Smooth curve

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smooth curve
piecewise smooth curve

A smooth curve is a continuously differentiable parameterized curve (here path ) with a non-vanishing derivative . This clearly means that the path does not stop at any point when running through the parameter or that the direction changes abruptly.

A curve in general is smooth if at least one smooth path has the curve to the image.

In contrast to these definitions, a smooth mapping must be infinitely differentiable.

Formal definition

Let be a curve in the parametric representation

  • is called smooth if for are continuously differentiable and holds for all .
  • is called piecewise smooth if there is a partition of such that on every interval for is smooth.


  • Kurt Endl, Wolfgang Luh : Analysis. An integrated representation; Study book for students of mathematics, physics and other natural sciences from the 1st semester. Volume 2. 7th revised edition. Aula-Verlag, Wiesbaden 1989, ISBN 3-89104-455-0 , p. 343.
  • Harro Heuser : Textbook of Analysis. Part 2. 5th revised edition. Teubner, Stuttgart 1990, ISBN 3-519-42222-0 , p. 365.

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