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ROAM again: wedgies

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I am trying to understand the ROAM algorithm and managed to do that till part 6 about the error metrics. Mark et.al. introduce the wedgies, and there is where the light goes off. I understand the concept of the wedgie: a volume around a triangle (and eventualy its children) with some tickness Et. They then state that w(v) = (vx, vy, z(v)) where (vx, vy) are the domain coordinates and z(v) is the hight at v. Then they introduce the affine heightmap for a bintree triangle T to be zt(x,y). I dont understand what this affine exactly means, and thus what the differences are between z(v) and zt(x,y). Is the only difference that the first takes a point and the other an (x,y) pair? Thanks already for the help.

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