# How to turn a complex geometric figure inside out (Win 10, 64-bit)?

This question falls under several categories {software, computer graphics, graphic design, art, cartography, geographic information, mathematical transforms}, but the software (specifically for Windows 10 [64-bit]) to perform this function is what I'm seeking.

I'm trying to shift geometric figures across a sphere:

As you can see from the picture above, all points are meant to follow meridional paths, stretching the figure outward around the surface's radial slices like an embedded rubber band until it crosses a great circle, then contracting back down to its original size at the far end of the sphere (so that the figure is now turned inside out).

The picture below shows the same intended transform on an object with a centroid that's outside of its borders:

Basically, it's a question of flipping a figure's borderline radial distances around its mean radius (everting dents and inverting bulges). I know what I want, but I don't know how to get there.

It's clearly not a rotation, translation (in the usual sense of being identical to a rotation), reflection (though that's close), scale, shear, or skew. The closest terminology that I've imagined so far is a "sleeve-transform on genus 0 topology", but I haven't found the right terms for a decent Google search on the underlying mathematical techniques with which to approach the transform.

Right now, I'm doing it by hand -- finding the inscribing and circumscribing circles, taking the centroid as a locus from which to measure the edge-coordinates, and flipping the annulus of circles around the mean annular radius. A pain in the ass, but functional for figures that contain their centroid within their boundary (simple polygons are easy, but not something complex like a country's border, and definitely no good for figures with an external centroid (e.g.: the majuscule letter "F", or a sans serif minuscule "f").

/// EDIT (PATH TO POSSIBLE SOLUTION): I have this question in two other websites, and in one of them someone suggested inverting a stereographic projection. I haven't had the time to pursue this possibility yet, but hopefully the suggestion could be of use to others. ///

• What are you looking to obtain from this? The geometrically accurate result of said operation? An approximation? Just a simple picture of it? An animation of the process? Blender might be able to do an animation of it, but not sure about it's mathematical accuracy Jul 9, 2018 at 0:48
• I'm hoping to turn the borders of geographical and political areas inside out. No animation is needed, just a static picture. Since I would use it only for an RPG (and possibly some fictional writing), accuracy isn't extremely necessary (as long as it's a reasonably close approximation of the "accurate" inversion, it's sufficient -- and of course, "accuracy" is relative, since the inversion would sacrifice either the border's curvatures or their radial distances). Jul 9, 2018 at 5:29
• You might be able to achieve this using Blender, but it will certainly require some investment to learn it Jul 9, 2018 at 5:49
• It's worth a shot, since I'm already working with several other programs for 2D stuff (mostly AutoREALM, Inkscape, and Paint.net) -- I haven't really messed with GIMP much, much less Krita or Blender, but I have them in my system. If nothing else comes up or seems to work, then I'll just have to hope that one of those will. :-) Jul 9, 2018 at 8:33