FamilyCircumferencesCylinder::usage="The plane π: x - 3y + 2z = 14 cut to the sphere ξ of radius 6 and center at the origin of coordinates in a circumference ζ. To find: (a) The center and radius of ζ. (b) The cartesian equation of the geometric locus described when moving ζ parallel to the plane π, keeping the center on the line that joins the centers of the circumference ζ and from the sphere ξ. Note: The equation of the straight crcular cylinder is, using the formula of distance from a point to a straight line, where

A
it is the director vector of the line (cylinder axis)​, P = {x, y, z} it's a point on the surface of cylinder and
P
0
it's a point on the cylinder axis. Then r is the radius of a cover. FractionBox[\( | \(\*OverscriptBox[\(A\), \(\)]\((P - \*SubscriptBox[\(P\), \(0\)]​)\)\)\(|\)\), \( | \*OverscriptBox[\(A\), \(\)]\(|\)\)]​ = r References: Internet: Repositorio.pucp.pe, 1.2 Superficies Notables en
3

​, Ejercicios propuestos, prob.- 5; Personal ";
(a) Obtaining the center and the radius of ζ :
ℒ={0,0,0}+t{1,-3,2};
In[]:=
Solve[t-3(-3t)+2(2t)==14,t]
Out[]=
{{t1}}
C={1,-3,2}
In[]:=
Solve[{
2
x
+
2
y
+
2
z
==36,x-3y+2z==14},{x,y}]
x
1
10
14-2z-3
2
82+28z-7
2
z
,y
1
10
-42+6z-
2
82+28z-7
2
z
,x
1
10
14-2z+3
2
82+28z-7
2
z
,y
1
10
-42+6z+
2
82+28z-7
2
z
;​​
In[]:=
ξ[z_]:=
1
10
14-2z-3
2
82+28z-7
2
z
,
1
10
-42+6z-
2
82+28z-7
2
z
,z;​​Reduce[82+28z-7
2
z
>=0,z]​​C1={1,-3,2};​​r=Norm[C1-ξ[0]]
Out[]=
1
7
(14-
770
)≤z≤
1
7
(14+
770
)
Out[]=
4+
2
-3+
1
10
(42+2
41
)
+
2
1+
1
10
(-14+6
41
)
Parametric equations of the circumference:
x=3y-2z+14;​​y=u;​​z=v;
In[]:=
{3,1,0}{1,-3,2}
{2,-6,-10};
ζ[t_]:={1,-3,2}+Cos[t]
4+
2
-3+
1
10
42+2
41

+
2
1+
1
10
-14+6
41

10
{3,1,0}+Sin[t]
4+
2
-3+
1
10
42+2
41

+
2
1+
1
10
-14+6
41

2
35
{2,-6,-10};
(b) The equation of the geometric locus of the cylinder:
In[]:=
{x-1,y+3,z-2}{1,-3,2}
Out[]=
{2y+3z,-2x+z,-3x-y}
In[]:=
Expand[
2
(2y+3z)
+
2
(-2x+z)
+
2
(-3x-y)
]
Out[]=
13
2
x
+6xy+5
2
y
-4xz+12yz+10
2
z
In[]:=
Norm[{1,-3,2}]
Out[]=
14
In[]:=
N
2
14
4+
2
-3+
1
10
42+2
41

+
2
1+
1
10
-14+6
41


Out[]=
308.
13
2
x
+6xy+5
2
y
-4xz+12yz+10
2
z
=308
In[]:=
Manipulate​​ShowParametricPlot3D[{6Cos[u]Sin[v],6Sin[u]Sin[v],6Cos[v]},{u,0,2π},{v,0,π},ColorFunction"LightTemperatureMap",MeshFalse,PlotStyleDirective[Specularity[White,30],Opacity[0.7]],PerformanceGoal->"Quality"],ContourPlot3D[13
2
x
+6xy+5
2
y
-4xz+12yz+10
2
z
==308,{x,-10,10},{y,-10,10},{z,-30,7.5t1},ContourStyle->Directive[Purple,Opacity[0.2],Specularity[White,30]],MeshNone,PerformanceGoal->"Quality"],ParametricPlot3D[{0,0,0}+t{1,-3,2},{t,-5,5},ColorFunction->"BrownCyanTones"]/.Line[pts_,rest___]:>Tube[pts,0.08,rest],​​ContourPlot3D[x-3y+2z==14,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle->Directive[LightRed,Opacity[0.2],Specularity[White,30]],MeshNone],Graphics3D[{Red,Ball[{0,0,0},0.2]}],​​Graphics3D[{Black,Ball[{1,-3,2},0.2]}],​​ParametricPlot3D{1,-3,2}+Cos[t]
4+
2
-3+
1
10
42+2
41

+
2
1+
1
10
-14+6
41

10
{3,1,0}+Sin[t]
4+
2
-3+
1
10
42+2
41

+
2
1+
1
10
-14+6
41

2
35
{2,-6,-10},{t,0,2π},PlotStyleBlack/.Line[pts_,rest___]:>Tube[pts,0.1,rest],TableParametricPlot3Dl{1,-3,2}+Cos[t]
4+
2
-3+
1
10
42+2
41

+
2
1+
1
10
-14+6
41

10
{3,1,0}+Sin[t]
4+
2
-3+
1
10
42+2
41

+
2
1+
1
10
-14+6
41

2
35
{2,-6,-10},{t,0,2π},PlotStyleYellow,{l,-4,t1,0.5}/.Line[pts_,rest___]:>Tube[pts,0.08,rest],​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{1.3,-2.4,-3},Style["Circumferences that form the cylinder",Bold,Large],{{t1,-4+0.00002,"Value (t1)"},-4+0.00001,4,0.5},ControlPlacementTop
Out[]=
​
Circumferences that form the cylinder
Value (t1)