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Inverse stereographic projection for 3d printing

Eddie Morris
Posted 4 years ago

Hi Folks, My first post here as a Mathematica newbie. I would like to 3d print the boolean operation of a logarithmic spiral projected onto a sphere, ie, the spiral is the cutter. The is idea is to to shine a light at the north pole of the sphere to see the spiral shadow on a plan(Wall). I have asked this question on Mathematica exchange here but did not get the final crucial steps so I thought I would ask here.

I have attached my latest attempt which outputs an error "reg1 must be a boolean function"

Any tips would be appreciated! Thanks
POSTED BY: Eddie Morris
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Eddie Morris
Eddie Morris
Posted 4 years ago
POSTED BY: Eddie Morris
POSTED BY: Henrik Schachner

OK, good - here is a much cleaner solution:

From above I use the expression spiralRegion; from this I calculate a 3D-spiral:

spiralRegion3D = DiscretizeRegion@
  RegionProduct[spiralRegion, DiscretizeRegion@Line[{{-.1}, {1.1}}]]

enter image description here

And then:

(* 3D-spiral cut out of a ball: *)
spiralBall = RegionDifference[DiscretizeRegion@Ball[], spiralRegion3D];
sphereThickness = .05;
(* sphere with thickness 'sphereThickness' as difference of two isocentric balls: *)
spiralSphere = RegionDifference[spiralBall, DiscretizeRegion@Ball[{0, 0, 0}, 1 - sphereThickness]];
(* lower half cut off: *)
spiralHalfSphere = RegionDifference[spiralSphere, Cuboid[{-2, -2, -2}, {2, 2, 0}]]

enter image description here

Now the cut is strictly parallel to an axis through the pole. The whole procedure seems to work only with the excessive use of DiscretizeRegion. Being a Region this structure should definitely be suitable for 3D-Printing.

Isn't Mathematica just great !?!
POSTED BY: Henrik Schachner

... 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]

enter image description here

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"]

enter image description here

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.
POSTED BY: Henrik Schachner
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