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Richa Nandra, Lovely Professional University Unit 19: Serret-Frenet Formulae

Unit 19: Serret-Frenet Formulae Notes

CONTENTS
Objectives

Introduction
19.1 Serret-Frenet Formulae
19.2 Summary
19.3 Keywords
19.4 Self Assessment

19.5 Review Questions
19.6 Further Readings

Objectives

After studying this unit, you will be able to:
Define serret-frenet formula

Explain serret frenet formula

Introduction

In the last unit, you have studied about space theory of curve. Depending on how the arc is
defined, either of the two end points may or may not be part of it. When the arc is straight, it is
typically called a line segment. The derivatives of the vectors t, p, and b can be expressed as a
linear combination of these vectors. The formulae for these expressions are called the Frenet-
Serret Formulae. This is natural because t, p, and b form an orthogonal basis for a three-
dimensional vector space.

19.1 Serret-Frenet Formulae

Given a curve f: ]a, b[  E (or f: [a, b]  E ) of class C , with p  n, it is interesting to consider
n
p
n
families (e (t), . . ., e (t)) of orthonormal frames. Moreover, if for every k, with 1  k  n, the kth
1
n
derivative f (t) of the curve f(t) is a linear combination of (e (t), . . ., e (t)) for every t ]a, b[, then
(k)
k
1
such a frame plays the role of a generalized Frenet frame. This leads to the following definition:
Lemma 1. Let f: ]a, b[  E (or f : [a, b]  E ) be a curve of class C , with p  n. A family (e (t), . .
p
n
n
1
., e (t)) of orthonormal frames, where each e : ]a, b[  E is C continuous for i = 1, . . ., n – 1 and
n
n–i
n
i
en is C -continuous, is called a moving frame along f. Furthermore, a moving frame (e (t), . . .,
1
1
e (t)) along f so that for every k, with 1  k  n, the kth derivative f (t) of f(t) is a linear
(k)
n
combination of (e (t), . . ., e (t)) for every t  ]a, b[, is called a Frenet n-frame or Frenet frame.
1
k
If (e (t), . . ., e (t)) is a moving frame, then
n
1
.
e (t) e(t) =  for all i, j, 1  i, j  n.
i j ij
LOVELY PROFESSIONAL UNIVERSITY 221