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 (geometry), focus for the special case of the squared eccentricity (geometry), 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 translation of axes, 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\ \\ &= \frac a^2\left, \int\, \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. ...
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