TangentDevelopableSurfaceHelix::usage="A ruled surface M is a tangent developable of a curve y if M can be parameterized by x[u, v] = y[u] +ν y'[u]. A tangent developable is a developable surface.References: Wolfram MathWorld: Tangent Developable; Personal.";
Development:
Let the developable tangent surface of a helix:
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ManipulateClear;ζ[t_]:=Cos[t],Sin[t],;[v_,t_]:=ζ[t]+vζ'[t];Show[{ParametricPlot3D[[v,t],{v,-10,10},{t,0,t1},PlotStyle->Opacity[0.3],ColorFunction->"AvocadoColors",BoundaryStyleDirective[Black,Thick],MaxRecursion->3],ParametricPlot3D[ζ[t],{t,0,t1},PlotStyle{Thick,Blue}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange10,ViewPoint{1.3,-2.4,2.},ImageSize500],Style["Tangent developable surface of a helix:",Bold,Large],{{t1,0.1,"Value (t1)"},0.01,4π,0.01},ControlPlacementTop
2
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Verification that the surface r[v, t] = ζ[t] + v ζ ‘[t] is a developable surface with the calculation of the distribution parameter:
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p[t]=Det[{ζ'[t],ζ'[t],ζ''[t]}]
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0