GeometricLocus::usage="Find and identify the equation of the geometric locus of a point that moves in such a way that the sum of its distances to the coordinate planes is always equal to its distance to the origin. References: Lehmann, Charles H. Geometría Analítica. Mexico, City: Limusa, p. 411,2003; Personal";
Applying the information:
Expand[
2
(|x|+|y|+|z|)
-(
2
x
+
2
y
+
2
z
)]
2xy+2xz+2yz=0;
xy+xz+yz=0;
In[]:=
ManipulateClearAll;Ifx1+y1==0,z1=-
1
2
x1,z1=-
x1y1
x1+y1
;d1=N[Abs[y1]];d2=N[Abs[x1]];d3=N[Abs[z1]];d4=N
2
x1
+
2
y1
+
2
z1
;​​w=Grid[{{
P-XZ
,
P-YZ
,
P-XY
},{d1,d2,d3}},FrameAll,ItemStyle->15,ItemStyle->15];​​q=Grid{
O-P
,NORM},d4,
2
d1
+
2
d2
+
2
d3
,FrameAll,ItemStyle->15,ItemStyle->15;​​Show[{ContourPlot3D[xy+xz+yz==0,{x,-20,20},{y,-20,20},{z,-20,20},Mesh->None,ContourStyleDirective[Green,Opacity[0.1],Specularity[White,30]],PerformanceGoal"Quality"],​​ContourPlot3D[y==0,{x,-20,20},{y,-20,20},{z,-20,20},Mesh->None,ContourStyleDirective[Blue,Opacity[0.1],Specularity[White,30]],PerformanceGoal"Quality"],​​ContourPlot3D[x==0,{x,-20,20},{y,-20,20},{z,-20,20},Mesh->None,ContourStyleDirective[Yellow,Opacity[0.1],Specularity[White,30]],PerformanceGoal"Quality"],​​ContourPlot3D[z==0,{x,-20,20},{y,-20,20},{z,-20,20},Mesh->None,ContourStyleDirective[Red,Opacity[0.1],Specularity[White,30]],PerformanceGoal"Quality"],​​Graphics3D[{Red,Ball[{x1,y1,z1},0.32]}],Graphics3D[{Black,Ball[{x1,0,z1},0.32]}],Graphics3D[{Black,Ball[{0,y1,z1},0.32]}],Graphics3D[{Black,Ball[{x1,y1,0},0.32]}],Graphics3D[{Black,Ball[{0,0,0},0.32]}],Graphics3D[{Black,Dashed,Thick,Line[{{x1,y1,z1},{x1,0,z1}}]}],Graphics3D[{Black,Dashed,Thick,Line[{{x1,y1,z1},{0,y1,z1}}]}],​​Graphics3D[{Black,Dashed,Thick,Line[{{x1,y1,z1},{x1,y1,0}}]}],Graphics3D[{Orange,Dashed,Thick,Line[{{x1,y1,z1},{0,0,0}}]}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{-0.5,-2,5}],Style["Cone:",Bold,Large],{{x1,-19.9,"Value (​
x
1
​)"},-20,20,0.1},Delimiter,{{y1,-19.9,"Value (​
y
1
​)"},-20,20,0.1},Delimiter,{{w,2,"Distance from point P to each coordinate plane"}},Delimiter,q,4,"Distance from point P to origin and the result of the norm of the vector sum of the distances of P to each coordinate plane",ControlPlacementTop
Out[]=
​
Cone:
Value (​
x
1
​)
Value (​
y
1
​)
Distance from point P to each coordinate plane
P-XZ
P-YZ
P-XY
20.
20.
10.
Distance from point P to origin and the result of the norm of the vector sum of the distances of P to each coordinate plane
O-P
NORM
30.
30.