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CATALAN SURFACE

Surface studied by Catalan
in 1855.
Eugene Charles Catalan (1814-1894): Franco-Belgian mathematician. Other name: ruled surface with directrix plane. |

Cartesian parametrization: ,
reunion of the lines D
passing through
and directed by
with .
_{u}Reduced Cartesian equation of Catalan surfaces with directrix plane xOy with only one line in each plane parallel to xOy:
(reunion of the lines ),
giving a conoid for . |

A Catalan surface is a ruled
surface the generatrices of which remain parallel to a fixed plane,
called the *directrix plane*, in other words, a ruled surface the
directrix cone of which is planar.

Examples: the cylinders, the conoids, the ruled helicoids with directrix plane.

The family of lines based on two given curves and parallel
to a given plane generates a surface of Catalan. Taking by

example *xOy* as plane, we get:

Cartesian parametrization : , meeting of lines going through and . |
Example of a Catalan surface based on two sinusoids. |

Do not mistake these for Catalan's
minimal surface.

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© Robert FERRÉOL 2019