Fish curve

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

[1] The parametric equations for a fish curve correspond to those of the associated ellipse.

For an ellipse with the parametric equations

x = a cos ⁡ ( t ) ,

a sin ⁡ ( t )

the corresponding fish curve has parametric equations

x = a cos ⁡ ( t ) −

sin

y = a cos ⁡ ( t ) sin ⁡ ( t )

When the origin is translated to the node (the crossing point), the Cartesian equation can be written as:[2][3]

The area of a fish curve is given by:

3 cos ⁡ ( t ) + cos ⁡ ( 3 t ) + 2

sin

{\displaystyle {\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:

{\displaystyle {\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:[2]

The arc length of the curve is given by

The curvature of a fish curve is given by:

+ 3 cos ⁡ ( t ) − cos ⁡ ( 3 t )

cos

sin

sin ⁡ ( t ) sin ⁡ ( 2 t )

and the tangential angle is given by:

ϕ ( t ) = π − arg ⁡

arg ⁡ ( z )

is the complex argument.