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


Tangential glissette of a sinusoid

 Cartesian parameterization for a glissette of X, Y coordinates in the Frenet coordinate system, the starting curve being parameterized by.

The tangential glissettes of a curve (C) are the traces in the fixed plane of the points of a plane related to a sliding tangent on the curve
(C); see glissette (case # 1 dual). In other words, these are the trajectories of the points having fixed coordinates in the Frenet coordinate system moving along the curve.

The normal line to the curve rolling without sliding on the evolute, the tangential glissettes are the roulettes of the movement "plane over plane" of base curve the evolute and rolling curve the normal line.

When the tracer point is on the tangent, the glissettes are the equitangential curves associated with (C), and when it is on the normal line, the glissettes are none other than the curves parallel to (C).

Cartesian paramétrization: .

Exemple of the ellipse :
Cartesian paramétrization:(biquartic).
Note that according to Holditch's theorem, the area swept by the tangent segment is .

The starting curve is a tractory of its equitangential curves.

So, more concretely, the equitangential curves are the curves of the front wheels of bicycles which one imposes on the rear wheel to follow the starting curve.

Below, case of the cycloïde and Bernoulli lemniscate.

See the syntractrixes, equitangential curves of the tractrix.
next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

© Robert FERRÉOL 2019