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 with fixed parameter is a ruling, describes the directrix and the vectors are all parallel to the directrix plane. The planarity of the vectors can be represented by