On the Directrix, the Pedal Curve of a Parabola is a Strophoid (top left). On the foot of the Directrix, it is a Right Strophoid (top middle). On reflection of the Focus in the Directrix, it is a Maclaurin Trisectrix (top right). On the Vertex, it is a Cissoid of Diocles (bottom left). On the Focus, it is a straight line (bottom right).
Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 94-97, 1972.