I'm not aware of any techniques or theorems that would give an answer to the following question.

Suppose we have a smooth surface $\mathcal{S} \subset \mathbb{R}^3$, and also suppose that for each ${p} = (x,y,z)^T \in \mathcal{S}$ we have associate a vector $v \in \mathcal{T}_p$ (tangent space/plane at $p$). Is it possible to define a parameterization of the surface using such vector field?

I assume there're probably conditions I'm not mentioning in order to make this valid.

Does this problem have a name so I can look it up?

Thank you

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    $\begingroup$ Vector fields and parameterizations are different beasts, and it's unclear to me what you might mean by "defining a parameterization of a surface using a vector field". Can you give an exact definition? Or an example of what you mean? $\endgroup$ – Lee Mosher Feb 11 at 17:02
  • $\begingroup$ If you want more details I can give them. I read in a paper, a while ago, about "conjugate direction field".In practice such direction field seems to be used to define a parameterization of a given surface, how exactly I'm not 100% sure. This is a mesh processing / geometric modelling problem, what I was wondering is if such problem has a mathematical setting in general. By parameterization I mean what you would expect namely some mapping from parameter space to the manifold. $\endgroup$ – user8469759 Feb 11 at 17:05
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    $\begingroup$ If this is a mesh processing / geometric modelling problem then your tags are not very well chosen. Those tags are going to attract mostly people who are as puzzled as I am. $\endgroup$ – Lee Mosher Feb 11 at 18:53
  • $\begingroup$ All the concepts I mentioned come from differential geometry. $\endgroup$ – user8469759 Feb 11 at 19:06
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    $\begingroup$ I'm just trying to give you some advice. Yes, the term "parameterization" and the term "vector field" are common in differential geometry. But without some further hints as to the source of your question, or the kind of parameterization you are seeking, the question is quite unclear. $\endgroup$ – Lee Mosher Feb 11 at 19:11

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