LugarGeometricoSuperficies8Parametricas::usage="It is about obtaining the equation of the geometric locus of a point that moves in such a way that the sum of the squares of its distances to the X and Y axes is always the same. References: Lehman, Charles H. Geometría Analítica. Ed. Limusa: p.437, 2003.";
Applying the information:
In[]:=
Expand
2
Norm[{1,0,0}{-x,-y,-z}]
Norm[{1,0,0}]
+
2
Norm[{0,1,0}{-x,-y,-z}]
Norm[{0,1,0}]
-4
-4+
2
Abs[x]
+
2
Abs[y]
+2
2
Abs[z]
;
2
x
+
2
y
+2
2
z
=4;
In[]:=
Manipulated1=N[4
2
(Cos[x1]Sin[x1/2])
+2
2
(Sin[x1])
];d2=N[4
2
(Cos[x1]Cos[x1/2])
+2
2
(Sin[x1])
];​​w=Grid
2
(P-X)
,
2
(P-Y)
,SUM,{d1,d2,d1+d2},FrameAll,ItemSize->15,ItemStyle->15;​​Show[ParametricPlot3D[2Cos[v]Cos[u],2Cos[v]Sin[u],
2
Sin[v],{u,0,π},{v,0,2π},Mesh->None,PlotStyleDirective[Green,Opacity[0.3],Specularity[White,30]],PerformanceGoal"Quality"],​​Graphics3D[Red,Ball[2Cos[x1]Cos[x1/2],2Cos[x1]Sin[x1/2],
2
Sin[x1],0.05]],Graphics3D[{Black,Ball[{2Cos[x1]Cos[x1/2],0,0},0.05]}],Graphics3D[{Black,Ball[{0,2Cos[x1]Sin[x1/2],0},0.05]}],Graphics3D[Black,Dashed,Thick,Line[2Cos[x1]Cos[x1/2],2Cos[x1]Sin[x1/2],
2
Sin[x1],{2Cos[x1]Cos[x1/2],0,0}]],Graphics3D[Black,Dashed,Thick,Line[2Cos[x1]Cos[x1/2],2Cos[x1]Sin[x1/2],
2
Sin[x1],{0,2Cos[x1]Sin[x1/2],0}]],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSizeFull,ViewPoint{1.3,-2.4,2.}],Style["The geometric locus is an ellipsoid (oblate spheroid)",Bold,Large],{{x1,1,"Value (​
x
1
​)"},-7,7,0.1},Delimiter,{{w,2,"Squared distance from point P to X axis, squared distance from point P to Y axis and the sum"}},ControlPlacementTop
Out[]=
​
The geometric locus is an ellipsoid (oblate spheroid)
Value (​
x
1
​)
Squared distance from point P to X axis, squared distance from point P to Y axis and the sum
2
(P-X)
2
(P-Y)
SUM
1.68454
2.31546
4.