How can I map a 2D flat picture on a curved surface whilst maintaining distances and angles between points of the flat picture as much as possible?
2026-03-26 06:26:24.1774506384
Mapping on a surface
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In general preserving all distances will not be possible at all, while preserving angles is possible for some kinds of surfaces. The uniformization theorem allows for some classification for the latter. Usually there is a balance between distortion of angles and distances: the more you preserve one, the more severely you will distort the other. So without further details on how to balence them against one another, it is impossible to come up with a formalization of what would make an optimal solution.
One exception would be developable surfaces which thanks to their zero Gaussian curvature are already intrinsically flat. Those can be mapped to the plane with perfect preservation of angles and distances.
In the case of a sphere as curved surface, the topic has been intensely studied for the purpose of map-making. See Wikipedia articles for map projections and list of map projections. The latter has distance-preserving maps listed as equidistant, angle-preserving maps as conformal, area-preserving as equal-area, and many characterized as compromise since they balance these goals in various ways. Note that while angle and area preservation are global properties, equidistant projections will only preserve some distances, e.g. those to a given point of reference (azimuthal).