Fitting a mesh to a density function

Here is one possible interpretation of your question.

Assume a probability density function $f$ is given. Is there a sequence of triangulations $T_n$ with $\varepsilon_n$-equilateral triangles such that counting probability measure on nodes converges to $f$ and $\varepsilon_n\to 0$ as $n\to\infty$.

(Say a triangle is $\varepsilon_n$-equilateral if the ratio of maximum side length and minimum side length is $\le 1+\varepsilon$.)

I am almost sure that the answer is "YES" if and only if $f$ is conformal factor of a flat metric; i.e., if and only if $f=e^{2{\cdot}\phi}$ and $\Delta \phi\equiv 0$.

There is an analogy in mechanics that might help: think of the nodes of the mesh as being connected by springs, which have tension proportional to something meaningful, e.g. the integral of $f(x)$ along the segment $[X_i,X_j]$. Then, if you let it stabilize, you will get a mesh with nodes distributed roughly according to $f$; if you pre-process $f$ to make is smooth enough so that it does not change much on every initial triangle, you should end up with roughly equilateral triangles, too.