RuledSurfaceAHyperbolicParaboloid::usage=" Prove that the given equation represents a ruled surface: xy - x - y - z + 1 = 0 . References: Lehmann, Charles H. Geometría Analítica, Mexico city: Limusa, p.418, 2003; Personal";
Development:
Parametric equations of the generatrix:
Parametric equations of the generatrix:
ky-k-z=0;x-1=k;
Cross[{0,k,-1},{1,0,0}]
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{0,-1,-k}
k+1,,z;
k+z
k
Doing z = 0:
ℒ:{k+1,1,0}+t{0,-1,-k};
Therefore it is a ruled surface.
Let’s find out with the generatrix if it’s a ruled surface:
Let’s find out with the generatrix if it’s a ruled surface:
ℒ:{u+1,1,0}+v{0,-1,-u};γ[u_]:={u+1,1,0};ω[u_]:={0,-1,-u};p[u]=Det[{γ'[u],ω[u],ω'[u]}]
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1
It is not a developable surface.We need a directrix for the animation, therefore we will obtain the striction line:
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β[u]=γ[u]-ω[u]
(ω'[u]ω[u]).(γ'[u]ω[u])
2
Norm[ω'[u]ω[u]]
{1+u,1,0};
Animation:
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SwatchLegend[{Green,Yellow,Blue},{"Ruled surface: A hyperbolic paraboloid","Straightline directrix (Striction line)","Straightline generatrix"}]Manipulate[β[u_]:={1+u,1,0};ℒ[t2_]:={1+t2,1,0}+t2{0,-1,-t2};σ[u_,v_]:={u+1,1,0}+v{0,-1,-u};Show[{Graphics3D[{Blue,Thickness[0.004],InfiniteLine[{β[t2],ℒ[t2]}]}],ParametricPlot3D[β[u],{u,-10,10},PlotStyleYellow]/.Line[pts_,rest___]:>Tube[pts,0.06,rest],Graphics3D[{Yellow,Ball[β[t2],0.2]}],ParametricPlot3D[σ[u,v],{v,-10,10},{u,-10,t2},PlotStyle->Directive[Opacity[0.6],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality",PlotPoints->60,ColorFunction->"BlueGreenYellow",BoundaryStyleDirective[Black,Thick]]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange11,ImageSize900,ViewPoint{1.3,-2.4,2.}],Style["Ruled surface: A hyperbolic paraboloid",Bold,Medium],{{t2,-10+0.00002,"Value (t2)"},-10+0.00001,10,0.00001},ControlPlacementLeft]
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View of the all angles (below):
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gr=Show[β[u_]:={1+u,1,0};ℒ[t2_]:={1+t2,1,0}+t2{0,-1,-t2};σ[u_,v_]:={u+1,1,0}+v{0,-1,-u};{Graphics3D[{Blue,Thickness[0.004],InfiniteLine[{β[10],ℒ[10]}]}],ParametricPlot3D[β[u],{u,-10,10},PlotStyleYellow]/.Line[pts_,rest___]:>Tube[pts,0.06,rest],Graphics3D[{Yellow,Ball[β[10],0.2]}],ParametricPlot3D[σ[u,v],{v,-10,10},{u,-10,10},PlotStyle->Directive[Opacity[0.6],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality",PlotPoints->60,ColorFunction->"BlueGreenYellow",BoundaryStyleDirective[Black,Thick]]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange11,ImageSize900,ViewPoint{1.3,-2.4,2.}]
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