next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

INVOLUTE OF A PLANE CURVE

Notion studied by Fontenelle in 1712 (who gave it its name), then by Cesaro in 1888.

 
If M0 is the current point on (G0), the current point M of an involute is the locus of the points , which yields:
Cartesian parametrization: .
Complex parametrization: .
Radius of curvature: .

The involutes of a plane curve (G0) are the curves traced by the end of a wire tightened along (G0) and winding itself along (G0). In other words, they are the traces on the plane of a point on a line pivoting without slipping on (G0) (therefore, they are special cases of roulettes).

Thus, they also are the orthogonal trajectories of the family of the tangents to the curve, or also the curves for which the initial curve is the evolute.

The involutes of a curve are parallel to one another.

It will be noticed (cf. figure above) that an inflexion point on the initial curve yields a cuspidal point of the second kind on the involute, as the Marquis de l'Hospital had already understood in 1696 (see this text).

Examples: (see also at evolute)
 
initial curve origin involute, or one of them
circle any point involute of a circle!
point any point circle
cycloid vertex the same cycloid, translated
cardioid vertex similar cardioid with ratio 3
nephroid vertex similar nephroid with ratio 2
nephroid cuspidal point Cayley sextic
epicycloid with parameter q vertex similar epicycloid with ratio (q + 2)/q
deltoid vertex similar deltoid with ratio 1/3
astroid vertex similar astroid with ratio 1/2, Maltese cross
hypocycloid with parameter q vertex similar hypocycloid with ratio (q-2)/q
logarithmic spiral centre logarithmic spiral 
catenary vertex tractrix
logarithmic any point involute of an exponential
involute of a circle cuspidal point Norwich spiral

See the generalisation to 3D curves.
 
next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

© Robert FERRÉOL 2017