SurfaceformedbythelineparalleltotheYZplane::usage="A ruled surface is one that can be generated by the movement of a straight line in motion, in any of its positions is called surface generatrix.A ruled surface is generated by a line that moves in such a way that it always remains parallel to the YZ plane and cuts to the linex + z = 1, y = 0 and the parabola
2
y
= x, z = 0.Answer:
2
y
​(x - 1
2
)
= x(x + z - 1
2
)
​Reference: Charles H. Lehmann, Geometría Analítica Ed. Limusa (2003).Prob.- 22, pag.- 418.";
Development:
If the line moves parallel to the YZ plane (x = 0), the abscisses of the extreme points must be equal:
2
y
==1-z;​​z==1-
2
y
;
Parametric equations of the surface:
σ[u_,V_]:={
2
u
,0,1-
2
u
}+v{0,u,
2
u
-1};
Cartesian equation of the surface:
In[]:=
Eliminate[{x==
2
u
,y==vu,z==1-
2
u
+v(
2
u
-1)},{u,v}]
-x+2
2
x
-
3
x
+
2
y
-2x
2
y
+
2
x
2
y
+2xz-2
2
x
z-x
2
z
0;
Verification:
In[]:=
Expand[
2
y
2
(x-1)
==x
2
(x+z-1)
]
2
y
-2x
2
y
+
2
x
2
y
x-2
2
x
+
3
x
-2xz+2
2
x
z+x
2
z
;​​-x+2
2
x
-
3
x
+
2
y
-2x
2
y
+
2
x
2
y
+2xz-2
2
x
z-x
2
z
==0;
It’s exactly the same equation.
Animation:
In[]:=
​​SwatchLegend[{Green,Gray,Orange,Blue},{"Ruled surface","Straight line directrix","Parabolic directrix","Straight line generatrix"}]​​Manipulate[ζ[u_]:={
2
u
,u,0};​​β[u_]:={
2
u
,0,1-
2
u
};​​σ[u_,V_]:={
2
u
,0,1-
2
u
}+v{0,u,
2
u
-1};​​​​Show[{​​ParametricPlot3D[β[u],{u,-1,1},PlotStyleGray]/.Line[pts_,rest___]:>Tube[pts,0.005,rest],​​ParametricPlot3D[ζ[u],{u,-1,1},PlotStyleOrange]/.Line[pts_,rest___]:>Tube[pts,0.005,rest],​​ParametricPlot3D[σ[u,v],{u,-1,t1},{v,-1,1},PlotStyle->Directive[Opacity[0.6],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality",PlotPoints->60,ColorFunction->"BlueGreenYellow",BoundaryStyleDirective[Black,Thick]],​​Graphics3D[{Blue,Thickness[0.005],InfiniteLine[{β[t1],ζ[t1]}]}],Graphics3D[{Gray,Ball[β[t1],0.015]}],Graphics3D[{Orange,Ball[ζ[t1],0.015]}]},​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{15,-10,8}],Style["Ruled surface:",Bold,Medium],{{t1,-1+0.00002,"Valor (t1)"},-1+0.00001,1,0.00001},ControlPlacementLeft]
Out[]=
Ruled surface
Straight line directrix
Parabolic directrix
Straight line generatrix
Out[]=
​
Ruled surface:
Valor (t1)
View from all angles (below):
In[]:=
gr=​​Show[ζ[u_]:={
2
u
,u,0};​​β[u_]:={
2
u
,0,1-
2
u
};​​σ[u_,V_]:={
2
u
,0,1-
2
u
}+v{0,u,
2
u
-1};{​​ParametricPlot3D[β[u],{u,-1,1},PlotStyleGray]/.Line[pts_,rest___]:>Tube[pts,0.005,rest],​​ParametricPlot3D[ζ[u],{u,-1,1},PlotStyleOrange]/.Line[pts_,rest___]:>Tube[pts,0.005,rest],​​ParametricPlot3D[σ[u,v],{u,-1,1},{v,-1,1},PlotStyle->Directive[Opacity[0.6],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality",PlotPoints->60,ColorFunction->"BlueGreenYellow",BoundaryStyleDirective[Black,Thick]],​​Graphics3D[{Blue,Thickness[0.005],InfiniteLine[{β[1],ζ[1]}]}],Graphics3D[{Gray,Ball[β[1],0.015]}],Graphics3D[{Orange,Ball[ζ[1],0.015]}]},​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{15,-10,8}]
Out[]=