Page 315 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Richa Nandra, Lovely Professional University

Notes Unit 25: Curvature

CONTENTS
Objectives
Introduction
25.1 Curvature of Plane Curves

25.1.1 Signed Curvature
25.1.2 Local Expressions
25.1.3 Curvature of a Graph
25.2 Curvature of Space Curves
25.2.1 Local Expressions
25.2.2 Curvature from Arc and Chord Length
25.3 Curves on Surfaces
25.4 Summary
25.5 Keywords
25.6 Self Assessment

25.7 Review Question
25.8 Further Readings

Objectives

After studying this unit, you will be able to:

Discuss the Curvature of plane curves

Explain the Curvature of a graph

Define Signed curvature

Describe Curvature of space curves

Explain Curves on surfaces

Introduction

In mathematics, curvature refers to any of a number of loosely related concepts in different areas
of geometry. Intuitively, curvature is the amount by which a geometric object deviates from
being flat, or straight in the case of a line, but this is defined in different ways depending on the
context. There is a key distinction between extrinsic curvature, which is defined for objects
embedded in another space (usually a Euclidean space) in a way that relates to the radius of
curvature of circles that touch the object, and intrinsic curvature, which is defined at each point
in a Riemannian manifold. This unit deals primarily with the first concept. The canonical example
of extrinsic curvature is that of a circle, which everywhere has curvature equal to the reciprocal
of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature
of a smooth curve is defined as the curvature of its osculating circle at each point.

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