geometry - How can I sample uniformly from a surface?

just to develop what Fedja said above:

say you have a map $phi(x,y)=(u(x,y),v(x,y),w(x,y))$ from $D = [a;b]^2$ to $mathbb{R}^3$ that represents a surface (Klein bottle in your initial example). The infinitesimal surface $(x;x+dx) times (y;y+dy)$ is mapped to a surface of area $|partial_x Phi wedge partial_y Phi| dx dy = A(x,y) dx dy$. So it would suffice to sample points inside $D$ distributed according to a probability distribution $p(x,y) propto A(x,y) dx dy$.

1: as Fedja mentioned it, you can use a rejection sampling approach - you just need to find a bound on $|A|_{infty}$, which should be very easy since $D$ is simple.

2: you can use a MCMC approach, which is equally easily implemented (e.g. independence sampler) - might be a little bit faster.

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