From a high level, what are we trying to do when we interpolate? Are we trying to map a point on the triangle in the 3-D model to a 2-D point in the texture map? If so, would the map be the same for all triangles, or different for each triangle?
I found this article to be helpful in understanding interpolation. https://www.scratchapixel.com/lessons/mathematics-physics-for-computer-graphics/interpolation and also this quora post https://www.quora.com/What-does-interpolating-in-computer-graphics-mean. So I don't think it necessarily means we are mapping to a different dimension, but we are more using approximation to figure out some points between some known values