CylinderConeOfRevolution::usage"A sphere of equation
2
x
+
2
y
+
2
z
= 16 and a plane x - 2y + 5z = 0 they intersect to form a circle that will be the directrix of a straight circular cylinder and straight circular cone. In the case of the cylinder, the generatrix are parallel to its axis. In the case of the cone it has its vertex at the point V​
2
3
​, -
4
3
​,
10
3
​. Obtain the parametric equations of the circumference and the cartesian equations of the cone and the cylinder. Reference: Personal.";
(a) Cylinder:
In[]:=
Solve[{
2
x
+
2
y
+
2
z
==16,x-2y+5z==0},{x,y}]
In[]:=
x
1
5
-5z-2
10
8-3
2
z
,y
1
5
-5z-2
10
8-3
2
z
,x
1
5
-5z+2
10
8-3
2
z
,y
1
5
10z+
10
8-3
2
z
;​​Reduce[8-3
2
z
>=0,z]
Out[]=
-2
2
3
≤z≤2
2
3
In[]:=
ξ[z_]:=
1
5
-5z+2
10
8-3
2
z
,
1
5
10z+
10
8-3
2
z
,z;​​ξ[0]
Out[]=

8
5
,
4
5
,0
Parametric equations of the directrix (circumference):
In[]:=
x=2y-5z;​​y=u;​​z=v;​​{2,1,0}{1,-2,5}​​
Out[]=
{5,-10,-5}
ζ[t_]:=Cos[t]
4
5
{2,1,0}+Sin[t]
4
130
{5,-10,-5};
Cylinder’s axis:
ℒ:t{1,-2,5}
Generatrix of cylinder:
ℒ
1
:
8
5
,
4
5
,0+t{1,-2,5};
Cartesian equation of the cylinder:
Norm[{1,-2,5}{x,y,z}]
Norm[{1,-2,5}]
==4
2
Abs[2x+y]
+
2
Abs[-5y-2z]
+
2
Abs[5x-z]
30
4
In[]:=
Expand
2

2
(2x+y)
+
2
(-5y-2z)
+
2
(5x-z)

==
2

30
4

29
2
x
+4xy+26
2
y
-10xz+20yz+5
2
z
480;
(b) Cone:
Generatrix of the cone:
ℒ
2
:
8
5
,
4
5
,0+t
2
3
,-
4
3
,
10
3
-
8
5
,
4
5
,0;​​
ℒ
2
:
8
5
,
4
5
,0+t
2
3
-
8
5
,-
4
3
-
4
5
,
10
3
;
It is calculated Cos[θ]:
In[]:=
Cos[θ]==

2
3
-
8
5
,-
4
3
-
4
5
,
10
3
.{1,-2,5}
Norm
2
3
-
8
5
,-
4
3
-
4
5
,
10
3
Norm[{1,-2,5}]
In[]:=
SimplifyCos[θ]
52
3
-
8
5
-2-
4
3
-
4
5
30
100
9
+
2
4
3
+
4
5
+
2
-
2
3
+
8
5

Out[]=
Cos[θ]
5
11
Cartesian equation of the cone:
In[]:=
5
11
==
{x,y,z}-
2
3
,-
4
3
,
10
3
.{1,-2,5}-
2
3
,-
4
3
,
10
3

Norm{x,y,z}-
2
3
,-
4
3
,
10
3
Norm{1,-2,5}-
2
3
,-
4
3
,
10
3

In[]:=
Simplify
5
11

3
10
1
3
-
2
3
+x-
2
3
4
3
+y+
5
3
-
10
3
+z
2
Abs-
2
3
+x
+
2
Abs
4
3
+y
+
2
Abs-
10
3
+z

x-2y+5(-4+z)
30
2
-
2
3
+x
+
2
4
3
+y
+
2
-
10
3
+z

5
11
In[]:=
Expand
2
5
30
2
-
2
3
+x
+
2
4
3
+y
+
2
-
10
3
+z
==
2

11
(x-2y+5z-20)

2000-200x+150
2
x
+400y+150
2
y
-1000z+150
2
z
4400-440x+11
2
x
+880y-44xy+44
2
y
-2200z+110xz-220yz+275
2
z
;​​139
2
x
+106
2
y
-125
2
z
+44xy-110xz+220yz+240x-480y+1200z-2400==0;​​
Animation of the cylinder:
In[]:=
Manipulateζ[t_]:=Cos[t]
4
5
{2,1,0}+Sin[t]
4
130
{5,-10,-5};​​ShowRevolutionPlot3D
8
5
+t,
4
5
-2t,5t,{t,-2,2},{θ,0,θ1},RevolutionAxis{1,-2,5},PlotStyle->Opacity[0.4],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow",PlotPoints->40,ParametricPlot3D[t{1,-2,5},{t,-2,2},ColorFunction->"DeepSeaColors"]/.Line[pts_,rest___]:>Tube[pts,0.07,rest],Graphics3D[{Black,Ball[{0,0,0},0.16]}],ParametricPlot3D[ζ[-t],{t,0,θ1},PlotStyle->Yellow]/.Line[pts_,rest___]:>Tube[pts,0.08,rest],​​ParametricPlot3D[{4Cos[u]Sin[v],4Sin[u]Sin[v],4Cos[v]},{u,0,π},{v,0,2π},MeshNone,PerformanceGoal"Quality",PlotStyleDirective[Red,Opacity[0.6],Specularity[White,20]]],ContourPlot3D[x-2y+5z==0,{x,-7,7},{y,-7,7},{z,-7,7},ContourStyle->Directive[Blue,Opacity[0.05],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality"],Graphics3D[{Yellow,Ball[ζ[-θ1],0.16]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-3,3,5},ImageSize900,Style["Cylinder with directrix circumference and generatrix parallel to its axis:",Bold,Large],{{θ1,0.00002,"Value (θ1)"},0.00001,2π,0.00001},ControlPlacementTop
Out[]=
​
Cylinder with directrix circumference and generatrix parallel to its axis:
Value (θ1)
animation of the cone: