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Fish curve

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Source updated September 11, 2026

A fish curve is an ellipse negative pedal curve that is shaped like a fish. In a fish curve, the pedal point is at the focus for the special case of the squared eccentricity $e^{2}={\tfrac {1}{2}}$ . The parametric equations for a fish curve correspond to those of the associated ellipse.

01Equations

For an ellipse with the parametric equations $\textstyle {x=a\cos(t),\qquad y={\frac {a\sin(t)}{\sqrt {2}}}},$ the corresponding fish curve has parametric equations $\textstyle {x=a\cos(t)-{\frac {a\sin ^{2}(t)}{\sqrt {2}}},\qquad y=a\cos(t)\sin(t)}.$

When the origin is translated to the node (the crossing point), the Cartesian equation can be written as: $\left(2x^{2}+y^{2}\right)^{2}-2{\sqrt {2}}ax\left(2x^{2}-3y^{2}\right)+2a^{2}\left(y^{2}-x^{2}\right)=0.$

02Properties

Area

The area of a fish curve is given by: ${\begin{aligned}A&={\frac {1}{2}}\left|\int {\left(xy'-yx'\right)dt}\right|\\&={\frac {1}{8}}a^{2}\left|\int {\left[3\cos(t)+\cos(3t)+2{\sqrt {2}}\sin ^{2}(t)\right]dt}\right|,\end{aligned}}$ so the area of the tail and head are given by: ${\begin{aligned}A_{\text{Tail}}&=\left({\frac {2}{3}}-{\frac {\pi }{4{\sqrt {2}}}}\right)a^{2},\\A_{\text{Head}}&=\left({\frac {2}{3}}+{\frac {\pi }{4{\sqrt {2}}}}\right)a^{2},\end{aligned}}$ giving the overall area for the fish as: $A={\frac {4}{3}}a^{2}.$

Curvature, arc length, and tangential angle

The arc length of the curve is given by $a{\sqrt {2}}\left({\frac {1}{2}}\pi +3\right).$

The curvature of a fish curve is given by: $K(t)={\frac {2{\sqrt {2}}+3\cos(t)-\cos(3t)}{2a\left[\cos ^{4}t+\sin ^{2}t+\sin ^{4}t+{\sqrt {2}}\sin(t)\sin(2t)\right]^{\frac {3}{2}}}},$ and the tangential angle is given by: $\phi (t)=\pi -\arg \left({\sqrt {2}}-1-{\frac {2}{\left(1+{\sqrt {2}}\right)e^{it}-1}}\right),$ where $\arg(z)$ is the complex argument.

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Sources and credits

This article is adapted from the Wikipedia article “Fish curve”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Fish curve.svg by Krishnavedala, CC BY-SA 3.0

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