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ZINDLER'S CONOID

Konrad Zindler (1866 -1934): Austrian mathematician. |

Cylindrical equation: .
Cartesian parametrization: (). Cartesian equation: . Rational ruled cubic surface with double line Oz. |

Zindler's conoid is the right conoid
with directrix a cylindrical
tangent wave with 4 branches (here, the wave )
and axis the axis of this wave.

Zindler's conoid is also the right conoid with directrix
a cubic of the type
located in a plane parallel to its axis.

Any cylinder with generatrix *Oz* and directrix a
rectangular hyperbola perpendicular to the axis cuts Zindler's conoid at
hyperbola (in red) plus the axis *Oz* (in blue).

The complex scaling transforms it into Plücker's conoid, but these two conoids are images of one another by a real homography.

this can be generalized to any cylindrical tangent wave, which gives the cylindrical equation: ;

The case n = 1 gives the rectangular hyperbolic
paraboloid . |

Case *n* = 3

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© Robert FERRÉOL L.G. VIDIANI 2017