Conoid

In geometry a conoid (Greek: κωνος cone and -ειδης similar) is a ruled surface, whose rulings (lines) fulfill the additional conditions

Because of (1) any conoid is a Catalan surface and can be represented parametrically by

Any curve ${\displaystyle \mathbf {x} (u_{0},v)}$ with fixed parameter ${\displaystyle u=u_{0}}$ is a ruling, ${\displaystyle \mathbf {c} (u)}$ describes the directrix and the vectors ${\displaystyle \mathbf {r} (u)}$ are all parallel to the directrix plane. The planarity of the vectors ${\displaystyle \mathbf {r} (u)}$ can be represented by

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