Page 237 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Richa Nandra, Lovely Professional University

Notes Unit 20: Curves

CONTENTS
Objectives

Introduction
20.1 Preliminaries
20.2 Local Theory for Curves in  3
20.3 Plane Curves
20.3.1 Local Theory

20.3.2 Global Theory
20.3.3 Convexity
20.4 Fenchel’s Theorem

20.5 Summary
20.6 Keywords
20.7 Self Assessment
20.8 Review Questions

20.9 Further Readings

Objectives

After studying this unit, you will be able to:

Define Preliminaries

Explain Plane Curves

Identify Fenchel’s Theorem

Introduction

In last unit, you have studied about serret-frenet formula. In this unit, you will read about
curves.

20.1 Preliminaries

.
n

Definition 1. A parametrized curve is a smooth (C ) function g : I ®  A curve is regular if
' g ¹ 0.
When the interval I is closed, we say that  is C on I if there is an interval J and a C function 


on J which agrees with  on I.
n
n
Definition 2. Let : Ig ®  be a parametrized curve, and let : Jb ®  be another parametrized
curve. We say that  is a reparametrization (orientation preserving reparametrization) of  if
there is a smooth map : Jt ® I with ' 0t > such that b = g t .

230 LOVELY PROFESSIONAL UNIVERSITY