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 $e^2=\tfrac$. 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:
:$\left(2x^2+y^2\right)^2-2 \sqrt ax\left(2x^2-3y^2\right)+2a^2\left(y^2-x^2\right)=0.$

Area

The area of a fish curve is given by:

:$A=\frac \left, \int\$

:$=\frac a^2\left, \int\$,

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

:$A_=\left(\frac -\frac \right)a^2$

:$A_=\left(\frac +\frac \right)a^2$

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.

References

{{Reflist

Plane curves