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SPHERICAL SINUSOID

Curve studied by Chasles in 1875. |

Spherical equation: . Cartesian parametrization:
n > 0.
Algebraic curve when n is rational. |

The spherical sinusoids are the spherical curves for which the central projection of the center of the sphere on a cylinder circumscribed to the sphere is a cylindrical sine wave (which in turn develops onto a sinusoid). With the parameters given above, the equation of this sinusoid is , where . Therefore, they also are the intersections between a half sinusoidal cone and a sphere centered on its vertex.

The central projections with center *O* on the planes perpendicular to *Oz*, with equation *y = b*, are the epispirals: , where .

When *n* is an integer, we get a curve with *n* arches.

For *n* = 1, we get a great circle of the sphere (intersection between the sphere and the plane *z = kx*); the above parametrization provides a parametrization of the great circles of the sphere, except the meridians.

For n = 2, we get a curve that could be the seam line of a tennis ball, intersection between the sphere with the half-sinusoidal cone: . |

Compare to the satellite curves.

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© Robert FERRÉOL 2018