title: Fish curve
text: 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 . The parametric equations for a fish curve correspond to those of the associated ellipse.
  
  Equations
  For an ellipse with the parametric equations
  
  the corresponding fish curve has parametric equations
  
  When the origin is translated to the node (the crossing point), the Cartesian equation can be written as:
  
  Area
  The area of a fish curve is given by:
  
  ,
  
  so the area of the tail and head are given by:
  
  giving the overall area for the fish as: 
  
  .
  
  Curvature, arc length, and tangential angle
  The arc length of the curve is given by .
  
  The curvature of a fish curve is given by:
  
  ,
  
  and the tangential angle is given by:
  
  where  is the complex argument.
  
  References
  
  Plane curves