The problem sounds trivial: put a flat design onto a captured three-dimensional surface so an artist can see it on the actual subject. Every graphics toolkit has a decal projector for this. Ours produced results that looked correct in preview and wrong to anyone who knew the subject.
What a box projector does
The standard approach casts the design along a single axis, like a slide projector pointed at an object. On a flat or gently curved surface this is indistinguishable from correct. On a surface that curves in two directions at once it stretches the design in proportion to how steeply the surface turns away from the projection axis.
The distortion is not uniform, and it is worst exactly where the geometry is most interesting — which is, reliably, where the artist placed the important part of the design.
A projector answers "where does this ray land". The artist is asking "how far is it across the surface". Those are different questions.
Geodesic distance
The correct question is the distance travelled *along* the surface, not through the space around it. That is a shortest-path problem across the mesh, and it is solvable directly: treat the mesh as a graph, weight each edge by its real length, and compute distance from the placement point outward.
The design is then laid out in that distance field rather than in screen space. Two points a centimetre apart in the design end up a centimetre apart across the surface, whichever way the surface happens to be turning at that moment. Which is what the artist meant.
The costs
It is more expensive than a projector — a shortest-path traversal per placement rather than a single matrix. On a dense captured mesh that is enough to be felt, so the field is computed once per placement and cached, then re-used for every subsequent adjustment of the design within that placement.
It also depends entirely on mesh quality. A projector does not care whether your geometry has holes; a geodesic traversal cares enormously, because a hole is a discontinuity in the surface it is measuring across. Capture quality stopped being a nice-to-have and became a hard input to the feature.
Why keep the projector
Because it is right often enough to be useful. On a nearly flat region the two agree, and the projector is instant. The system uses geodesic placement as the default and falls back where the surface is provably flat enough for the difference to be below the tolerance anyone can perceive.