ConeCircumscribedSphere::usage="The axis of a cone 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
)
​, with A, B ≠ 0 and r = h + t A > 0 , because the radius of each element of the family is r.For t > -
h
A
.The cone is circumscribed to the family of spheres. That is, the cone 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;​​Solve[
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
==
2
(h+tA)
,
∂
t
(
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
-
2
(h+tA)
)==0,{x,y}]​​x
1
50
182t-35z-
624
2
t
+260tz-75
2
z
,y
1
50
-26t+5z-7
624
2
t
+260tz-75
2
z
,x
1
50
182t-35z+
624
2
t
+260tz-75
2
z
,y
1
50
-26t+5z+7
624
2
t
+260tz-75
2
z
;​​Eliminatex==
1
50
182t-35z-
624
2
t
+260tz-75
2
z
,y==
1
50
-26t+5z-7
624
2
t
+260tz-75
2
z
,z==z,t​​23
2
x
-14xy-25
2
y
+70xz-10yz-
2
z
0;​​​​t>0;{Graphics3D[{Opacity[0.8],Red,Specularity[White,40],Sphere[{h+t1A,k+t1B,l+t1C1},h+t1A]}],ParametricPlot3D[{h,k,l}+t2{A,B,C1},{t2,-8,8},PlotStyle{DarkGreen,Thick}],ContourPlot3D[23
2
x
-14xy-25
2
y
+70xz-10yz-
2
z
0,{x,-30,30},{y,-30,30},{z,-25,30},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{{-30,30},{-30,30},{-25,30}},ImageSize900,ViewPoint{6,-30,2.}
Out[]=
Example 2:
In[]:=
Showh=2;k=-5;l=8;A=1;B=1;C1=1;t1=9;​​Solve[
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
==
2
(h+tA)
,
∂
t
(
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
-
2
(h+tA)
)==0,{x,y}]​​x
1
2
10+2t-
26+4t-3z
-6+z
-z,y-2+t+
1
2
26+4t-3z
-6+z
-
z
2
,x
1
2
10+2t+
26+4t-3z
-6+z
-z,y
1
2
-4+2t-
26+4t-3z
-6+z
-z;​​Eliminatex==
1
2
10+2t-
26+4t-3z
-6+z
-z,y==-2+t+
1
2
26+4t-3z
-6+z
-
z
2
,z==z,t​​169-2x+
2
x
+26y-2xy+
2
y
-26z-2xz-2yz+
2
z
0;​​t>-2;Graphics3D[{Opacity[0.8],Red,Specularity[White,40],Sphere[{h+t1A,k+t1B,l+t1C1},h+t1A]}],Graphics3DOpacity[0.8],Red,Specularity[White,40],Sphereh+
t1
9
A,k+
t1
9
B,l+
t1
9
C1,h+
t1
9
A,ParametricPlot3D[{h,k,l}+t2{A,B,C1},{t2,-12,20},PlotStyle{DarkGreen,Thick}],ContourPlot3D[169-2x+
2
x
+26y-2xy+
2
y
-26z-2xz-2yz+
2
z
0,{x,-30,30},{y,-30,30},{z,-15,30},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,-10,4.}
Out[]=