EllipsoidCircumscribedSphere::usage="The major axis of an ellipsoid 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
b
2
a
​​
2
a
- (h + t A
2
)
​)​, with A, B ≠ 0 and -
2
a
2
A
-
h
A
≤ t ≤
2
a
2
A
-
h
A
​, because the radius of each element of the family is r =
b
2
a
-
2
(h+tA)
a
with a > b.​The ellipsoid is circumscribed to the family of spheres. That is, the ellipsoid is the envelope of the family of spheres.Reference: Personal. ";
Example 1:
In[]:=
Showa=4;b=2;h=2;k=0;l=-7;A=1;B=-1;C1=2;t1=-2;​​Solve
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
==
2
b
2
a
(
2
a
-
2
(h+tA)
),
∂
t
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
-
2
b
2
a
(
2
a
-
2
(h+tA)
)==0,{x,y}​​​​x
1
8
-38+25t-8z-
-4388+2700t-425
2
t
-1312z+400tz-96
2
z
,y
1
8
54-25t+8z-
-4388+2700t-425
2
t
-1312z+400tz-96
2
z
,x
1
8
-38+25t-8z+
-4388+2700t-425
2
t
-1312z+400tz-96
2
z
,y
1
8
54-25t+8z+
-4388+2700t-425
2
t
-1312z+400tz-96
2
z
;​​Eliminatex==
1
8
-38+25t-8z-
-4388+2700t-425
2
t
-1312z+400tz-96
2
z
,y==
1
8
54-25t+8z-
-4388+2700t-425
2
t
-1312z+400tz-96
2
z
,z==z,t​​721-192x+21
2
x
+92y+8xy+21
2
y
+166z-16xz+16yz+9
2
z
0;​​Reduce[
2
a
-
2
(h+tA)
>=0,t]​​-6≤t≤2;​​​​Graphics3DOpacity[0.8],Red,Specularity[White,40],Sphere{h+t1A,k+t1B,l+t1C1},
b
2
a
-
2
(h+t1A)
a
,ParametricPlot3D[{h,k,l}+t2{A,B,C1},{t2,-8,8},PlotStyle{DarkGreen,Thick}],ContourPlot3D[721-192x+21
2
x
+92y+8xy+21
2
y
+166z-16xz+16yz+9
2
z
0,{x,-5,5},{y,-5,8},{z,-20,-1},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{{-5,5},{-5,8},{-20,-1}},ImageSize700,ViewPoint{4,2.4,2.}
Out[]=
Example 2:
In[]:=
Showa=3;b=2;h=2;k=-5;l=3;A=7;B=-1;C1=5;t1=0;​​Solve
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
==
2
b
2
a
(
2
a
-
2
(h+tA)
),
∂
t
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
-
2
b
2
a
(
2
a
-
2
(h+tA)
)==0,{x,y}​​x
1
450
2237+6097t-315z-
-63931-332722t-366691
2
t
+41490z+78390tz-6075
2
z
,y
1
450
-2441-871t+45z-7
-63931-332722t-366691
2
t
+41490z+78390tz-6075
2
z
,x
1
450
2237+6097t-315z+
-63931-332722t-366691
2
t
+41490z+78390tz-6075
2
z
,y
1
450
-2441-871t+45z+7
-63931-332722t-366691
2
t
+41490z+78390tz-6075
2
z
;​​Eliminatex==
1
450
2237+6097t-315z-
-63931-332722t-366691
2
t
+41490z+78390tz-6075
2
z
,y==
1
450
-2441-871t+45z-7
-63931-332722t-366691
2
t
+41490z+78390tz-6075
2
z
,z==z,t​​8301+792x+215
2
x
+3993y+63xy+431
2
y
-803z-315xz+45yz+323
2
z
0;​​Reduce[
2
a
-
2
(h+tA)
>=0,t]​​-
5
7
≤t≤
1
7
;​​​​Graphics3DOpacity[0.8],Red,Specularity[White,40],Sphere{h+t1A,k+t1B,l+t1C1},
b
2
a
-
2
(h+t1A)
a
,Graphics3DOpacity[0.8],Red,Specularity[White,40],Sphereh+t1-
4.5
7
A,k+t1-
4.5
7
B,l+t1-
4.5
7
C1,
b
2
a
-
2
h+t1-
4.5
7
A
a
,ParametricPlot3D[{h,k,l}+t2{A,B,C1},{t2,-8,8},PlotStyle{DarkGreen,Thick}],ContourPlot3D[8301+792x+215
2
x
+3993y+63xy+431
2
y
-803z-315xz+45yz+323
2
z
0,{x,-5,5},{y,-7,0},{z,-6,5},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{{-5,5},{-7,0},{-6,5}},ImageSize900,ViewPoint{4,5,2.}
Out[]=