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The pedal curve of the parabola with parametric equations
(1)
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(2)
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with pedal point is
(3)
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(4)
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On the conic section directrix, the pedal curve of a parabola is a strophoid (top left). On the foot of the conic section directrix, it is a right strophoid (top middle). On reflection of the focus in the conic section directrix, it is a Maclaurin trisectrix (top right). On the parabola vertex, it is a cissoid of Diocles (bottom left; Gray 1997, p. 119). On the focus, it is a straight line (bottom right; Hilbert and Cohn-Vossen 1999, pp. 26-27). On the symmetry axis for a parabola with , it is a conchoid of de Sluze (H. Smith, pers. comm., Aug. 4, 2004). The following table summarizes these special cases.
pedal point | pedal curve |
directrix | strophoid |
foot of directrix | right strophoid |
reflection of focus in directrix | Maclaurin trisectrix |
parabola vertex | cissoid of Diocles |
focus | line |
axis of a parabola with | conchoid of de Sluze |