InversePedalNegativePedalCylinders::usage="In this notebook for each cylinder, the rotational and the quadratic, its inversion will be made with respect to a sphere with center at the origin and radius 4. Once you have the inverse you will get your pedal and negative-pedal surfaces and they will be explained. Reference: Personal.";
Rotational cylinder:
Animation of Pedal Surface
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SwatchLegend[{Yellow,Green,Gray},{"Cylinder","Inversion surface (Toroid)","Pedal surface of inversion surface"}]Manipulate[σ[u_,v_]:={Cos[u],Sin[u],v};α=ResourceFunction["InversionSurface"][{0,0,0},4,σ[u,v],{u,v}];=ResourceFunction["PedalSurface"][α,{0,0,0},{u,v}];Show[{ParametricPlot3D[σ[u,v],{u,0,2π},{v,-5,5},PlotStyle{Yellow,Opacity[1]},PerformanceGoal->"Quality",PlotPoints60,MeshFalse],ParametricPlot3D[α,{u,0,2π},{v,-5,5},PlotStyle{Green,Opacity[0.1]},PerformanceGoal->"Quality",PlotPoints60],ParametricPlot3D[,{u,0,t1},{v,-5,5},PlotStyle{Gray,Opacity[0.2]},PerformanceGoal->"Quality",PlotPoints60]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{1.3,-2.4,1},PlotRangeAll,ImageSize700],{t1,0.00001,2π,0.00001},SaveDefinitions->True,ControlPlacement->Top]
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Explanation
Inversion surface (toroid):
Making a cut with the plane y = 0:
That’s why the pedal surface of the inversion surface of the rotational cylinder is that way
Animation of Negative Pedal Surface
Explanation
That’s why the negative-pedal surface of the inversion surface of the rotational cylinder is that way
Quadratic cylinder:
Animation of Pedal Surface
Explanation
Inversion surface (toroid):
Making a cut with the plane x = 0:
That’s why the pedal surface of the inversion surface of the quadratic cylinder is that way
Animation of Negative Pedal Surface
Explanation
That’s why the negative-pedal surface of the inversion surface of the rotational cylinder is that way