Fish curve
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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 = 1 2 {\displaystyle e^{2}={\tfrac {1}{2}}}. The parametric equations for a fish curve correspond to those of the associated ellipse.
Equations
For an ellipse with the parametric equations x = a cos ( t ) , y = a sin ( t ) 2 , {\displaystyle \textstyle {x=a\cos(t),\qquad y={\frac {a\sin(t)}{\sqrt {2}}}},} the corresponding fish curve has parametric equations x = a cos ( t ) − a sin 2 ( t ) 2 , y = a cos ( t ) sin ( t ) . {\displaystyle \textstyle {x=a\cos(t)-{\frac {a\sin ^{2}(t)}{\sqrt {2}}},\qquad y=a\cos(t)\sin(t)}.}
When the origin is translated to the node (the crossing point), the Cartesian equation can be written as: ( 2 x 2 + y 2 ) 2 − 2 2 a x ( 2 x 2 − 3 y 2 ) + 2 a 2 ( y 2 − x 2 ) = 0. {\displaystyle \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.}
Properties
Area
The area of a fish curve is given by: A = 1 2 | ∫ ( x y ′ − y x ′ ) d t | = 1 8 a 2 | ∫ [ 3 cos ( t ) + cos ( 3 t ) + 2 2 sin 2 ( t ) ] d t | , {\displaystyle {\begin{aligned}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{aligned}}} so the area of the tail and head are given by: A Tail = ( 2 3 − π 4 2 ) a 2 , A Head = ( 2 3 + π 4 2 ) a 2 , {\displaystyle {\begin{aligned}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{aligned}}} giving the overall area for the fish as: A = 4 3 a 2 . {\displaystyle A={\frac {4}{3}}a^{2}.}
Curvature, arc length, and tangential angle
The arc length of the curve is given by a 2 ( 1 2 π + 3 ) . {\displaystyle a{\sqrt {2}}\left({\frac {1}{2}}\pi +3\right).}
The curvature of a fish curve is given by: K ( t ) = 2 2 + 3 cos ( t ) − cos ( 3 t ) 2 a [ cos 4 t + sin 2 t + sin 4 t + 2 sin ( t ) sin ( 2 t ) ] 3 2 , {\displaystyle 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}}}},} and the tangential angle is given by: ϕ ( t ) = π − arg ( 2 − 1 − 2 ( 1 + 2 ) e i t − 1 ) , {\displaystyle \phi (t)=\pi -\arg \left({\sqrt {2}}-1-{\frac {2}{\left(1+{\sqrt {2}}\right)e^{it}-1}}\right),} where arg ( z ) {\displaystyle \arg(z)} is the complex argument.