RuledSurfaceACone::usage=" It is shown that the given equation represents a conical surface demonstrating that its geometric locus is a ruled surface whose generatrix are all concurrent.:
2
y
- 4 x z = 0. Reference: Lehmann, Charles H. Geometría Analítica​, Mexico City: Limusa, p.418, 2003.; Personal";
Development:
Parametric equations of the generatrix:
​​k=
y
2x
;k=
2z
y
;2kx-y=0;ky-2z=0;​​
In[]:=
Cross[{2k,-1,0},{0,k,-2}]
Out[]=
{2,4k,2
2
k
}
In[]:=
Solve[{2kx-y==0,ky-2z==0},{x,y}]
Out[]=
x
z
2
k
,y
2z
k

In[]:=
{x,y}=
z
2
k
,
2z
k


z
2
k
,
2z
k
;
Doing z = 0, it has:
{0,0,0};
It is a point where all the generatrix are concurrent, therefore it is a conical surface.
ℒ:t{2,4k,2
2
k
};
It is a ruled surface.
W will find out if it is a developable surface, with the generatrix:
In[]:=
ℒ:=v{2,4u,2
2
u
};​​γ[u_]:={0,0,0};​​ω[u_]:={2,4u,2
2
u
};​​p[u]=Det[{γ'[u],ω[u],ω'[u]}]​​
Out[]=
0
As we expected, it is a developable surface. Its recoil edge must be the vertex::
In[]:=
β[u]=γ[u]-
(ω'[u]ω[u]).(γ'[u]ω[u])
2
Norm[ω'[u]ω[u]]
ω[u]
Out[]=
{0,0,0}
We need to find a directrix::
For this we use the plane z = - x + 1, that intersects the surface in an ellipse that can serve as a directrix for the animation:
2
y
-4x(-x+1)==0;z==-x+1;​​4
2
x
-4x+
2
y
==0;z==-x+1;​​4
2
x
-x+
1
4
+
2
y
==1;z==-x+1;​​4
2
x-
1
2
+
2
y
==1;z==-x+1;​​
2
x-
1
2

1
4
+
2
y
==1;z==-x+1;​​
1
2
Cos[θ]+
1
2
,Sin[θ],-
1
2
Cos[θ]+
1
2
+1;​​
1
2
Cos[θ]+
1
2
,Sin[θ],-
1
2
Cos[θ]+
1
2
;​​
Animation:
In[]:=
​​SwatchLegend[{Green,Yellow,Blue},{"Rukled surface: The cone","Elliptical directrix","Rectilinear generatrixa"}]​​ManipulateP={0,0,0};​​ζ[t_]:=
1
2
Cos[t]+
1
2
,Sin[t],-
1
2
Cos[t]+
1
2
;​​σ[v_,t_]:=vζ[t];​​Show[{​​ParametricPlot3D[σ[v,t],{v,-1,1},{t,0,t1},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[{P,ζ[t1]}]}],​​ParametricPlot3D[ζ[t],{t,0,2π},PlotStyleYellow]/.Line[pts_,rest___]:>Tube[pts,0.0075,rest],Graphics3D[{Yellow,Ball[ζ[t1],0.026]}]},​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{-3,3,5}],Style["Rule surface: The cone",Bold,Medium],{{t1,0.00002,"Value (t1)"},0.00001,2π,0.00001},ControlPlacementTop
Out[]=
Rukled surface: The cone
Elliptical directrix
Rectilinear generatrixa
Out[]=
​
Rule surface: The cone
Value (t1)
View from all angles (below):