ConeDirectrixSpherePlane::usage="Find the cartesian equation of the straight circular cone whose directrix is the circumference:Γ: + + - 2z - 8 = 0 and x + y + z - 4 = 0and whose vertex belongs to the plane π: 4x + 2y - 3z - 3 = O.References: Internet: Repositorio.pucp.pe, 1.2 Superficies Notables en , Cono Circular Recto,Ejercicios propuestos, prob.- 6.; Personal. ";
2
x
2
y
2
z
3
Obtaining the cone axis:
2
x
2
y
2
(z-1)
ℒ={0,0,1}+t{1,1,1};
Obtaining the vertex:
In[]:=
Solve[4(t)+2(t)-3(1+t)-3==0,t]
Out[]=
{{t2}}
V={2,2,3}
Obtaining the cosine:
In[]:=
ca=++
2
(2-1)
2
(2-1)
2
(3-2)
Out[]=
3
In[]:=
2
h
2
3
2
3
Out[]=
2
h
Cos[θ]==;
3
2
3
1
2
Cone equation:
In[]:=
ExpandNorm[{x-2,y-2,z-3}]Norm[{1,1,1}]
2
1
2
Out[]=
3
4
2
Abs[-2+x]
3
4
2
Abs[-2+y]
3
4
2
Abs[-3+z]
In[]:=
Expand[]
2
({x-2,y-2,z-3}.{1,1,1})
Out[]=
49-14x+-14y+2xy+-14z+2xz+2yz+
2
x
2
y
2
z
In[]:=
Expand++-(49-14x+-14y+2xy+-14z+2xz+2yz+)
3
4
2
(-2+x)
3
4
2
(-2+y)
3
4
2
(-3+z)
2
x
2
y
2
z
-+11x-+11y-2xy-+-2xz-2yz-==0;++-44x-44y-38z+8xy+8xz+8yz+145==0;
145
4
2
x
4
2
y
4
19z
2
2
z
4
2
x
2
y
2
z
Obtaining the parametric equations of the circumference to trace the graph:
In[]:=
x+y+z-4==0;x=-y-z+4;y=u;z=v;{-1,1,0}{1,1,1}
Out[]=
{1,1,-2}
In[]:=
Solve[{++-2z-8==0,x+y+z-4==0},{x,y}]
2
x
2
y
2
z
Out[]=
x4-z-4-z+4-z+4-z-
1
2
3
4z-
,y2
z
1
2
3
-((-4+z)z)
,x1
2
3
4z-
,y2
z
1
2
3
-((-4+z)z)
In[]:=
ξ[z_]:=4-z-4-z+>=0,z]
1
2
3
4z-
,2
z
1
2
3
-((-4+z)z)
,z;Reduce[4z-2
z
Out[]=
0≤z≤4
In[]:=
Solve[t+t+1+t-4==0,t]
Out[]=
{{t1}}
In[]:=
C1={1,1,2};r1=N[Norm[C1-ξ[0]]]
Out[]=
2.44949
ζ[t_]:={1,1,2}+Cos[t]{-1,1,0}+Sin[t]{1,1,-2};
2.449489742783178`
Norm[{-1,1,0}]
2.449489742783178`
Norm[{1,1,-2}]
Animation:
In[]:=
Manipulateζ[t_]:={1,1,2}+Cos[t]{-1,1,0}+Sin[t]{1,1,-2};Show[{ParametricPlot3D[{0,0,1}+t{1,1,1},{t,-6,6},ColorFunction->"DeepSeaColors"]/.Line[pts_,rest___]:>Tube[pts,0.09,rest],ParametricPlot3D[{3Cos[u]Sin[v],3Sin[u]Sin[v],3Cos[v]+1},{u,0,2π},{v,0,2π},MeshFalse,PlotStyle->Directive[Blue,Opacity[0.3],Specularity[White,30]],PerformanceGoal->"Quality"],ContourPlot3D[++-44x-44y-38z+8xy+8xz+8yz+145==0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle->Directive[Yellow,Opacity[0.4],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality",PlotPoints40],ContourPlot3D[x+y+z-4==0,{x,-5,5},{y,-5,5},{z,-5,5},ContourStyle->Directive[Green,Opacity[0.4],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality"],ContourPlot3D[4x+2y-3z-3==0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle->Directive[Purple,Opacity[0.2],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality"],Graphics3D[{Black,Ball[{0,0,1},0.2]}],Graphics3D[{Black,Ball[{2,2,3},0.2]}],Graphics3D[{Black,Ball[{1,1,2},0.2]}],ParametricPlot3D[ζ[t],{t,0,t1},PlotStyle->Yellow]/.Line[pts_,rest___]:>Tube[pts,0.09,rest]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{-3,7,-1}],Style["Cone with vertex in the plane:",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,2π,0.00001},ControlPlacementTop
2.449489742783178`
Norm[{-1,1,0}]
2.449489742783178`
Norm[{1,1,-2}]
2
x
2
y
2
z
Out[]=
View from different angles (below):
In[]:=
gr=Show[{ParametricPlot3D[{0,0,1}+t{1,1,1},{t,-6,6},ColorFunction->"DeepSeaColors"]/.Line[pts_,rest___]:>Tube[pts,0.09,rest],ParametricPlot3D[{3Cos[u]Sin[v],3Sin[u]Sin[v],3Cos[v]+1},{u,0,2π},{v,0,2π},MeshFalse,PlotStyle->Directive[Blue,Opacity[0.3],Specularity[White,30]],PerformanceGoal->"Quality"],ContourPlot3D[++-44x-44y-38z+8xy+8xz+8yz+145==0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle->Directive[Yellow,Opacity[0.4],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality",PlotPoints40],ContourPlot3D[x+y+z-4==0,{x,-5,5},{y,-5,5},{z,-5,5},ContourStyle->Directive[Green,Opacity[0.4],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality"],ContourPlot3D[4x+2y-3z-3==0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle->Directive[Purple,Opacity[0.2],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality"],Graphics3D[{Black,Ball[{0,0,1},0.2]}],Graphics3D[{Black,Ball[{2,2,3},0.2]}],Graphics3D[{Black,Ball[{1,1,2},0.2]}],ParametricPlot3D[ζ[t],{t,0,2π},PlotStyle->Yellow]/.Line[pts_,rest___]:>Tube[pts,0.09,rest]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{-3,7,-1}]
2
x
2
y
2
z