What area(s) of mathematics to uv parameterise a 3D triangulation?

I'm a CAD developer and have a surface made up of 3D triangles. I want to split a region of this surface up into a number of patches where each patch has 4 corners. Here's a screen capture of the region of the surface. (triangles not displayed, as they're too dense). .

The circles/spheres on the boundary lines I am given. They are the corners of the patches on the boundaries. All I need to find is the positions of the corners of the patches inside the region. The black marks I've added on the picture hopefully depict what I'm trying to find. (Although the corners on the boundary lines are equally spaced, that will not necessarily be the case).

EDIT: The corners need to be positioned proportionately. For example, in the picture the top boundary line has a green node/corner adjacent to the blue one which happens to be $\frac{1}{8}$ along the top boundary. The corresponding corner below (i.e. the leftmost black one I've drawn) also needs to be $\frac{1}{8}$ away from the left boundary. It also needs to be $\frac{1}{7}$ of the way "down" from the top boundary line. The actual proportions depend on the positioning of the corners/nodes on the boundary lines.

I've been trying to do accomplish this for many weeks! The "tools" I've been using so far is to create planes, section using the plane, cut out cross section relevant to the region, and walk a proportion along the line. This approach has been plagued with issues. A particular angled plane is not suitable along the entire height of the surface due to surface curvature changes, etc. I would like to approach this entirely differently (i.e. properly!)

Q: What area of mathematics will help me to do this?

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