I am currently working with a triangulated surface, represented by a list of vertices (x, y, z coordinates) and triangles defined by three vertices each. Notably, the triangles vary in size, and their centroids do not form a regular lattice. My objective is to approximate the surface using equi-sized and equi-spaced squares. I aim to accomplish this through a two-step process:
Step 1: Cubic Lattice Generation I intend to establish a cubic lattice with lattice points defined as $$ \vec{x}_{n_1, n_2, n_3} = a (n_1 \hat{x} + n_2 \hat{y}+n_3 \hat{z}) $$ where $a$ denotes the lattice spacing, and $n_1, n_2, n_3 \in \mathbb{N}$.
Step 2: Surface Discretization For each triangle, my plan is to filter out lattice points within an $\epsilon$-neighbourhood. Subsequently, I will construct a small square (side length $\dfrac{a}{\sqrt{2}}$) centered on each selected lattice point, oriented parallel to the original triangle.
I welcome any insights, suggestions, or improvements regarding this proposed approach, particularly if there is an efficient method for executing Step 2. Additionally, I am open to exploring alternative methods for achieving discretization with squares. Thank you in advance for your assistance!