CatenaryProjectedOrthogonally::usage="
Referencia:e.math.cornell.edu,Belk,Jim.DifferentialGeometry,CornellUniversity.Exam2PracticeProblems,ParameterizingSpaceCurves,Prob.-2.;Personal.";
Development:
Normal line to plane y = 3z that passes through the points of the catenary of the XY plane:
ℒ
N
:{t,Cosh[t],0}+v{0,1,-3};
Intersection of the straight line
ℒ
N
with the plane y = 3z:
Cosh[t]+v-3(-3v)==0;​​v=-
1
10
Cosh[t];​​t,
9
10
Cosh[t],
3
10
Cosh[t];
In[]:=
ManipulateClearAll;c[t_]:={t,Cosh[t],0};​​ξ[t_]:=t,
9
10
Cosh[t],
3
10
Cosh[t];​​Show[{ParametricPlot3D[c[t],{t,-2.8,t1},PlotStyleBlue]/.Line[pts_,rest___]:>Tube[pts,0.08,rest],ParametricPlot3D[ξ[t],{t,-2.8,t1},PlotStyleRed]/.Line[pts_,rest___]:>Tube[pts,0.08,rest],Graphics3D[{Red,Thick,Ball[ξ[t1],0.15]}],​​Graphics3D[{Blue,Thick,Ball[c[t1],0.15]}],ContourPlot3D[z==0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyleDirective[Yellow,Opacity[0.5],Specularity[White,30]],MeshFalse,PerformanceGoal"Quality"],​​ContourPlot3D[y-3z==0,{x,-10,10},{y,-10,10},{z,-5,5},ContourStyleDirective[Gray,Opacity[0.2],Specularity[White,30]],MeshFalse,PerformanceGoal"Quality"],Graphics3D[{Black,Dashed,Thick,Line[{c[t1],ξ[t1]}]}]​​},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange8,ImageSize900,ViewPoint{-3,3,-1}],Style["Catenary projected orthogonally in the plane y - 3z = 0:",Bold,Large],{{t1,-2.7777,"Value (t1)"},-2.8+0.00001,2.8,0.00001},ControlPlacementTop
Out[]=
​
Catenary projected orthogonally in the plane y - 3z = 0:
Value (t1)