Subdivide a triangle along its UVs
You can use barycentric coordinates to calculate points inside the triangle with a determinate value for uv.

The process is easy =>

Let's say that the UV coordinates of each vertex of the triangle are:

vertex 0 => u0,v0

vertex 1 => u1,v1

vertex 2 => u2,v2

A point inside the triangle has UV coordinates value expressed in function of its barycentric coordinates:

u coord = p0 * u0 + p1 * u1 + p2 * u2

v coord = p0 * v0 + p1 * v1 + p2 * v2

where p0,p1,p2 are the barycentric coordinates of this point.

As barycentric coordinates has the property:

p0 + p1 + p2 = 1

you can calculate the barycentric coordinates of a point with a specific value for UV just by solving these equations. I mean, for example, you want to know which point inside the triangle has

UV = (1,2)

To calculate such a point just do:

1 = p0 * u0 + p1 * u1 + p2 * u2

2 = p0 * v0 + p1 * v1 + p2 * v2

p0 + p1 + p2 = 1

as you know the values for (u0,v0), (u1,v1) and (u2,v2),  you can easily calculate the barycentric coordinates of this point, (p0, p1, p2).

Once you have the barycentric coordinates, you can calculate the coordinates (px,py,pz) of the point:

(px,py,pz) = p0 * vertex0 coordinates + p1 * vertex 1 coordinates + p2 * vertex 2 coordinates