Page 265 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Richa Nandra, Lovely Professional University

Notes Unit 22: Bertrand Curves

CONTENTS
Objectives

Introduction
22.1 Special Frenet Curves in E n
22.2 Bertrand Curves in E n
22.3 (1, 3)-Bertrand Curves in E 4
22.4 An Example of (1, 3)-Bertrand Curve

22.5 Summary
22.6 Keyword
22.7 Self Assessment

22.8 Review Questions
22.9 Further Readings

Objectives

After studying this unit, you will be able to:

Discuss Special Frenet Curves in E n

Describe Bertrand Curves in E n

Explain (1, 3)-Bertrand Curves in E 4

Introduction

We denote by E a 3-dimensional Euclidean space. Let C be a regular C - curve in E , that is, a
3
3

C -mapping c : L  E (s  c(s)). Here L  R is some interval, and s ( L) is the arc-length
3

parameter of C. Following Wong and Lai [7], we call a curve C a C -special Frenet curve if there

exist three C -vector fields, that is, the unit tangent vector field t, the unit principal normal

vector field n, the unit binormal vector field b, and two C -scalar functions, that is, the curvature

function k(> 0), the torsion function T( 0). The three vector fields t, n and b satisfy the Frenet
equations. A C -special Frenet curve C is called a Bertrand curve if there exist another C -special


Frenet curve C and a C -mapping  : C  C such that the principal normal line of C at c(s) is

collinear to the principal normal vector n(s). It is a well-known result that a C -special Frenet

curve C in E is a Bertrand curve if and only if its curvature function K and torsion function T
3
satisfy the condition aK(s) + bT(s) = 1 for all s  L, where a and b are constant real numbers.
n


In an n-dimensional Euclidean space E , let C be a regular C -curve, that is a C -mapping c : L 
E (s  c(s)), where s is the arc-length parameter of C. Then we can define a C -special Frenet
n

curve C. That is, we define t(s) = c’(s), n (s) = (1/||c”(s)||). c”(s), and we inductively define n (s)
1
k
(k = 2, 3, ..., n – 1) by the higher order derivatives of c (see next section, in detail). The n vector
fields t, n , ..., n along C satisfy the Frenet equations with positive curvature functions k , ...,
1
n-1
1
258 LOVELY PROFESSIONAL UNIVERSITY