Tensor-product NURBS patch maps and rational spline spaces

Let $d\in \{2,3\}$ and let the reference patch be $\widehat{\mathrm{\Omega }}=(0,1{)}^d$. For each coordinate $r$, fix a degree $p_r\in {\mathbb{N}}_0$, a finite clamped knot vector ${\mathrm{\Xi }}_r$ on $[0,1]$, and its one-dimensional B-spline basis $B_{i_r,p_r,{\mathrm{\Xi }}_r}^{(r)}$ from B-spline spaces and knot conventions. For a multi-index $\alpha =(i_1,\dots ,i_d)$ in the finite product index set $\mathcal{I}$, define the tensor-product basis function $N_{\alpha }({\xi }_1,\dots ,{\xi }_d):={\prod }_{r=1}^d{B}_{i_r,p_r,{\mathrm{\Xi }}_r}^{(r)}({\xi }_r)\mathrm{.}$

Choose finitely many control points $P_{\alpha }\in {\mathbb{R}}^d$ and positive weights $w_{\alpha }$, and write $0<w_{-}:={\min }_{\alpha }{w}_{\alpha }\le {\max }_{\alpha }{w}_{\alpha }=:w_{+}<\mathrm{\infty }$. The weight denominator, rational basis, and associated rational spline space are $W(\xi ):={\sum }_{\alpha \in \mathcal{I}}{w}_{\alpha }{N}_{\alpha }(\xi ),R_{\alpha }(\xi ):=\frac{w_{\alpha }{N}_{\alpha }(\xi )}{W(\xi )},{\mathcal{N}}_h:=\mathrm{span}\{R_{\alpha }:\alpha \in \mathcal{I}\}\mathrm{.}$ The corresponding NURBS geometry map is $F(\xi ):={\sum }_{\alpha \in \mathcal{I}}{R}_{\alpha }(\xi )P_{\alpha }\mathrm{.}$ Equivalently, let $\tilde{F}(\xi ):={\sum }_{\alpha \in \mathcal{I}}(w_{\alpha }{P}_{\alpha },w_{\alpha })N_{\alpha }(\xi )\in {\mathbb{R}}^{d+1}$ and let $\mathrm{\Pi }(y,z):=y\mathrm{/}z$ for $z>0$. Then $F=\mathrm{\Pi }\circ \tilde{F}$ wherever $W>0$. The basis properties in B-spline basis properties and polynomial reproduction give nonnegative tensor factors summing to one. Consequently the weights' convex combination satisfies $W\ge w_{-}>0$; the later item lem-positive-nurbs-basis-and-quotient-derivative records the full rational-basis consequences.

An admissible NURBS patch map is a continuous, globally injective, bi-Lipschitz map $F:\overline{\widehat{\mathrm{\Omega }}}\to \bar{\mathrm{\Omega }}$ whose restriction to every nonempty knot cell is a $C^1$ diffeomorphism onto its image. There are constants $0<j_{-}\le j_{+}<\mathrm{\infty }$ and $1\le L<\mathrm{\infty }$ such that, almost everywhere on every cell, $j_{-}\le J_F:=\det DF\le j_{+},\mathrm{\parallel }DF{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le L,\mathrm{\parallel }D{F}^{-1}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le L\mathrm{.}$ The physical image $\mathrm{\Omega }$ is required to be a bounded Lipschitz domain. The images of the finite knot cells form a face-to-face partition with the piecewise $C^1$ boundary presentation of Specified finite piecewise C1 boundary presentations; the map agrees on each shared cell face.

A finite multipatch geometry consists of finitely many such patch maps with disjoint physical interiors, a connected bounded Lipschitz union, and matching full physical faces wherever two patches meet. Positive weights only define a positive denominator and a convex rational basis; they do not assert global injectivity, positive Jacobian, or interface matching. Exact NURBS representation means exact representation of the specified CAD map and does not assert that this map is the exact geometry of a target physical domain.