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PARABOLIC CONOID

A parabolic conoid is a conoid a directrix of which is a parabola; it is therefore the reunion of the lines supported by a line (D), a parabola (C) and parallel to a plane (P).

When the conoid is right ((D) and (P) perpendicular) and the axis of the parabola is parallel to (D), we get Whitney's umbrella.

When the conoid is right, and the axis of the parabola orthogonal to (D), we get the following surface, the name of which I am looking for.
Cartesian equation: .
Ruled cubic surface.

This surface is projectively (and really) equivalent to Whitney's umbrella, and therefore to Plücker's conoid .

When the parabola and the axis of the conoid intersect, the surface is a hyperbolic paraboloid.
 
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© Robert FERRÉOL 2017