CylinderCircumscribedSphere::usage="The axis of a cylinder is the line:ℒ: {h, k, l} + t {A, B, C1}. The family of spheres that has its center on that axis is:​(x - (h + t A)​
2
}
+ {y - (k + t B)​
2
}
+ {z - (l + t C1)​
2
}
=
2
r
​, with A, B ≠ 0 and r > 0 , because the radius of each element of the family is r.For t ϵ .The cylinder is circumscribed to the family of spheres. That is, the cylinder is the envelope of the family of spheres.Reference: Personal. ";
Example 1:
In[]:=
Showh=0;k=0;l=0;A=7;B=-1;C1=5;t1=1;r=3;​​Solve[
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
==
2
r
,
∂
t
(
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
-
2
r
)==0,{x,y}]​​x
1
10
105t-7z-
3
6-25
2
t
+10tz-
2
z
,y
1
10
-15t+z-7
3
6-25
2
t
+10tz-
2
z
,x
1
10
105t-7z+
3
6-25
2
t
+10tz-
2
z
,y
1
10
-15t+z+7
3
6-25
2
t
+10tz-
2
z
;​​Eliminatex==
1
10
105t-7z-
3
6-25
2
t
+10tz-
2
z
,y==
1
10
-15t+z-7
3
6-25
2
t
+10tz-
2
z
,z==z,t​​-675+26
2
x
+14xy+74
2
y
-70xz+10yz+50
2
z
0;{Graphics3D[{Opacity[0.8],Red,Specularity[White,40],Sphere[{h+t1A,k+t1B,l+t1C1},r]}],ParametricPlot3D[{h,k,l}+t2{A,B,C1},{t2,-8,8},PlotStyle{DarkGreen,Thick}],ContourPlot3D[-675+26
2
x
+14xy+74
2
y
-70xz+10yz+50
2
z
0,{x,-15,15},{y,-15,15},{z,-15,15},PerformanceGoal->"Quality",PlotPoints->60,MeshFalse,ContourStyle{Opacity[0.2],Green,Specularity[White,40]}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{{-15,15},{-15,15},{-15,15}},ImageSize900,ViewPoint{1.3,-2.4,2.}
Out[]=
Example 2:
In[]:=
Showh=2;k=-5;l=8;A=1;B=1;C1=1;r=4;t1=9;​​Solve[
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
==
2
r
,
∂
t
(
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
-
2
r
)==0,{x,y}]​​x
1
2
12+3t-z-
-160-48t-3
2
t
+48z+6tz-3
2
z
,y
1
2
-2+3t-z+
-160-48t-3
2
t
+48z+6tz-3
2
z
,x
1
2
12+3t-z+
-160-48t-3
2
t
+48z+6tz-3
2
z
,y
1
2
-2+3t-z-
-160-48t-3
2
t
+48z+6tz-3
2
z
;​​Eliminatex==
1
2
12+3t-z-
-160-48t-3
2
t
+48z+6tz-3
2
z
,y==
1
2
-2+3t-z+
-160-48t-3
2
t
+48z+6tz-3
2
z
,z==z,t​​103-x+
2
x
+20y-xy+
2
y
-19z-xz-yz+
2
z
0;Graphics3D[{Opacity[0.8],Red,Specularity[White,40],Sphere[{h+t1A,k+t1B,l+t1C1},r]}],Graphics3DOpacity[0.8],Red,Specularity[White,40],Sphereh+
t1
9
A,k+
t1
9
B,l+
t1
9
C1,r,ParametricPlot3D[{h,k,l}+t2{A,B,C1},{t2,-8,20},PlotStyle{DarkGreen,Thick}],ContourPlot3D[103-x+
2
x
+20y-xy+
2
y
-19z-xz-yz+
2
z
0,{x,-30,30},{y,-30,20},{z,0,25},PerformanceGoal->"Quality",PlotPoints->60,MeshFalse,ContourStyle{Opacity[0.2],Green,Specularity[White,40]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{4,-2.4,4.}
Out[]=