Page 205 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Sachin Kaushal, Lovely Professional University

Notes Unit 18: Theory of Space Curves

CONTENTS
Objectives
Introduction
18.1 Arc Length
18.2 Curvature and Fenchel’s Theorem
18.3 The Unit Normal Bundle and Total Twist
18.4 Moving Frames

18.5 Curves at a Non-inflexional Point and the Frenet Formulas
18.6 Local Equations of a Curve
18.7 Plane Curves and a Theorem on Turning Tangents
18.8 Plane Convex Curves and the Four Vertex Theorem
18.9 Isoperimetric Inequality in the Plane
18.10 Summary
18.11 Keywords
18.12 Self Assessment
18.13 Review Questions
18.14 Further Readings

Objectives

After studying this unit, you will be able to:
Define Arc Length

Discuss Curvature and Fenchel’s Theorem

Explain The Unit Normal Bundle and Total Twist

Define Moving Frames

Describe Curves at a Non-inflexional Point and the Frenet Formulas

Explain Plane Convex Curves and the Four Vertex Theorem

Introduction

In the last unit, you have studied about Curvilinear coordinates. These coordinates may be
derived from a set of Cartesian coordinates by using a transformation that is locally invertible
(a one-to-one map) at each point. The term curve has several meanings in non-mathematical
language as well. For example, it can be almost synonymous with mathematical function or
graph of a function. An arc or segment of a curve is a part of a curve that is bounded by two
distinct end points and contains every point on the curve between its end points. 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.

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