Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 66 / 4 / iv / Solution

Let the univariate linear precision sites be ${\xi }_j,{\eta }_k$, meaningThen their Cartesian-product grid is a linear precision set for the tensor-product surface basis:Indeed, the first coordinate is $({\sum }_j{\xi }_j{\varphi }_j)({\sum }_k{\psi }_k)=u$, and the second is $({\sum }_j{\varphi }_j)({\sum }_k{\eta }_k{\psi }_k)=v$. More generally, take controls $P_{jk}=A+{\xi }_{j}B+{\eta }_{k}C$. The resulting parametric surface is exactlyThis proves linear precision of a geometric basis and affine map reproduction, including arbitrary changes of the embedding plane. In fact the products also reproduce the bilinear coordinate $uv$, since ${\sum }_{j,k}{\xi }_j{\eta }_k{\varphi }_j{\psi }_k=uv$; affine precision is the requested consequence.