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InversionOfParaboloidOnSphere::usage="In this case it is the inversion of a circular paraboloid on a sphere that may or may not be at the origin. Example is the sphere:
2
x
+
2
y
+
2
z
=
2
R
​, the coordinates of the new points after the the inversion are: (x, y, z)  FractionBox[\(\*SuperscriptBox[\(R\), \(2\)] . \((x, y, z)\)\), \(\*SuperscriptBox[\(x\), \(2\)] + \*SuperscriptBox[\(y\), \(2\)] + \*SuperscriptBox[\(z\), \(2\)]\)]​. References: Wolfram MathWorld: Inversion; Personal. ";
In[]:=
​​Manipulate[δ=ResourceFunction["InversionSurface"][{3,4,5},4,{uCos[v],uSin[v],
2
u
},{u,v}];Show[ParametricPlot3D[δ,{u,0,θ1},{v,0,θ1},PlotStyleOpacity[0.3],ColorFunction->"RedBlueTones",PerformanceGoal->"Quality",PlotPoints->60],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,PlotRangeAll,ViewPoint{7,-2.4,-2.},ImageSize900,BoxRatiosAutomatic],Style["Inversion of a paraboloid on a sphere :",Bold,Large],{{θ1,1,"Parameter (θ)"},0.01,3π,0.01},ControlPlacementTop]
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​
Inversion of a paraboloid on a sphere :
Parameter (θ)