . ., e (t)) is the distinguished Frenet frame, the row vector
n
1
(e (t),...,e (t)) can be expressed in terms of the row vector (e (t), . . ., e (t)) via a skew symmetric
'
'
n
i
1
n
matrix , as shown below:
'
'
(e (t),...,e (t))   (e (t),...,e (t)) (t) ,

1
n
i
n
where
 0  12 
   
 12 0 23 
 =   23 0  
 
    n 1n 

 
  n 1n 0 

The next lemma shows the effect of a reparametrization and of a rigid motion.
Lemma 4. Let f: ]a, b[  E (or f: [a, b]  E ) be curve of class C , with p  n, so that the derivatives
p
n
n
f (t), . . ., f (n–1) (t) of f(t) are linearly independent for all t  ]a, b[ . Let h : E  E be a rigid motion,
n
(1)
n
and assume that the corresponding linear isometry is R. Let f = h o f. The following properties

hold:
(1) For any moving frame (e (t), . . ., e (t)), the n-tuple (e (t), . . . , e (t)), where e (t) = R(e (t)),



n
i
i
1
n
1
is a moving frame along ef, and we have
  ij (t) =  ij (t)) and  f'(t)  f'(t) .
(2) For any orientation-preserving diffeomorphism  : ]c, d[  ]a, b[ (i.e., ’(t) > 0 for all

t  ]c, d[ ), if we write f  f r, then for any moving frame (e (t), . . ., e (t)) on f, the n-tuple

n
1
(e (t), . . . , e (t)), where e (t) = e ((t)), is a moving frame on f.




i
n
1
i
More on Frenet . . .

Furthermore, if f'(t)  0, then

  ij (t)   ij ( (t)) .
f'(t) f'( (t))


The proof is straightforward and is omitted.
More on Frenet . . .
The above lemma suggests the definition of the curvatures  , . . .,  .
n–1
1
Lemma 5. Let f : ]a, b[  E (or f: [a, b]  E ) be a curve of class C , with p  n, so that the
n
p
n
derivatives f (t), . . ., f (n–1) (t) of f(t) are linearly independent for all t ]a, b[ . If (e (t), . . ., e (t)) is
(1)
1
n
the distinguished Frenet frame associated with f, we define the ith curvature,  , of f, by
i
ii 1
 (t) =  f'(t) (t) ,

i
with 1  i  n – 1.
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