thumb|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 <math>e^2=\tfrac{1}{2}</math>.{{r|lockwood}} The parametric equations for a fish curve correspond to those of the associated ellipse.
==Equations== For an ellipse with the parametric equations <math display="block">\textstyle {x=a\cos(t), \qquad y=\frac {a\sin(t)}{\sqrt {2}}},</math> the corresponding fish curve has parametric equations <math display="block">\textstyle {x=a\cos(t)-\frac {a\sin^2 (t)}{\sqrt 2}, \qquad y=a\cos(t)\sin(t)}.</math>
When the origin is translated to the node (the crossing point), the Cartesian equation can be written as:{{r|fish|book}} <math display="block">\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.</math>
== Properties == === Area === The area of a fish curve is given by: <math display="block"> \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} </math> so the area of the tail and head are given by: <math display="block"> \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} </math> giving the overall area for the fish as:{{r|fish}} <math display="block"> A = \frac {4}{3}a^2.</math>
=== Curvature, arc length, and tangential angle === The arc length of the curve is given by <math display="block"> a\sqrt {2}\left(\frac {1}{2}\pi+3\right). </math>
The curvature of a fish curve is given by: <math display="block"> 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}},</math> and the tangential angle is given by: <math display="block"> \phi(t)=\pi-\arg\left(\sqrt {2}-1-\frac {2}{\left(1+\sqrt {2}\right)e^{it} -1}\right), </math> where <math>\arg(z)</math> is the complex argument.
==References== {{Reflist|refs=
<ref name=book>{{cite book | last = Lockwood | first = E. H. | title = A Book of Curves | location = Cambridge, England | publisher = Cambridge University Press | page = 157 | year = 1967 }}</ref>
<ref name=fish>{{cite web | last = Weisstein | first = Eric W. | title = Fish Curve | publisher = MathWorld | url = http://mathworld.wolfram.com/FishCurve.html | accessdate = May 23, 2010 }}</ref>
<ref name=lockwood>{{cite journal | last = Lockwood | first = E. H. | title = Negative Pedal Curve of the Ellipse with Respect to a Focus | journal = Math. Gaz. | pages = 254–257 | volume = 41 | year = 1957 | doi = 10.1017/S0025557200037293 | s2cid = 125623811 }}</ref>
}}
Category:Quartic curves