Cyclid::usage="In this case, the inversion of a certain surface will be made on a sphere that may or may not be at the origin. Example is the sphere: + + = , the coordinates of the new points will be after the inversion: (x, y, z) FractionBox[\(\*SuperscriptBox[\(R\), \(2\)] . \((x, y, z)\)\), \(\*SuperscriptBox[\(x\), \(2\)] + \*SuperscriptBox[\(y\), \(2\)] + \*SuperscriptBox[\(z\), \(2\)]\)]. References: Wolfram MathWorld: ResourceFunction, InversionSurface; Personal. ";
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Manipulate[δ=ResourceFunction["InversionSurface"][{3,4,5},4,{(6+2Cos[u])Cos[v],(6+2Cos[u])Sin[v],2Sin[u]},{u,v}];Show[ParametricPlot3D[δ,{u,0,θ1},{v,0,θ1},PlotStyleOpacity[0.8],ColorFunction->"RedBlueTones",PerformanceGoal->"Quality"],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ImageSize900,PlotRangeAll,BoxRatiosAutomatic,ViewPoint{1.3,-2.4,32.}],Style["Cyclid: inversion of a toroid over a sphere :",Bold,Large],{{θ1,1,"Parameter (θ)"},0.01,2π,0.01},ControlPlacementTop]
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