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Unit 22: Bertrand Curves

k of C and positive or negative curvature function k of C. We call n the Frenet j-normal Notes
j
n-1
n-2
vector field along C, and the Frenet j-normal line of C at c(s) is a line generated by n(s) through
j
c(s) (j = 1, 2, ..., n – 1). The Frenet (j, k)-normal plane of C at c(s) is a plane spanned by n(s) and n (s)
k
j
through c(s) (j, k = 1, 2, ... , n – 1; j  k). A C -special Frenet curve C is called a Bertrand curve if

there exist another C -special Frenet C and a C -mapping  : C  C such that the Frenet 1-


normal lines of C and C at corresponding points coincide. Then we obtain
Theorem A. If n  4, then no C -special Frenet curve in E is a Bertrand curve.
n

This is claimed with different viewpoint, thus we prove the above Theorem.
We will show an idea of generalized Bertrand curve in E . A C -special Frenet curve C in E is
4
4

called a (1, 3)-Bertrand curve if there exist another C -special Frenet curve C and a C -mapping


 : C  C such that the Frenet (1, 3)-normal planes of C and C at corresponding points coincide.
Then we obtain
Theorem B. Let C be a C -special Frenet curve in E with curvature functions k , k , k . Then C is
4

1
2
3
a (1, 3)-Bertrand curve if and only if there exist constant real numbers , , ,  satisfying
k (s) – k (s)  0 (a)
2 3
k (s) + {k (s) – k (s)} = 1 (b)
3
2
1
k (s) – k (s) = k (s) (c)
1
3
2
( – 1)k (s)k (s) + {(k (s)) – (k (s)) – (k (s)) }  0 (d)
2
2
2
2
3
2
1
1
2
for all s  L.
This Theorem is proved in subsequent section.
We remark that if the Frenet j-normal vector fields of C and C are not vector fields of same
meaning then we can not consider coincidence of the Frenet 1-normal lines or the Frenet (1, 3)-
normal planes of C and C . Then we consider only special Frenet curves.
Give an example of (1, 3)-Bertrand curve.
We shall work in C -category.

22.1 Special Frenet Curves in E n
Let E be an n-dimensional Euclidean space with Cartesian coordinates (x , x ,..., x ). By a
n
2
n
1
parametrized curve C of class C , we mean a mapping c of a certain interval I into E given by
n

1
 x (t)
 x (t) 
2
c(t)     t I.

 : 
 n 
  x (t) 
1
If dc(t)  dc(t) dc(t) 2  0 for all t  I, then C is called a regular curve in En. Here .,.
,
dt dt dt
denotes the Euclidean inner product on E .
n
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