RuledSurfaceTangentDevelopable::usage="A ruled surface M is a tangent developable of a curve y if M can be parametrized by x[u, v] = y[u] + ν y'[u]. A tangent developable is a developable surface.surace.References: Wolfram MathWorld: Tangent Developable; Personal.";
Development:
Tangent developable of the Viviani’s curve:
In[]:=
Reduce
25
2
-
25
2
Cos[t]>=0,t
1
2
(-1+

2
)

2
∈&&tπ

2
||(

1
∈&&tπ(1+2

1
))||(

1
∈&&Re[t]π+2π

1
)||(

2
∈&&tπ

2
)||t∈;
In[]:=
ManipulateClear;​​ζ[t_]:=
5
2
+
5
2
Cos[t],
5
2
Sin[t],
25
2
-
25
2
Cos[t]
;​​
ζ
1
[t_]:=
5
2
+
5
2
Cos[t],
5
2
Sin[t],-
25
2
-
25
2
Cos[t]
;​​
r
1
[v_,t_]:=ζ[t]+vζ'[t];
r
2
[v_,t_]:=
ζ
1
[t]+v
ζ
1
'[t];​​Show[{ParametricPlot3D[
r
1
[v,t],{v,-10,10},{t,0,t1},PlotStyle->Opacity[0.3],ColorFunction->"DeepSeaColors",PerformanceGoal"Quality",BoundaryStyleDirective[Black,Thick]],​​ParametricPlot3D[
r
2
[v,t],{v,-10,10},{t,0,t1},PlotStyle->{Opacity[0.3],Yellow},PerformanceGoal"Quality",BoundaryStyleDirective[Black,Thick]],​​ParametricPlot3D[ζ[t],{t,0,t1},PlotStyle{Thick,Blue}],​​ParametricPlot3D[
ζ
1
[t],{t,0,t1},PlotStyle{Thick,Blue}]}​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{{-25,25},{-25,25},{-35,35}},ViewPoint{-10,-30,30},ImageSize500],Style["Tangent developable of the Viviani's curve:",Bold,Large],{{t1,0.1,"Value (t1)"},0.01,2π,0.01},ControlPlacementTop​​
Out[]=
​
Tangent developable of the Viviani's curve:
Value (t1)
Checking that it is r[v, t] = ζ[t] + v ζ ‘[t] a developable surface with the distribution parameter:
In[]:=
p[u]=Det[{ζ'[u],ζ'[u],ζ''[u]}]
Out[]=
0