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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.

Cartesian equation : .
Ruled cubic surface. |
Cas |
b= 0 |

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.

Picture from this
website.

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© Robert FERRÉOL
2020