DEnepperSurface::usage="Interesting surface that intersects itself, it is also a minimal surface, that is, its average curvature is zero. We will draw the geodesic lines of this surface from the origin. Geodesic lines are the shortest and most direct paths between two points on a curved surface. In a plane the geodesic lines woulbe straight. References: Public domain; Personal."
Animation:
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Manipulateσ[u_,v_]:=u-
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;geo=ResourceFunction["Geodesic"][σ[u,v],{u,v},t,{0,0},θ];​​solg=Flatten(NDSolve[#1,{u,v},{t,-5,5}]&)/@Tablegeo,θ,
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,2π,0.1,1;​​ShowParametricPlot3D[σ[u,v],{u,-5,t1},{v,-5,t1},PlotStyleDirective[Orange,Opacity[0.7],Specularity[White,30]],MeshFalse,MaxRecursion->3],​​ParametricPlot3DEvaluateu[t]-
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/.solg,{t,-5,t1}​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange5,ViewPoint{-20,-10,-4.},ImageSize900,Style["D'Enneper Surface and Geodesic Lines:",Bold,Large],{{t1,-5+0.00002,"Value (t1)"},-5+0.00001,5,0.00001},ControlPlacementTop
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D'Enneper Surface and Geodesic Lines:
Value (t1)
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​​gr=Showσ[u_,v_]:=u-
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;geo=ResourceFunction["Geodesic"][σ[u,v],{u,v},t,{0,0},θ];​​solg=Flatten(NDSolve[#1,{u,v},{t,-5,5}]&)/@Tablegeo,θ,
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,2π,0.1,1;ParametricPlot3D[σ[u,v],{u,-5,5},{v,-5,5},PlotStyleDirective[Orange,Opacity[0.7],Specularity[White,30]],MeshFalse,MaxRecursion->3],​​ParametricPlot3DEvaluateu[t]-
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/.solg,{t,-5,5}​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange5,ViewPoint{-20,-10,-4.},ImageSize900
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ResourceFunction["SpinShow"][gr,30]
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