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I'm trying to create a geodesic dome with the stars of the northern night sky inscribed on the outside. The highest point of the dome would be the celestial north pole. How do I create a map that projects onto the "net" of the geodesic dome, such that when I assemble the dome, the stars are in the correct positions? I have an equirectangular map that I have rotated 90 degrees north so it is centered on the pole.

My existing map is a high-resolution jpeg file in an equirectangular projection. I have some experience creating and manipulating panoramic images, but much less experience converting between map projections! I would like to use a computer program to convert this projection into a printable template if possible.

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    $\begingroup$ thank you for your response! I will edit my question to be more clear. $\endgroup$
    – Rúnatál Davino
    Mar 29, 2019 at 12:26
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    $\begingroup$ I see, great. Now you don't have a spherical printer for spherical paper, so I think you will have to print out each face of the geodetic dome separately. Do you have the coordinates of the vertices of the mesh? Or at least the name of the polyhedron that represents the shape? $\endgroup$
    – uhoh
    Mar 29, 2019 at 12:31
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    $\begingroup$ It would most likely be a two-frequency geodesic dome based on an icosahedron! $\endgroup$
    – Rúnatál Davino
    Mar 29, 2019 at 12:35
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    $\begingroup$ I'm voting to close this question as off-topic because it is a question about geometric projection and computer algorithms. $\endgroup$
    – Carl Witthoft
    Mar 29, 2019 at 14:55
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    $\begingroup$ This question is on topic. Projecting the celestial sphere on to non-optimal approximations of a hemisphere is planetariums 101. There might be other sites where it is also on topic (thus my earlier comment about it being a cross-disciplinary question). After reading answer(s) here you can always ask a follow-up question on another site, say Mathematics SE if necessary. $\endgroup$
    – uhoh
    Mar 29, 2019 at 23:01

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The mathematically ideal scheme would be a separate gnomonic projection (see Wikipedia or IMO) for each tile of the dome. Many planetarium programs implement this projection as an option. Fortunately, each tile boundary is a straight line on both adjoining maps. Unfortunately, continuity across tile boundaries requires some care in choosing each tile's scale and center of projection.

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