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Notion studied in 1873 by abbé Aoust (p. 119), in 2010 by Apostol and Mamikon under the name of tanvolute.
Other names : oblique involute, generalized involute.

If M0  is the current point of , the current point M of an inverse evolutoid making an angle  with the tangents is the locus of points  where  is defined by the differential equation:  or  (see the notations).
Solution of this equation :  where 

The inverse evolutoids under an angle  of a plane curve are the curves  of which  is the evolutoid of angle . Those are
so the curves crossing with an angle  the tangents to the curve .
For , we find the involutes of .

Cartezian paramétrization :  with .

Animation for several values of k


Oblique involute of a circle under an angle of 45 °.

  The curve has an asymptote branch to the circle of radius , for , and
  a branch "approaching" a logarithmic spiral (one of the logarithmic spirals crossing at an angle  the lines from O).

Pour  proche de 90°, la branche pour t > 0 commence avec des spires peu ou prou équidistantes, comme pour une spirale d'Archimède, puis les spires s'écartent, comme pour une spirale logarithmique.

For  close to 90 °, the branch for  t > 0 starts with coils more or less equidistant, as for an Archimedes spiral, then the turns move apart, as for a logarithmic spiral.

One will find in the article or the book of Apostol/Mamikon, a study on the oblique involutes of cycloidal curves. Opposite, example of the cardioid.

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© Robert FERRÉOL  2020