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Medians, Altitudes, Bisectors: Praxis I Geometry

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Parmananda Gupta Laxmi Publications PDF

2

Curvature and Torsion

1. INTRODUCTION

For curves in space, the concepts of curvature and torsion are of fundamental importance.

We know that line segments are uniquely determined by their lengths, circles by their radii, triangles by side-angle-side etc. In geometry, we look for geometric quantities which distinguish one figure from another. The importance of curvature and torsion can easily be estimated from the fact that it can be proved that a curve is uniquely determined (except for its position in space) if its curvature and torsion are given as continuous functions of arc length ‘s’.

2. CURVATURE OF A CURVE

Let r = r(s) be a regular curve C of class Cm(m ≥ 2), where s is the parameter ‘arc length’.

The vector r″(s) is called the curvature vector on the curve C at the point r(s) and it is denoted by t(s)

κ(s) (or by κ). The magnitude of the curvature vector r(s) is called the curvature of the curve C at the point r(s) and it is denoted by κ(s) (or by κ).

C k(s)

κ(s) = |r″(s)|

Also t(s) = r′(s), so we have

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