Hi Eddie,

I actually had the wrong idea about stereographic projection - sorry! I thought the light source were located at an infinite distance above the north pole.

Now I tried a hopefully correct approach - but failed! Here is what I did:

I first calculated a "focused" version of this 3D-spiral:

spiralRegion3D = DiscretizeRegion@RegionProduct[spiralRegion, DiscretizeRegion@Line[{{-1}, {1}}]];
polygs = MeshPrimitives[spiralRegion3D, 2];
invStTr[x_, y_, z_] := With[{mag = 1. - (z + 1.)/2.}, {x mag, y mag, z}]
polygsTrans = polygs /. Polygon[cl_] :> Polygon[invStTr @@@ cl];
focusSpiralRegion3D = DiscretizeGraphics@Graphics3D[polygsTrans]

The tip of this structure is located at {0,0,1}, i.e. at the north pole of the sphere, (Normally the above transformation is supposed to be done using TransformedRegion, but I could not make that work!)

OK, the simple remaining task is to calculate the difference of this and - as a simplified case - a ball, but this does not work either:

RegionUnion does work and at least shows my idea:

If I am doing something wrong I have no idea what my mistake could be! So - either anyone can help us here, or we have to wait for an improved version of Mathematica. Again: Sorry!

... a logarithmic spiral projected onto a sphere, ie, the spiral is the cutter.

Eddie,

I hope I understand your question correctly. My approach is to start with the wanted shadow. So I begin by constructing the respective region:

logSpiral[t_] := {Sin[t], Cos[t]} Exp[.1 t];
thickn = .02;
img = AlphaChannel[ppl = ParametricPlot[logSpiral[t], {t, -10 Pi, 0}, 
     PlotStyle -> {Black, Thickness[thickn]}, Axes -> False, 
     PlotRangePadding -> thickn .75]];
plRange = PlotRange /. AbsoluteOptions[ppl, PlotRange];
spiralRegion = RegionResize[DiscretizeRegion[ImageMesh[img]], plRange];
shadowRegion = DiscretizeRegion @ RegionDifference[DiscretizeRegion@Disk[], spiralRegion]

Now a spherical function can simple be plotted onto that region:

Plot3D[Sqrt[1 - x^2 - +y^2], {x, y} \[Element] shadowRegion, 
 PlotPoints -> 100, ImageSize -> Large, Boxed -> False, Axes -> False,
 BoxRatios -> Automatic,
 PlotTheme -> "ThickSurface"]

Notice the option PlotTheme -> "ThickSurface" witch makes it suitable for 3D printing.

Well, at first glance this looks good, but in fact it does not really work the way you probably want. The problem is that now the sphere is cut perpendicular to the surface - and not along the z-axis. I tried with RegionProduce and RegionDilation and ShellRegion, etc., but I could not get a better result. Nevertheless I hope this is helpful.