Geometriclocusofthemovementofalinethatcreatesaconoid::usage="A ruled surface is one that can be generated by the movement of a straight line. The straight line in motion, in any of its positions, is called the surface generatrix. A ruled surface is generated by a straight line that moves in such a way that it always remains parallel to the XZ plane and cuts to the line z = 2, x = 0 and the circumference
2
x
+
2
y
= 4, z = 0. Answer: 4
2
x
= (4 -
2
y
​)​(2 - z
2
)
. References: Lehmann, Charles H. Geometría Analítica. Ciudad de México: Ed. Limusa, p. 419, prob. 24, (2003).";
Development:
If the line moves parallel to the plane XZ (y = 0), the value of y of the coordinates of its extreme points must be equal:
y==2Sin[θ];
Parametrics quations of the surface:
σ[θ_,V_]:={0,2Sin[θ],2}+v{2Cos[θ],0,-2};
Cartesian equation of the surface:
In[]:=
Eliminate[{x==2vCos[θ],y==2Sin[θ],z==2-2v},{θ,v}]
Out[]=
(-4+
2
y
)
2
(-2+z)
4x
-x&&
2
(-4+
2
y
)
2
(-2+z)
4
2
x
4-
2
y
&&(16-4
2
y
)z+(-4+
2
y
)
2
z
16-4
2
x
-4
2
y
(-4+
2
y
)
2
(-2+z)
4x
-x;​​4
2
x
==(4-
2
y
)
2
(2-z)
;
Animation:
In[]:=
​​SwatchLegend[{Green,Gray,Orange,Blue},{"Ruled surface: Conoid","Straight line directrix","Circular directrix","Straight line generatrix"}]​​Manipulate[ζ[θ_]:={0,2Sin[θ],2};​​β[θ_]:={2Cos[θ],2Sin[θ],0};​​σ[θ_,V_]:={0,2Sin[θ],2}+v{2Cos[θ],0,-2};​​Show[{​​ParametricPlot3D[β[θ],{θ,0,2π},PlotStyleOrange]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],​​ParametricPlot3D[ζ[θ],{θ,0,2π},PlotStyleGray]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],​​ParametricPlot3D[σ[θ,v],{θ,0,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[0.004],InfiniteLine[{β[t1],ζ[t1]}]}],Graphics3D[{Orange,Ball[β[t1],0.033]}],Graphics3D[{Gray,Ball[ζ[t1],0.033]}]},​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{5,-10,5}],Style["Ruled surface: Conoid:",Bold,Medium],{{t1,0.00002,"Value (t1)"},0.00001,2π,0.00001},ControlPlacementTop]
Out[]=
Ruled surface: Conoid
Straight line directrix
Circular directrix
Straight line generatrix
Out[]=
​
Ruled surface: Conoid:
Value (t1)
View from all angles (below):
In[]:=
gr=​​Show[ζ[θ_]:={0,2Sin[θ],2};​​β[θ_]:={2Cos[θ],2Sin[θ],0};​​σ[θ_,V_]:={0,2Sin[θ],2}+v{2Cos[θ],0,-2};​​{ParametricPlot3D[β[θ],{θ,0,2π},PlotStyleOrange]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],​​ParametricPlot3D[ζ[θ],{θ,0,2π},PlotStyleGray]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],​​ParametricPlot3D[σ[θ,v],{θ,0,2π},{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[0.004],InfiniteLine[{β[2π],ζ[2π]}]}],Graphics3D[{Orange,Ball[β[2π],0.033]}],Graphics3D[{Gray,Ball[ζ[2π],0.033]}]},​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{5,-10,5}]
Out[]=