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CYLINDRICAL SINE WAVE

Homemade name. |

Cartesian parametrization:
n > 0.
Cylindrical equation: ; in the following, we take a = b.
Curvilinear abscissa: . Radius of curvature: , radius of torsion: . |

The cylindrical sine waves are the coiling of a sinusoid around a cylinder (in other words, if we make a cylindrical sine wave roll on a plane, we get a sinusoid).

They are special cases of 3D Lissajous curves.

The projections on the planes containing *Oz* are the planar Lissajous curves, with parameter *n* if *n* > 1, 1/*n*
otherwise.

For *n* = 1, we get an ellipse, for *n*
= 1/2, Viviani's curve and for
*n*
= 2, the pancake curve.

For integral values of *n*, the number of arches is equal to *n*. The curve with three arches is used to represent the Borromean rings.

When we apply a horizontal projection onto the sphere with center O and radius a, the cylindrical sine wave becomes the clelia: . |
Figure made by Alain Esculier |

See also the spherical sinusoids, the 3D basins, the cylindrical tangent waves and the sine tori.

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© Robert FERRÉOL,
Jacques MANDONNET 2018