Fish Curve

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The fish curve is a term coined in this work for the ellipse negative pedal curve with pedal point at the focus for the special case of the eccentricity . For an ellipse with parametric equations

			(1)
			(2)

the corresponding fish curve has parametric equations

			(3)
			(4)

The Cartesian equation is

(5)

which, when the origin is translated to the node, can be written

(6)

(Lockwood 1957).

The area of a curve is given by

			(7)
			(8)

so the areas of the tail and head are given by

			(9)
			(10)

giving an overall area for the fish as

(11)

(Lockwood 1957).

The arc length of the curve is given by

			(12)
			(13)
			(14)

(Lockwood 1957).

The curvature and tangential angle are given by

			(15)
			(16)

where is the complex argument.

The Tschirnhausen cubic, illustrated above, also resembles a fish, as does the trefoil curve.

SEE ALSO: Burleigh's Oval, Ellipse Negative Pedal Curve, Folium, Talbot's Curve, Trefoil Curve, Tschirnhausen Cubic

REFERENCES:

Lockwood, E. H. "Negative Pedal Curve of the Ellipse with Respect to a Focus." Math. Gaz. 41, 254-257, 1957.

Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 157, 1967.

CITE THIS AS:

Weisstein, Eric W. "Fish Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/FishCurve.html