SINUSOID

 Curve studied for the first time by Roberval in 1636, under the name "companion of the roulette".

 Cartesian equation: . Transcendental curve. Length on a period, given by an elliptic integral of the second kind: . In the case a = b: .

The sinusoid is the trajectory of a movement composed of a sinusoidal motion (i.e. the projection on a line of a uniform circular motion) and of a motion of uniform translation:

If the plane of the sinusoid is winded into a cylinder of revolution with generatrix Oy and radius nb, then we get the cylindric sine wave.
When = 1, it is an ellipse of eccentricity , and therefore, the expansion of a planar section of a cylinder of revolution is a sinusoid: concretely, the trace of the edge of a bevelled candle, rolling on a plane, is a sinusoid:

When n = 2, the cylindric sine wave is a pancake curve:
When n = 1/ 2, it is a Viviani curve.

The orthogonal projection of a circular helix on a plane parallel to its axis is a sinusoid.

See also the egg box, the revolution of the sinusoid, the sine surface.