For an ellipse with parametric equations
(1)
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(2)
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the negative pedal curve with respect to the origin has parametric equations
(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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where
(9)
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is the distance between the ellipse center and one of its foci and
(10)
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is the eccentricity. For , the base curve is a circle, whose negative pedal curve with respect to the origin is also a circle. For , the curve becomes a "squashed" ellipse. For , the curve has four cusps and two ordinary double points and is known as Talbot's curve (Lockwood 1967, p. 157).
Taking the pedal point at a focus (i.e., ) gives the negative pedal curve
(11)
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(12)
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Lockwood (1957) terms this family of curves Burleigh's ovals. As a function of the aspect ratio of an ellipse, the neagtive pedal curve varies in shape from a circle (at ) to an ovoid (for ) to a fish-shaped curve with a node and two cusps to a line plus a loop to a line plus a cusp.
The special case of the negative pedal curve for pedal point and (i.e., ) is here dubbed the fish curve.