Fish curve
title: "Fish curve" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["quartic-curves"] topic_path: "general/quartic-curves" source: "https://en.wikipedia.org/wiki/Fish_curve" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::figure[src="https://upload.wikimedia.org/wikipedia/commons/c/ce/Fish_curve.svg" caption="The fish curve with scale parameter ''a'' = 1"] ::
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.
Equations
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.
Properties
Area
The area of a fish curve is given by: \begin{align} 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{align} so the area of the tail and head are given by: \begin{align} 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{align} 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.
References
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