Geometry pp 247-267 | Cite as

Curves, Envelopes, Evolutes

  • Michèle Audin
Part of the Universitext book series (UTX)


Curves appear in a wide variety of mathematical problems and in very many ways. Here is a list of examples (in a real affine space):
  • in kinematics, e.g., as integral curves of vector fields, in other words as solutions of differential equations;

  • a variant: as envelopes of families of lines, or of families of other curves;

  • they also appear, in space, at the intersection of two surfaces (e.g., of a surface and a plane), therefore, they are useful for the study of surfaces (see Chapter VIII);

  • as the solutions of an algebraic equation P( x, y) = 0 (as conics in the plane, for instance), or of several algebraic equations (P1 (x, y, z) = P2 (x, y, z) = 0 in space(1) of dimension 3, etc.).


Singular Point Parametrized Curve Plane Curve Curvature Center Perpendicular Bisector 
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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Michèle Audin
    • 1
  1. 1.IRMAUniversité Louis PasteurStrasbourgFrance

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