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I am aware of many methods (like the marching cubes algorithm) which, given a surface in $\mathbb{R}^3$ described by an implicit function $f:\mathbb{R}^3 \rightarrow \{0\} \subset \mathbb{R}$, convert the surface to a triangle mesh.

However, I have been unable to find any literature documenting methods for representing arbitrary triangle meshes by implicit surface equations. Has this problem of converting triangle meshes to implicit surfaces been addressed in any papers to date? If so, what methods have been used to solve this problem (or attempt to solve it)?

EDIT: I am looking for methods that approximate the input triangle mesh, like the method mentioned in paper "Implicit Surfaces That Interpolate" by G. Turk, H. Dinh, et al., in addition to methods that reconstruct the mesh exactly in a way possibly similar to some of the answers to question Is there any equation for triangle?.

FabrizzioMuzz
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    Such a conversion (triangular mesh to a surface) is not always possible. – Moishe Kohan Mar 03 '23 at 15:34
  • Could you elaborate? – FabrizzioMuzz Mar 04 '23 at 00:09
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    There are many obstructions. The simplest example is the graph which is the union of three triangles sharing a common edge. Once you fill in the triangles (by 2-dimensional faces), you realize that the result is not a surface; it cannot be even embedded in a surface. Another simple example is two linked triangles. Of course, maybe there are some unstated assumptions in your question to rule out such examples... – Moishe Kohan Mar 04 '23 at 00:37
  • Are you referring to the union of three triangles sharing an edge in $\mathbb{R}^3$? Is the union not a non-manifold surface? – FabrizzioMuzz Mar 04 '23 at 04:09
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    Yes, this is what I wrote. – Moishe Kohan Mar 04 '23 at 10:23

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I was going down the same rabbit hole myself recently, and found this review paper. hope it's a starting point for others looking for some overview on the matter:

Jones, M. W., Bærentzen, J. A., & Sramek, M. (2006). 3D distance fields: A survey of techniques and applications. IEEE Transactions on Visualization and Computer Graphics, 12(4), 518–599. https://doi.org/10.1109/TVCG.2006.56

FlashDD
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    Just dropping a (not freely available) reference without any other comment does not help much. At what pages is the question addressed? – Alex M. Mar 03 '23 at 15:29
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    The question asked for papers, and most people looking for papers either have university access or know about SciHub. I would assume putting the pdf here is not very legal. the question is addressed directly from page 3 onwards. besides, this question was left without an answer for 6 months, a reference to a decent review paper is a great starting point for anyone looking for info, and it's def better than nothing. if you have a better answer, please do post it... – FlashDD Mar 03 '23 at 15:40