Lissajous curves are shaped to obtain various configurations. The shapes created include bent plane, cylinder, parabolic hyperboloid, infinite sign.
Vary n
n=10;​​c3=ParametricPlot3D[{Sin[(n)x],Cos[(n+1)x],0},{x,0,2Pi},PlotStyle->{Cyan,Tube[0.01]}][[1]];​​Graphics3D[{c3},ViewPoint->{0,0,100}]
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In[]:=
n=24;​​c3=ParametricPlot3D[{Sin[nx],Cos[(n-1)x],0},{x,0,2Pi},PlotStyle->{Cyan,Tube[0.004]}][[1]];​​Graphics3D[{c3},ViewPoint->{0,0,100}]
Bending
n=10;​​bend=1.4;​​c3=ParametricPlot3D[{Sin[nx],Cos[(n+1)x],bendSin[xn]Sin[xn]},{x,-Pi,Pi},PlotStyle->{Cyan,Tube[0.02]}][[1]];​​Graphics3D[{c3},Boxed->False,SphericalRegion->True]
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Parabolic hyperboloid
n=14;​​form=2;​​flat=2;​​c3=ParametricPlot3D[{formSin[(n)x],formCos[(n+1)x],flatSin[(n)x]Cos[(n+1)x]},{x,-Pi,Pi},PlotStyle->{Yellow,Tube[0.03]}][[1]];​​Graphics3D[{c3},Boxed->False,SphericalRegion->True]
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Cylinder
n=40;​​comp=1;​​c5=ParametricPlot3D[{compSin[nx],Cos[(n+1)x],Cos[nx]},{x,-Pi,Pi},PlotStyle->{LightGreen,Tube[0.02]}][[1]];​​Graphics3D[{c5},Boxed->False,SphericalRegion->True]
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Infinite sign
n=80;​​widen=2;​​c5=ParametricPlot3D[{2Sin[(n)x],Cos[(n+1)x],widenCos[xn]Sin[xn]},{x,-Pi,Pi},PlotStyle->{LightBlue,Tube[0.01]}][[1]];​​Graphics3D[{c5},Boxed->False,SphericalRegion->True]
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Holder
n=60;​​c5=ParametricPlot3D[{Sin[(n)x],Cos[(n+1)x],Cos[2x]},{x,-Pi,Pi},PlotStyle->{LightGreen,Tube[0.05]}][[1]];​​Graphics3D[{c5},Boxed->False,SphericalRegion->True]
Cross
n=60;​​c5=ParametricPlot3D[{Sin[(n)x],Cos[(n+1)x],Cos[4x]},{x,-Pi,Pi},PlotStyle->{LightGreen,Tube[0.05]}][[1]];​​Graphics3D[{c5},Boxed->False,SphericalRegion->True]
Tetrahedron
n=50;​​c5=ParametricPlot3D[{Sin[(n)x],Cos[(n+1)x],Sin[Sin[x]]},{x,-Pi,Pi},PlotStyle->{LightRed,Tube[0.02]}][[1]];​​Graphics3D[{c5},Boxed->False,SphericalRegion->True]

CITE THIS NOTEBOOK

Shaping of Lissajous curve​
by Sándor Kabai​
Wolfram Community, STAFF PICKS, September 8, 2023
​https://community.wolfram.com/groups/-/m/t/3008641