You always, either directly or implicitly, work in three-dimensional homogeneous space. That is why all coordinates, even texture coordinates, are four dimensional. In three-dimensional homogeneous space, your texture coordinates as well as the interpolation are linear. The non-linearity comes from the division, that is correct, but the division happens after interpolation.

You always, either directly or implicitly, work in three-dimensional homogeneous space. That is why all coordinates, even texture coordinates, are four dimensional. In three-dimensional homogeneous space, your texture coordinates and the interpolation are linear. The non-linearity comes from the division, that is correct, but the division happens after interpolation.