Fish curve explained

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 $\textstyle,$ the corresponding fish curve has parametric equations $\textstyle .$

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

Properties

Area

The area of a fish curve is given by: $\begin A &= \frac \left|\int\right| \\ &= \frac a^2\left|\int\right|,\end$ so the area of the tail and head are given by: $\begin A_ &= \left(\frac -\frac \right)a^2, \\ A_ &= \left(\frac +\frac \right)a^2, \end$ giving the overall area for the fish as: $A = \frac a^2.$

Curvature, arc length, and tangential angle

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

The curvature of a fish curve is given by: $K(t) = \frac,$ and the tangential angle is given by: $\phi(t)=\pi-\arg\left(\sqrt -1-\frac \right),$ where

\arg(z)

is the complex argument