ParaboloidCircumscribedSphere::usage="The axis of a circular paraboloid 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
}
= 4 (h + t A)​, with A, B ≠ 0 and t ≥ -
h
A
​, because the radius of each element of the family is r = 2
h+tA
.The paraboloid is circumscribed to the family of spheres. That is, the paraboloid 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))
==4(h+tA),
∂
t
(
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
-4(h+tA))==0,{x,y}]​​x
1
50
-98+525t-35z-
-196+2100t-1875
2
t
-140z+750tz-75
2
z
,y
1
50
14-75t+5z-7
-196+2100t-1875
2
t
-140z+750tz-75
2
z
,x
1
50
-98+525t-35z+
-196+2100t-1875
2
t
-140z+750tz-75
2
z
,y
1
50
14-75t+5z+7
-196+2100t-1875
2
t
-140z+750tz-75
2
z
;​​Eliminatex==
1
50
-98+525t-35z-
-196+2100t-1875
2
t
-140z+750tz-75
2
z
,y==
1
50
14-75t+5z-7
-196+2100t-1875
2
t
-140z+750tz-75
2
z
,z==z,t​​-98-98x+13
2
x
+14y+7xy+37
2
y
-70z-35xz+5yz+25
2
z
0;​​​​Reduce[h+tA>=0,t]​​t>=0;Graphics3DOpacity[0.8],Red,Specularity[White,40],Sphere{h+t1A,k+t1B,l+t1C1},2
h+t1A
,ParametricPlot3D[{h,k,l}+t2{A,B,C1},{t2,-8,8},PlotStyle{DarkGreen,Thick}],ContourPlot3D[-98-98x+13
2
x
+14y+7xy+37
2
y
-70z-35xz+5yz+25
2
z
0,{x,-15,15},{y,-15,15},{z,-3,25},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{{-15,15},{-15,15},{-3,25}},ImageSize700,ViewPoint{1.3,-2.4,2.}
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))
==4(h+tA),
∂
t
(
2
(x-(h+tA))
+
2
(y-(k+tB))
+
2
(z-(l+tC1))
-4(h+tA))==0,{x,y}]​​x
1
2
10+3t-z-
-148-36t-3
2
t
+44z+6tz-3
2
z
,y
1
2
-4+3t-z+
-148-36t-3
2
t
+44z+6tz-3
2
z
,x
1
2
10+3t-z+
-148-36t-3
2
t
+44z+6tz-3
2
z
,y
1
2
-4+3t-z-
-148-36t-3
2
t
+44z+6tz-3
2
z
;​​Eliminatex==
1
2
10+3t-z-
-148-36t-3
2
t
+44z+6tz-3
2
z
,y==
1
2
-4+3t-z+
-148-36t-3
2
t
+44z+6tz-3
2
z
,z==z,t​​123-3x+
2
x
+18y-xy+
2
y
-21z-xz-yz+
2
z
0;​​​​Reduce[h+tA>=0,t]​​t>=-2;Graphics3DOpacity[0.8],Red,Specularity[White,40],Sphere{h+t1A,k+t1B,l+t1C1},2
h+t1A
,Graphics3DOpacity[0.8],Red,Specularity[White,40],Sphereh+
t1
9
A,k+
t1
9
B,l+
t1
9
C1,2
h+
t1
9
A
,ParametricPlot3D[{h,k,l}+t2{A,B,C1},{t2,-5,20},PlotStyle{DarkGreen,Thick}],ContourPlot3D[123-3x+
2
x
+18y-xy+
2
y
-21z-xz-yz+
2
z
0,{x,-30,30},{y,-30,20},{z,0,35},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700,ViewPoint{4,-2.4,4.}
Out[]=