Homogeneous knot insertion preserves an admissible NURBS patch

Write the homogeneous control array as ${\tilde{P}}_{\alpha }=(w_{\alpha }{P}_{\alpha },w_{\alpha })$. The tensor spline $\tilde{F}(\xi )={\sum }_{\alpha \in \mathcal{I}}{\tilde{P}}_{\alpha }{N}_{\alpha }(\xi )$ has first $d$ components $W(\xi )F(\xi )$ and last component $W(\xi )$ by [F3]. Fix the coordinate direction in which $u$ is inserted and fix a multi-index of all other coordinates. Grouping the finite sum over the remaining index writes the corresponding slice as a B-spline curve with controls in ${\mathbb{R}}^{d+1}$. By [F2], knot insertion leaves that slice unchanged for every value of the insertion coordinate. Multiplication by the fixed transverse tensor basis function and summation over all transverse indices show that the entire updated homogeneous spline ${\tilde{F}}^{\mathrm{\prime }}$ equals $\tilde{F}$ pointwise.

In the active range, each updated last coordinate is ${\alpha }_i{w}_i+(1-{\alpha }_i)w_{i-1}$ by [F1]; it is positive because both old weights are positive and $0<{\alpha }_i<1$. Copied weights are positive as well. Thus all updated weights are positive and [F4] gives $W^{\mathrm{\prime }}(\xi )>0$ on the closed patch. Since ${\tilde{F}}^{\mathrm{\prime }}=\tilde{F}$, their first $d$ components and last components agree, and division by this positive common last component gives $F^{\mathrm{\prime }}(\xi )=F(\xi )$ for every $\xi \in \overline{\widehat{\mathrm{\Omega }}}$.