TwoLeafCircumscribedSphere::usage="The axis of a two-leaf hyperboloid 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
}
= h + t A
2
)
-
2
a
​, with A, B ≠ 0 and t ∈  - -
2
a
2
A
-
h
A
​,
2
a
2
A
-
h
A
​​, because the radius of each element of the family is r =
2
(h+tA)
-
2
a
.The two-leaf hyperboloid is circumscribed to the family of spheres. That is, the two-leaf hyperboloid is the envelope of the family of spheres.Reference: Personal. ";
Example 1:
In[]:=
Showa=4;h=2;k=0;l=-7;A=1;B=-1;C1=2;t1=-8;​​Solve[
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
==
2
(h+tA)
-
2
a
,
∂
t
(
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
-(
2
(h+tA)
-
2
a
))==0,{x,y}]​​x
1
2
-12+5t-2z-
-378+160t-15
2
t
-92z+20tz-6
2
z
,y
1
2
16-5t+2z-
-378+160t-15
2
t
-92z+20tz-6
2
z
,x
1
2
-12+5t-2z+
-378+160t-15
2
t
-92z+20tz-6
2
z
,y
1
2
16-5t+2z+
-378+160t-15
2
t
-92z+20tz-6
2
z
;​​Eliminatex==
1
2
-12+5t-2z-
-378+160t-15
2
t
-92z+20tz-6
2
z
,y==
1
2
16-5t+2z-
-378+160t-15
2
t
-92z+20tz-6
2
z
,z==z,t​​129-48x+4
2
x
+28y+2xy+4
2
y
+14z-4xz+4yz+
2
z
0;​​Reduce[
2
(h+tA)
-
2
a
>=0,t]​​t≤-6||t≥2;​​​​Graphics3DOpacity[0.8],Red,Specularity[White,40],Sphere{h+t1A,k+t1B,l+t1C1},
2
(h+t1A)
-
2
a
,ParametricPlot3D[{h,k,l}+t2{A,B,C1},{t2,-15,12},PlotStyle{DarkGreen,Thick}],ContourPlot3D[129-48x+4
2
x
+28y+2xy+4
2
y
+14z-4xz+4yz+
2
z
0,{x,-25,25},{y,-25,25},{z,-40,20},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{{-25,25},{-25,25},{-40,20}},ImageSize700,ViewPoint{4,2.4,2.}
Out[]=
Example 2:
In[]:=
Showa=4;h=6;k=4;l=-3;A=1;B=1;C1=1;t1=5;​​Solve[
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
==
2
(h+tA)
-
2
a
,
∂
t
(
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
-(
2
(h+tA)
-
2
a
))==0,{x,y}]​​x
1
2
3+2t-z-
-59+36t-30z+4tz-3
2
z
,y
1
2
-1+2t-z+
-59+36t-30z+4tz-3
2
z
,x
1
2
3+2t-z+
-59+36t-30z+4tz-3
2
z
,y
1
2
-1+2t-z-
-59+36t-30z+4tz-3
2
z
;​​Eliminatex==
1
2
3+2t-z-
-59+36t-30z+4tz-3
2
z
,y==
1
2
-1+2t-z+
-59+36t-30z+4tz-3
2
z
,z==z,t​​81-22x+
2
x
-14y-2xy+
2
y
+14z-2xz-2yz+
2
z
0;​​​​Reduce[
2
(h+tA)
-
2
a
>=0,t]​​t≤-10||t≥-2;​​​​Graphics3DOpacity[0.8],Red,Specularity[White,40],Sphere{h+t1A,k+t1B,l+t1C1},
2
(h+t1A)
-
2
a
,Graphics3DOpacity[0.8],Red,Specularity[White,40],Sphere{h-(t1+10)A,k-(t1+10)B,l-(t1+10)C1},
2
(h-(t1+10)A)
-
2
a
,ParametricPlot3D[{h,k,l}+t2{A,B,C1},{t2,-30,20},PlotStyle{DarkGreen,Thick}],ContourPlot3D[81-22x+
2
x
-14y-2xy+
2
y
+14z-2xz-2yz+
2
z
0,{x,-40,40},{y,-40,40},{z,-45,40},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{{-40,40},{-40,40},{-45,40}},ImageSize700,ViewPoint{1.3,-2.4,2.}
Out[]=