If you have an image sheet of facets and data to indicate which point on each facet should connect with other points on the other facets, is it possible to position those facets in 3D space based on the connecting points? Essentially turning 2D images into 3D objects. Similar to paper craft folding.

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No. You are missing the convex/concave information.

Each 3d point can be determined by solving a simultaneous equation. Unfortunately it will have two solutions, an ‘out of the page’ or ‘into the page’ solution.

With this additional information, its still not a trivial problem. you need to determine the order of the folds, calculate the required fold angle and apply rotational matrix transformations to the joined vertices recursively.

Also, your paper craft example is the simplest version of the problem, with each fold on a single axis. Some paper folds rely on the bendyness of the paper eg inside/outside reverse folds

http://www.origami-instructions.com/origami-inside-and-outside-reverse-folds.html

My feeling is that, if you include enough information for a unique solution in the ‘instructions’ file, you would basically have just invented a rather long winded 3d file format, rather than achieve you goal of (presumably) automatically generate 3D models from paper folding instructions

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