Page 268 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 22: Bertrand Curves

And Notes

dn (s)
(C n 1 ) : k n 1 (s)  n 2 s ,n n 1 (s)  0 for all s L,


d



where the unit vector field n along C is determined by the fact that the frame {t, n , ..., n } is
n–1
n–1
1
of orthonormal and of positive orientation. We remark that the functions k , ..., k are of
1
n–2
positive and the function k is of non-zero. Such a curve C is called a special Frenet curve in E .
n
n–1
The term “special” means that the vector field n i+1 is inductively defined by the vector fields n i
and n and the positive functions k and k . Each function k is called the i-curvature function of
i
i–1
i
i–1
C (i = 1, 2, ... , n – 1). The orthonormal frame {t, n , ..., n } along C is called the special Frenet
1
n–1
frame along C.
Thus, we obtain the Frenet equations
dt(s)
 k (s) . n (s)
ds
1
1
dn (s)
1   k (s) . t(s) k (s) . n (s)

ds 1 2 2
. . .
dn (s)
   k (s) . n 1  k 1 (s) . n 1 (s)
ds       
. . .
dn (s)
n 2

ds   k n 2 (s) . n n 3 (s) k n 1 (s) . n n 1 (s)





dn (s)
n 1   k n 1 (s) . n n 2 (s)

ds


for all s  L. And, for j = 1, 2, ... , n – 1, the unit vector field n along C is called the Frenet j-normal
j
vector field along C. A straight line is called the Frenet j-normal line of C at c(s) (j = 1, 2, ... n – 1
and s  L), if it passes through the point c(s) and is collinear to the j-normal vector n(s) of C at c(s).
j
Remark. In the case of Euclidean 3-space, the Frenet 1-normal vector fields n1 is already called
the principal normal vector field along C, and the Frenet 1-normal line is already called the
principal normal line of C at c(s).
For each point c(s) of C, a plane through the point c(s) is called the Frenet (j, k)-normal plane of
C at c(s) if it is spanned by the two vectors n(s) and n (s) (j, k = 1, 2, ... , n – 1; j < k).
k
j
Remark. In the case of Euclidean 3-space, 1-curvature function k is called the curvature of C,
1
2-curvature function k is called the torsion of C, and (1, 2)-normal plane is already called the
2
normal plane of C at c(s).
22.2 Bertrand Curves in E n
A C -special Frenet curve C in E (c : L  E ) is called a Bertrand curve if there exist a C -special
n
n


d (s)

n

Frenet curve C (c : L  E ), distinct from C, and a regular C -map  : L  L (s   (s),  0
ds
for all s  L) such that curves C and C have the same 1-normal line at each pair of corresponding
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