Fish Curve
The fish curve is a term coined in this work for the ellipse negative pedal curve with pedal point at the focus for the special case of the eccentricity e^2=1/2. For an ellipse with parametric equations
the corresponding fish curve has parametric equations
The Cartesian equation is
| -2a^4sqrt(2)a^3x-2a^2(x^2-5y^2)+(2x^2+y^2)^2+2sqrt(2)ax(2x^2+5y^2)=0 |
(5)
|
which, when the origin is translated to the node, can be written
| (2x^2+y^2)^2-2sqrt(2)ax(2x^2-3y^2)+2a^2(y^2-x^2)=0 |
(6)
|
(Lockwood 1957).
The interior of the curve is not consistently oriented in the above parametrization, with the fish's head being on the left of the curve and the tail on the right as the curve is traversed. Treating the two pieces separately then gives the areas of the tail and head as
giving an overall area for the fish as
| A=4/3a^2 |
(9)
|
(Lockwood 1957).
The arc length of the curve is given by
(Lockwood 1957).
The curvature and tangential angle are given by
![画像:(2sqrt(2)+3cost-cos(3t))/(2a[cos^4t+sin^2t+sin^4t+sqrt(2)sintsin(2t)]^(3/2))](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/FishCurve/Inline31.svg){kind=link}
where arg(z) is the complex argument.
The Tschirnhausen cubic, illustrated above, also resembles a fish, as does the trefoil curve.
See also
Burleigh's Oval, Ellipse Negative Pedal Curve, Folium, Talbot's Curve, Trefoil Curve, Tschirnhausen CubicExplore with Wolfram|Alpha
More things to try:
References
Lockwood, E. H. "Negative Pedal Curve of the Ellipse with Respect to a Focus." Math. Gaz. 41, 254-257, 1957.Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 157, 1967.Referenced on Wolfram|Alpha
Fish CurveCite this as:
Weisstein, Eric W. "Fish Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FishCurve.html