For an ellipse with parametric equations
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(1)
|
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(2)
|
the negative pedal curve with respect to the origin has parametric equations
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(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)
|
where
 |
(9)
|
is the distance between the ellipse center and one of its foci and
 |
(10)
|
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)
|
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.
