Linear precision of a geometric basis (${\sum }_j{\varphi }_j(u)=1,{\sum }_j{\xi }_j{\varphi }_j(u)=u$)

A basis with the displayed identities reproduces affine functions from their values at the precision sites ${\xi }_j$. For two such bases, the tensor-product precision sites are $({\xi }_j,{\eta }_k)$: summation independently recovers $u$ and $v$. Affine images of this site grid therefore produce exact planar parameterizations, independent of the chosen control-grid representation.