# Plotting gauss sinus and cosinus lemniscaticus

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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]