Conoid

In geometry a conoid (Greek: κωνος cone and -ειδης similar) is a ruled surface, whose rulings (lines) fulfill the additional conditionshyperbolic paraboloidPlücker conoidWhitney umbrella 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 x ( u 0 , v ) {displaystyle mathbf {x} (u_{0},v)} with fixed parameter u = u 0 {displaystyle u=u_{0}} is a ruling, c ( u ) {displaystyle mathbf {c} (u)} describes the directrix and the vectors r ( u ) {displaystyle mathbf {r} (u)} are all parallel to the directrix plane. The planarity of the vectors r ( u ) {displaystyle mathbf {r} (u)} can be represented by The term conoid was already used by Archimedes in his treatise On conoids and spheroides.