Given some arbitrary curve or surface in 3D, if I wanted to obtain parameterize function. What are modern (and by hand) analytic methods do so?.
2026-03-26 16:05:45.1774541145
What are analytic methods used define a parametric equation for an arbitrary curve or surface in 3D?
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If you know your curve or surface by having samples (i.e., many points on the surface), then various kinds of splines give good approximate fittings to data. If you know it as an implicit surface, then the implicit function theorem gives ways to parametrize it, although it may involve root-finding, which is computationally intractable.
Finally, there are a great many papers from the last 10 years or so in Computer Graphics on the subject of "parameterization" in which surfaces represented by polygonal meshes are given parametric descriptions, although usually the parametric description is of some smooth-ish surface that's well approximated by the input poly-mesh. See recent proceedings of the SIGGRAPH conference, for instance, as a starting point for many papers on this topic.