In[]:=
ParaboloidCircumscribedCone::usage="Find the equation of a cone that has the vertex at the point M{-1, 0, 0} and is circumscribed around the paraboloid 2
2
y
+
2
z
= 4x.References: Fedenko, A.S. Problemas de Geometría Diferencial. Editorial Mir: p.74, Moscú, 1981. ";
Development:
The equation of the cone generatrix is:
In[]:=
x+1
a
==
y
b
==
z
c
==t;
The coordinates of any point of the generatrix are:
In[]:=
{-1+at,bt,ct};
Substituting in the paraboloid equation:
2
2
b
2
t
+
2
c
2
t
==4(-1+at);​​(2
2
b
+
2
c
)
2
t
-4at+4==0;
Due to the condition of tangency between the paraboloid and the cone, the discriminant of the second degree equation is zero:
2
(-4a)
-4(2
2
b
+
2
c
)(4)==0;​​16
2
a
-16(2
2
b
+
2
c
)==0;​​
2
a
-(2
2
b
+
2
c
)==0
Then we have (see the equation of the cone generatrix):
2
(x+1)
-(2
2
y
+
2
z
)==0;
Then, the equation of the cone is:
2
2
y
+
2
z
-
2
(x+1)
==0;
In[]:=
​​ManipulateVertex={-1,0,0};​​ContactPoint=1,
2
Cos[t1],2Sin[t1];​​Generatrix=InfiniteLine[{Vertex,ContactPoint}];​​ShowParametricPlot3Dv-1,
2
2
vCos[u],vSin[u],{u,0,t1},{v,-5,5},PlotStyle->{Opacity[0.4],Yellow},MeshFalse,PerformanceGoal"Quality",PlotPoints->60,ContourPlot3D[2
2
y
+
2
z
-4x==0,{x,-4,4},{y,-4,4},{z,-4,4},ContourStyle->{Opacity[0.8],Red},MeshFalse,PerformanceGoal"Quality"],​​ParametricPlot3D[1,
2
Cos[u],2Sin[u],{u,0,t1},PlotStyle->{Blue,Thick}],Graphics3D[{Black,Ball[Vertex,0.1],Ball[ContactPoint,0.1],Thick,Generatrix}],​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-3,3,-10},ImageSize500,Style["Cone:",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,2π,0.000001},ControlPlacementTop
Out[]=
​
Cone:
Value (t1)