CylindersIntersections::usage=" Be the cylinders σ[u, v] = { 4 Cos[u], 4 Sin[u], v} and the right circular cylinder whose axis passes through the origin and is parallel to the generatrix ℒ: {0, 4, 0} + t {1, 1, 1}. Find the parametric equations of your intersection curve.Reference: Personal "
Development:
Equation of the second cylinder:
In[]:=
r=
Norm[{0,4,0}{1,1,1}]
Norm[{1,1,1}]
Out[]=
4
2
3
In[]:=
4
2
3
==
Norm[{x,y,z}{1,1,1}]
Norm[{1,1,1}]
Out[]=
4
2
3

2
Abs[x-y]
+
2
Abs[y-z]
+
2
Abs[-x+z]
3
In[]:=
Expand
2

2
(x-y)
+
2
(y-z)
+
2
(-x+z)

==
2
4
2


2
2
x
-2xy+2
2
y
-2xz-2yz+2
2
z
32;​​
2
x
+
2
y
+
2
z
-xy-xz-yz-16==0;
Cartesian equation of the first cylinder:
In[]:=
Eliminate[{x==4Cos[u],y==4Sin[u],z==v},{u,v}]
16-
2
y

2
x
;
Parametric equations of the intersection curves:
In[]:=
Solve[{
2
x
+
2
y
+
2
z
-xy-xz-yz-16==0,16-
2
y

2
x
},{x,y}]
In[]:=
Simplifyx-
z
2
-
1
2
16+3
2
z
-
8-2
2
z
-
16z
16+3
2
z
-
3
3
z
16+3
2
z
2
,y
1
2
2
z
-
3
z
-
4
2
z
16+3
2
z
-
3
4
z
4
16+3
2
z
-2
16+3
2
z
-
9
8
2
z
16+3
2
z
+
1
8
3/2
(16+3
2
z
)
-2
2
8-2
2
z
-
16z
16+3
2
z
-
3
3
z
16+3
2
z
+
3
2
z
8-2
2
z
-
16z
16+3
2
z
-
3
3
z
16+3
2
z
2
+
z
16+3
2
z
8-2
2
z
-
16z
16+3
2
z
-
3
3
z
16+3
2
z
2
2
+
3/2
8-2
2
z
-
16z
16+3
2
z
-
3
3
z
16+3
2
z
2
2
,x-
z
2
-
1
2
16+3
2
z
+
8-2
2
z
-
16z
16+3
2
z
-
3
3
z
16+3
2
z
2
,y
1
2
2
z
-
3
z
-
4
2
z
16+3
2
z
-
3
4
z
4
16+3
2
z
-2
16+3
2
z
-
9
8
2
z
16+3
2
z
+
1
8
3/2
(16+3
2
z
)
+2
2
8-2
2
z
-
16z
16+3
2
z
-
3
3
z
16+3
2
z
-
3
2
z
8-2
2
z
-
16z
16+3
2
z
-
3
3
z
16+3
2
z
2
-
z
16+3
2
z
8-2
2
z
-
16z
16+3
2
z
-
3
3
z
16+3
2
z
2
2
-
3/2
8-2
2
z
-
16z
16+3
2
z
-
3
3
z
16+3
2
z
2
2
,x-
z
2
+
1
2
16+3
2
z
-
8-2
2
z
+
16z
16+3
2
z
+
3
3
z
16+3
2
z
2
,y
1
2
2
z
-
3
z
+
4
2
z
16+3
2
z
+
3
4
z
4
16+3
2
z
+2
16+3
2
z
+
9
8
2
z
16+3
2
z
-
1
8
3/2
(16+3
2
z
)
-2
2
8-2
2
z
+
16z
16+3
2
z
+
3
3
z
16+3
2
z
+
3
2
z
8-2
2
z
+
16z
16+3
2
z
+
3
3
z
16+3
2
z
2
-
z
16+3
2
z
8-2
2
z
+
16z
16+3
2
z
+
3
3
z
16+3
2
z
2
2
+
3/2
8-2
2
z
+
16z
16+3
2
z
+
3
3
z
16+3
2
z
2
2
,x-
z
2
+
1
2
16+3
2
z
+
8-2
2
z
+
16z
16+3
2
z
+
3
3
z
16+3
2
z
2
,y
1
2
2
z
-
3
z
+
4
2
z
16+3
2
z
+
3
4
z
4
16+3
2
z
+2
16+3
2
z
+
9
8
2
z
16+3
2
z
-
1
8
3/2
(16+3
2
z
)
+2
2
8-2
2
z
+
16z
16+3
2
z
+
3
3
z
16+3
2
z
-
3
2
z
8-2
2
z
+
16z
16+3
2
z
+
3
3
z
16+3
2
z
2
+
z
16+3
2
z
8-2
2
z
+
16z
16+3
2
z
+
3
3
z
16+3
2
z
2
2
-
3/2
8-2
2
z
+
16z
16+3
2
z
+
3
3
z
16+3
2
z
2
2

