Given a curve and a fixed point called the pedal point, then for a point on , draw a line perpendicular to . The envelope of these lines as describes the curve is the negative pedal of . It can be constructed by considering the perpendicular line segment for a curve parameterized by . Since one end of the perpendicular corresponds to the point , . Another end point can be found by taking the perpendicular to the line, giving
(1)
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or
(2)
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(3)
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Plugging into the two-point form of a line then gives
(4)
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or
(5)
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Solving the simultaneous equations and then gives the equations of the negative pedal curve as
(6)
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(7)
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If a curve is the pedal curve of a curve , then is the negative pedal curve of (Lawrence 1972, pp. 47-48).
The following table summarizes the negative pedal curves for some common curves.
curve | pedal point | negative pedal curve |
cardioid negative pedal curve | origin | circle |
cardioid negative pedal curve | point opposite cusp | cissoid of Diocles |
circle negative pedal curve | inside the circle | ellipse |
circle negative pedal curve | outside the circle | hyperbola |
ellipse negative pedal curve with | center | Talbot's curve |
ellipse negative pedal curve with | focus | ovoid |
ellipse negative pedal curve with | focus | two-cusped curve |
line | any point | parabola |
parabola negative pedal curve | origin | semicubical parabola |
parabola negative pedal curve | focus | Tschirnhausen cubic |