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Local theory of curves

  • Marco Abate
  • Francesca Tovena
Part of the UNITEXT book series (UNITEXT)

Abstract

Elementary geometry gives a fairly accurate and well-established notion of what is a straight line, whereas is somewhat vague about curves in general. Intuitively, the difference between a straight line and a curve is that the former is, well, straight while the latter is curved. But is it possible to measure how curved a curve is, that is, how far it is from being straight? And what, exactly, is a curve? The main goal of this chapter is to answer these questions. After comparing in the first two sections advantages and disadvantages of several ways of giving a formal definition of a curve, in the third section we shall show how Differential Calculus enables us to accurately measure the curvature of a curve. For curves in space, we shall also measure the torsion of a curve, that is, how far a curve is from being contained in a plane, and we shall show how curvature and torsion completely describe a curve in space. Finally, in the supplementary material, we shall present (in Section 1.4) the local canonical shape of a curve; we shall prove a result (Whitney’s Theorem 1.1.7, in Section 1.5) useful to understand what cannot be the precise definition of a curve; we shall study (in Section 1.6) a particularly well-behaved type of curves, foreshadowing the definition of regular surface we shall see in  Chapter 3; and we shall discuss (in Section 1.7) how to deal with curves in ℝ n when n ≥ 4.

Keywords

Tangent Vector Parametrized Curve Plane Curve Rigid Motion Global Parametrization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag Italia 2012

Authors and Affiliations

  • Marco Abate
    • 1
  • Francesca Tovena
    • 2
  1. 1.Department of MathematicsUniversity of PisaItaly
  2. 2.Department of MathematicsUniversity of Rome Tor VergataItaly

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