Out[]=
x
1
2
-z-
16+3
2
z
-
2
8-2
2
z
-z
16+3
2
z
,y-
z
2
-
8
16+3
2
z
-
3
2
z
2
16+3
2
z
+
4-
2
z
-
1
2
z
16+3
2
z
,x
1
2
-z-
16+3
2
z
+
2
8-2
2
z
-z
16+3
2
z
,y-
z
2
-
8
16+3
2
z
-
3
2
z
2
16+3
2
z
-
4-
2
z
-
1
2
z
16+3
2
z
,x
1
2
-z+
16+3
2
z
-
2
8-2
2
z
+z
16+3
2
z
,y-
z
2
+
8
16+3
2
z
+
3
2
z
2
16+3
2
z
+
4-
2
z
+
1
2
z
16+3
2
z
,x
1
2
-z+
16+3
2
z
+
2
8-2
2
z
+z
16+3
2
z
,y-
z
2
+
8
16+3
2
z
+
3
2
z
2
16+3
2
z
-
4-
2
z
+
1
2
z
16+3
2
z

ζ[t_]:=
1
2
-t±
16+3
2
t
±
2
8-2
2
t
±t
16+3
2
t
,-
t
2
±
8
16+3
2
t
±
3
2
t
2
16+3
2
t
±
4-
2
t
±
1
2
t
16+3
2
t
,t;
In[]:=
Reduce4-
2
t
-
1
2
t
16+3
2
t
>=0,t
Out[]=
2(-2-
2
)≤t≤2(2-
2
)
In[]:=
Reduce4-
2
t
+
1
2
t
16+3
2
t
>=0,t
Out[]=
2(-2+
2
)≤t≤2(2+
2
)
Animation:
In[]:=
Manipulateζ[t_]:=
1
2
-t-
16+3
2
t
-
2
8-2
2
t
-t
16+3
2
t
,-
t
2
-
8
16+3
2
t
-
3
2
t
2
16+3
2
t
+
4-
2
t
-
1
2
t
16+3
2
t
,t;​​
ζ
1
[t_]:=
1
2
-t-
16+3
2
t
+
2
8-2
2
t
-t
16+3
2
t
,-
t
2
-
8
16+3
2
t
-
3
2
t
2
16+3
2
t
-
4-
2
t
-
1
2
t
16+3
2
t
,t;​​
ζ
2
[t_]:=
1
2
-t+
16+3
2
t
-
2
8-2
2
t
+t
16+3
2
t
,-
t
2
+
8
16+3
2
t
+
3
2
t
2
16+3
2
t
+
4-
2
t
+
1
2
t
16+3
2
t
,t;​​
ζ
3
[t_]:=
1
2
-t+
16+3
2
t
+
2
8-2
2
t
+t
16+3
2
t
,-
t
2
+
8
16+3
2
t
+
3
2
t
2
16+3
2
t
-
4-
2
t
+
1
2
t
16+3
2
t
,t;​​σ[u_,v_]:={4Cos[u],4Sin[u],v};​​ShowRevolutionPlot3D[{t,4+t,t},{t,-8,8},{θ,0,θ1},RevolutionAxis{1,1,1},PlotStyle->Opacity[0.4],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow",PlotPoints->40],ParametricPlot3D[t{1,1,1},{t,-8,12},ColorFunction->"DeepSeaColors"]/.Line[pts_,rest___]:>Tube[pts,0.07,rest],Graphics3D[{Black,Ball[{0,0,0},0.16]}],​​ParametricPlot3Dζ[t],t,2-2-
2

θ1
2π
,22-
2

θ1
2π
,PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.08,rest],​​ParametricPlot3D
ζ
1
[t],t,2-2-
2

θ1
2π
,22-
2

θ1
2π
,PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.08,rest],​​ParametricPlot3D
ζ
2
[t],t,2-2+
2

θ1
2π
,22+
2

θ1
2π
,PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.08,rest],​​ParametricPlot3D
ζ
3
[t],t,2-2+
2

θ1
2π
,22+
2

θ1
2π
,PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.08,rest],​​ParametricPlot3D[σ[u,v],{u,0,θ1},{v,-8,8},MeshNone,PerformanceGoal"Quality",ColorFunction"BlueGreenYellow",PlotPoints->40,PlotStyle->Opacity[0.4],MeshFalse,BoundaryStyleDirective[Black,Thick]],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-3,3,1},ImageSize900,Style["Intersection curves between cylinders :",Bold,Large],{{θ1,0.00002,"Value (θ)"},0.00001,2π,0.00001},ControlPlacementTop
View from different angles (below):