The intersection of a plane and a cone is a conic section and rotating the plane leads to a family of conics. What happens to the foci of these conics as the plane rotates? A classical result gives the locus of the foci as an oblique strophoid when the plane rotates about a tangent to the cone. The analogous curve when the plane intersects a cylinder, in which case all the sections are ellipses, is a right strophoid. This article discusses both results and provides elementary geometric proofs. Rotation about a different axis, such as one meeting the axis of the cone or cylinder, gives a very different curve. We consider how the resulting curve relates to the classical one by analyzing the family of curves obtained as the axis of rotation moves.