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Given a compact manifold $M$ in $R^n$, $M = f(x)$, f(x) is infinitely differentiable. $x$ $\in$ $R^n$, I want to find a bunch of samples on the manifold.

Currently, I'm setting up an SQP optimization problem to solve for samples on the manifold, with a set of random samples in $R^n$. The optimization goal is $min |x_r - x|$, the optimization constraint is f(x) = 0, where $x_r$ is a random sample.

Is there a better way to sample the manifold or project random points onto the manifold? I know that computation geometry works have some methods like Coxeter triangulation, which triangulates the manifold and could be used for sampling points on the manifold as well. But I'm not sure whether it will be faster than my current method since it also computes a triangulation, which I don't need.

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  • $\begingroup$ I think given your ambient scalar field (not sure how dense it has to be) the "Marching cubes", a popular isosurface extraction algorithm, will output a triangulation of the surface, the vertices being your random samples at a chosen sparsity $\endgroup$
    – rych
    Commented Aug 27, 2022 at 11:23
  • $\begingroup$ It very much depends on what your manifold is. $\endgroup$
    – Igor Rivin
    Commented Aug 27, 2022 at 15:18
  • $\begingroup$ @rych I think the marching cubes algorithm will not perform well in higher dimensions. My manifold is in $R^6$ or $R^7$. $\endgroup$
    – Robin Lee
    Commented Aug 28, 2022 at 5:53
  • $\begingroup$ @IgorRivin The manifold is closed and without boundary. And I can obtain the exact function f(x) that defines the manifold. What other properties of the manifold would change the problem? $\endgroup$
    – Robin Lee
    Commented Aug 28, 2022 at 5:59

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