CircumferenceDirectrixCylinder::usage="Find the equation of the straight circular cylinder whose directrix is the intersection of the spheres:: + - 4y + + 3 = 0, : + 2x + - 2y + - 2z + 2 = 0References: Internet: Repositorio.pucp.pe, 1.2 Superficies Notables en , Cilindro Circular Recto,Ejercicios propuestos, prob.- 2.; Personal. ";
ξ
1
2
x
2
y
2
z
ξ
2
2
x
2
y
2
z
3
Obtaining the cylinder axis (It is the line that passes through the centers of the spheres {0, 2, 0} and {-1, 1 1}):
ℒ={0,2,0}+t{-1,-1,1};
Obtaining the radical plane:
In[]:=
Simplify[+-4y++3-(+2x+-2y+-2z+2)]
2
x
2
y
2
z
2
x
2
y
2
z
Out[]=
1-2x-2y+2z
1-2x-2y+2z=0;
Obtaining the parametric equations of the circumference intersection of the two spheres:
In[]:=
x=-y+z+;y=u;z=v;{-1,1,0}{-2,-2,2}
1
2
Out[]=
{2,2,4}
In[]:=
Solve[{+-4y++3==0,+2x+-2y+-2z+2==0},{x,y}]
2
x
2
y
2
z
2
x
2
y
2
z
Out[]=
x-3+2z+5+2z--3+2z-5+2z+
1
4
-1+12z-12
,y2
z
1
4
-1+12z-12
,x2
z
1
4
-1+12z-12
,y2
z
1
4
-1+12z-12
2
z
In[]:=
ξ[z_]:=-3+2z+5+2z->=0,z]
1
4
-1+12z-12
,2
z
1
4
-1+12z-12
,z;Reduce[-1+12z-122
z
2
z
Out[]=
1
6
6
)≤z≤1
6
6
)In[]:=
Solve[1-2(-t)-2(2-t)+2(t)==0,t]
Out[]=
t
1
2
In[]:=
C1=-,,;
1
2
3
2
1
2
In[]:=
r1=NNormC1-ξ
1
2
Out[]=
0.5
ζ[t_]:=-,,+Cos[t]{-1,1,0}+Sin[t]{2,2,4};
1
2
3
2
1
2
1
2
Norm[{-1,1,0}]
1
2
Norm[{2,2,4}]
Steps to reach the cylinder equation:
In[]:=
Norm[{-1,-1,1}]
Out[]=
3
In[]:=
Norm{-1,-1,1}x+,y-,z-
1
2
3
2
1
2
In[]:=
Expand[++]
2
(2+x-y)
2
(2-y-z)
2
(x+z)
Out[]=
8+4x+2-8y-2xy+2-4z+2xz+2yz+2
2
x
2
y
2
z
8+4x+2-8y-2xy+2-4z+2xz+2yz+2
2
x
2
y
2
z
In[]:=
-8+
3
4
Out[]=
-
29
4
Cylinder equation:
4x+2-8y-2xy+2-4z+2xz+2yz+2=-
2
x
2
y
2
z
29
4
Animación:
In[]:=
Manipulateζ[t_]:=-,,+Cos[t]{-1,1,0}+Sin[t]{2,2,4};ShowParametricPlot3D[{Cos[u]Sin[v],Sin[u]Sin[v]+2,Cos[v]},{u,0,2π},{v,0,2π},ColorFunction"LightTemperatureMap",MeshFalse,PlotStyleDirective[Specularity[White,30],Opacity[0.3]],PerformanceGoal->"Quality"],ParametricPlot3D[{Cos[u]Sin[v]-1,Sin[u]Sin[v]+1,Cos[v]+1},{u,0,2π},{v,0,2π},ColorFunction"LightTemperatureMap",MeshFalse,PlotStyleDirective[Specularity[White,30],Opacity[0.5]],PerformanceGoal->"Quality"],ParametricPlot3D{0,2,0}+t{-1,-1,1},t,-1,,ColorFunction->"DeepSeaColors"/.Line[pts_,rest___]:>Tube[pts,0.02,rest],ContourPlot3D[1-2x-2y+2z==0,{x,-3,3},{y,-3,3},{z,-2,2},ContourStyle->Directive[Blue,Opacity[0.4],Specularity[White,30]],MeshNone],ContourPlot3D4x+2-8y-2xy+2-4z+2xz+2yz+2==-,{x,-3,3},{y,-3,3},z,-3,,ContourStyle->Directive[Green,Opacity[0.4],Specularity[White,30]],MeshNone,PerformanceGoal->"Quality",PlotPoints->40,Graphics3DBlack,Ball-,,,0.05,ParametricPlot3D[ζ[t],{t,0,t1},PlotStyleYellow]/.Line[pts_,rest___]:>Tube[pts,0.04,rest],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{-3,3,2},Style["Cylinder graph:",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,2π,0.00001},ControlPlacementTop
1
2
3
2
1
2
1
2
Norm[{-1,1,0}]
1
2
Norm[{2,2,4}]
t1
π
2
x
2
y
2
z
29
4
t1
π
1
2
3
2
1
2
Out[]=
View from different angles (below):
In[]:=
gr=ShowParametricPlot3D[{Cos[u]Sin[v],Sin[u]Sin[v]+2,Cos[v]},{u,0,2π},{v,0,2π},ColorFunction"LightTemperatureMap",MeshFalse,PlotStyleDirective[Specularity[White,30],Opacity[0.3]],PerformanceGoal->"Quality"],ParametricPlot3D[{Cos[u]Sin[v]-1,Sin[u]Sin[v]+1,Cos[v]+1},{u,0,2π},{v,0,2π},ColorFunction"LightTemperatureMap",MeshFalse,PlotStyleDirective[Specularity[White,30],Opacity[0.5]],PerformanceGoal->"Quality"],ParametricPlot3D[{0,2,0}+t{-1,-1,1},{t,-1,2},ColorFunction->"DeepSeaColors"]/.Line[pts_,rest___]:>Tube[pts,0.02,rest],ContourPlot3D[1-2x-2y+2z==0,{x,-3,3},{y,-3,3},{z,-2,2},ContourStyle->Directive[Blue,Opacity[0.4],Specularity[White,30]],MeshNone],ContourPlot3D4x+2-8y-2xy+2-4z+2xz+2yz+2==-,{x,-3,3},{y,-3,3},{z,-3,2},ContourStyle->Directive[Green,Opacity[0.4],Specularity[White,30]],MeshNone,PerformanceGoal->"Quality",PlotPoints->40,Graphics3DBlack,Ball-,,,0.05,ParametricPlot3D-,,+Cos[t]{-1,1,0}+Sin[t]{2,2,4},{t,0,2π},PlotStyleYellow/.Line[pts_,rest___]:>Tube[pts,0.04,rest],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{-3,3,2}
2
x
2
y
2
z
29
4
1
2
3
2
1
2
1
2
3
2
1
2
1
2
Norm[{-1,1,0}]
1
2
Norm[{2,2,4}]