Degree elevation and simple-knot insertion preserve the identity patch

The degree-two Bernstein functions are $B_0^2(t)=(1-t{)}^2$, $B_1^2(t)=2t(1-t)$, and $B_2^2(t)=t^2$. Thus ${\sum }_{i=0}^2{B}_i^2(t)=1$ and ${\sum }_{i=0}^2(i\mathrm{/}2)B_i^2(t)=t$. For $P_{ij}=(i\mathrm{/}2,j\mathrm{/}2)$, unit weights make the tensor denominator one; summing the first and second control coordinates yields $\xi$ and $\eta$. Hence the elevated representation is still exactly the identity map, with positive weights.

The midpoint $1\mathrm{/}2$ is new and lies strictly inside the single nonempty span of the clamped degree-two Bernstein knot vector. Apply [F3] to each fixed-coordinate control-net slice first in $\xi$, then in $\eta$. Each insertion preserves the entire homogeneous geometry and positive weights, so the final patch is exactly $F(\xi ,\eta )=(\xi ,\eta )$. The resulting degree-two univariate knot vectors have a simple interior knot; [F2] gives $C^1$ continuity in both coordinate directions. This is the stated k-refinement, and no general p/k geometry-preservation claim is needed.