Tractrix

Tractriz or curve of the dog , is the curve that describes an object (located in B) that is dragged by another (located in A), which is kept at a constant distance d and moves in a straight line. The name of tractriz comes from the Latin tractus which means to pull, pull, drag. Its evolute is the catenary .

The tractrix is ​​also known as the equitangential curve, this denomination is due to the property that any segment tangent to it from the point of tangency to the asymptote of the curve has constant length.

Summary

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  • 1 Story
  • 2 equations
  • 3 Properties of the tractrix
  • 4 See also
  • 5 Sources

History

Huygens studies the curve in [[1690, he was the first to use the term catenary in a letter to Leibniz, and later it was studied by Leibniz , Johann Bernoulli , Joseph Liouville and Eugenio Beltrami the latter found an unsuspected application of it in the pseudosphere.

If we rotate a tractrix around its asymptote, we obtain a surface of revolution that has negative curvature (curves inward) at all its points. It is the pseudosphere.

Pseudosphere

Equations

The Cartesian equation of the tractrix is:

The generic equation of the tractrix in parametric equations is:

x = a ln (cotφ / 2 – cosφ)

y = a senφ

Tractrix properties

  • The x-axis is an asymptote of the curve.
  • In the trajectory any tangent segment from the point of tangency to the asymptote is of constant length.
  • The envelope of the normals of the tractrix, that is, the evolution of the tractrix is ​​the catenary

 

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