Fish curve

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 $e^2=\frac{1}{2}$. The parametric equations for a fish curve correspond to those of the associated ellipse.

Equations

For an ellipse with the parametric equations

$x=a\cos (t),y=\frac{a\sin (t)}{\sqrt{2}},$

the corresponding fish curve has parametric equations

$x=a\cos (t)-\frac{a{\sin }^2(t)}{\sqrt{2}},y=a\cos (t)\sin (t).$

When the origin is translated to the node (the crossing point), the Cartesian equation can be written as:

${(2x^2+y^2)}^2-2\sqrt{2}ax(2x^2-3y^2)+2a^2(y^2-x^2)=0.$

Area

The area of a fish curve is given by:

$A=\frac{1}{2}|\int (x{y}^{\prime }-y{x}^{\prime })dt|$

$=\frac{1}{8}{a}^2|\int [3\cos (t)+\cos (3t)+2\sqrt{2}{\sin }^2(t)]dt|$,

so the area of the tail and head are given by:

$A_{\mathrm{T}\mathrm{a}\mathrm{i}\mathrm{l}}=(\frac{2}{3}-\frac{\pi }{4\sqrt{2}})a^2$

$A_{\mathrm{H}\mathrm{e}\mathrm{a}\mathrm{d}}=(\frac{2}{3}+\frac{\pi }{4\sqrt{2}})a^2$

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 $a\sqrt{2}(\frac{1}{2}\pi +3)$.

The curvature of a fish curve is given by:

$K(t)=\frac{2\sqrt{2}+3\cos (t)-\cos (3t)}{2a{[{\cos }^{4}t+{\sin }^{2}t+{\sin }^{4}t+\sqrt{2}\sin (t)\sin (2t)]}^{\frac{3}{2}}}$,

and the tangential angle is given by:

$\varphi (t)=\pi -\arg (\sqrt{2}-1-\frac{2}{(1+\sqrt{2})e^{it}-1})$

where $\arg (z)$ is the complex argument.