INVERSE EVOLUTOID OF A PLANE CURVE

 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 .

EXAMPLE :  INVERSE EVOLUTOID OF CERCLE

 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.