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Plotting gauss sinus and cosinus lemniscaticus

Posted 6 months ago
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Today is the birthday of Gauss (born 4/30/1777), who defined 1796 the double periodic sinus and cosinus lemniscaticus (lemniscatic elliptic function). Here is a way to visualize them.

w = 2 EllipticK[-1]; sinlem1 = 
 ParametricPlot[{EllipticF[ArcSin[r], -1], r}, {r, 0, 1}, 
  AspectRatio -> 1/GoldenRatio, PlotStyle -> Red];
sinlem2 = 
  ParametricPlot[{w - EllipticF[ArcSin[r], -1], r}, {r, -1, 1}, 
   AspectRatio -> 1/GoldenRatio, PlotStyle -> Red];
sinlem3 = 
  ParametricPlot[{2 w + EllipticF[ArcSin[r], -1], r}, {r, -1, 1}, 
   AspectRatio -> 1/GoldenRatio, PlotStyle -> Red];
coslem1 = 
  ParametricPlot[{w/2 - EllipticF[ArcSin[r], -1], r}, {r, -1, 1}, 
   AspectRatio -> 1/GoldenRatio, PlotStyle -> Blue];
coslem2 = 
  ParametricPlot[{3/2 w + EllipticF[ArcSin[r], -1], r}, {r, -1, 1}, 
   AspectRatio -> 1/GoldenRatio, PlotStyle -> Blue];
coslem3 = 
  ParametricPlot[{5/2 w - EllipticF[ArcSin[r], -1], r}, {r, 0, 1}, 
   AspectRatio -> 1/GoldenRatio, PlotStyle -> Blue];
scl = Show[sinlem1, sinlem2, sinlem3, coslem1, coslem2, coslem3, 
  PlotRange -> All]

enter image description here

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