Project a half-sphere onto a disk to get to cartesian coordinates

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I guess this is an age-old problem related to making maps of the world BUT

If I have latitude and longitudes of points on a hemisphere and want to get cartesian coordinates on the 2D plane out of that, can I do the following :

  • Flatten/press down the half hemisphere into a circle (of which radius ? Such that radial distances to the pole are conserved, or such that area is conserved ? Is direction from the pole always conserved ?)
  • Calculate a formula to go from long/lat to polar coordinates centered on the pole (is it a simple proportion ?)
  • Go to cartesian from there.