# Equal Area Sampling on Curved Surface:

I have a quantity $\beta(\mathbf{x}) \in \mathbb{R}$ that I wish to compute on a curved, smooth surface defined by $\{\mathbf{x}: \Gamma(\mathbf{x})=0\} \subset \mathbb{R}^{3}$. (This surface is approximated by a trimesh.) Since I want to display the quantity as a color on the surface, I need to sample, then bin via $\sum_{\mathbf{x} \in \text{bin cell}} \beta(\mathbf{x})/\text{bin cell area}$.

How does one efficiently choose the bin cells so that they all have the same area, and how does one choose the points at which to evaluate the quantity so that each point corresponds roughly to the same subarea of the bin cell? References and answers appreciated.