On the limits of non-uniform rational B-spline surfaces with varying weights

Abstract

The non-uniform rational B-spline is a mathematical model commonly used in computer-aided design and manufacturing. For a non-uniform rational B-spline surface, when a single weight approaches infinity, the surface tends to the corresponding control point. A natural question is that what happens if all of the weights approach infinity. In this article, we define the regular control surface, which is a kind of control structure of non-uniform rational B-spline surface, and prove that it is exactly the limiting position of the non-uniform rational B-spline surface when all of weights, multiplied by a certain one-parametric function with different values for each control point, go to infinity. It develops the geometric meaning of weights of non-uniform rational B-spline surface. Moreover, some examples are presented to show the application for the surface deformation by this property.

Introduction

Machine parts in the mechanical processing, such as cam, blade, and mold are usually composed of complex curves or surfaces.^1,2 Outlines of them cannot be represented by simple mathematical models. Since non-uniform rational B-spline (NURBS) surface can represent not only the standard analytic surface, such as conical surface, surface of revolution, or general quadric surface, but also the complex free surfaces, most of the outlines are represented by NURBS curves and surfaces. ³ Due to a series of unique advantages, NURBS is defined as the only one mathematical method for geometrical shapes of industrial products by International Standard Organization (ISO). Up to now, the NURBS method is effectively applied in many areas of computer-aided design and manufacturing (CAD/CAM), computer-aided geometric design (CAGD), and geometric modeling. Particularly, it has diverse applications in isogeometric analysis,^4,5 reverse engineering,^6–8 computer numerical control (CNC) machining,^9–11 and so on.

The shape modification of existing objects plays an important role not only in mechanical processing ¹² but also in geometric modeling due to its wide applications in industrial design and computer animation.^13,14 It is important to correlate the shape of the nominal CAD model with the shape of the manufactured part. Brujic et al. ¹⁵ presented a method to update the CAD model consisting of NURBS surfaces based on a set of unorganized measured points. Plenty of geometric modeling techniques have been proposed for modifying and deforming shape.^16–20 Usually, the shape modification of NURBS surfaces can be achieved by means of knot vectors, control points, and weights. Piegl ²¹ and Piegl and Tiller ²² presented the geometric meaning of a single weight of NURBS surface: if a weight approaches infinity, the surface tends to the corresponding control point. Based on this, weight-based modifications and control point–based modifications to the shape of NURBS surface were presented by Piegl. ²¹ If more weights are used, the methods will be more complicated. Juhász ²³ addressed the problem of shape modification of NURBS curves by altering several weights, and he also outlined how to carry over to surfaces the results for curves. However, fewer works focus on explaining the meaning when all of the weights tend to vary (even to infinity). We extend their methods to more general cases to explain what happens if all of the weights of NURBS surface approach infinity simultaneously.

Warren ²⁴ created the hexagonal Bézier surfaces and predicted that the further work incorporating toric variety may lead to practical methods for multi-sided patches. Sturmfels ²⁵ introduced the theory of toric varieties, toric ideals, and Gröbner bases in detail. In 2002, Krasauskas ²⁶ presented a kind of multi-sided surface patch, named toric surface patch, whose mathematical theory is based on toric variety from algebraic geometry and toric ideals from combinatorics. The toric surface patches are multi-sided generalization of classical Bézier curves and surfaces. Craciun et al. ²⁷ proposed the concept of lifting function to get regular triangulation, control structures, and limiting patches of the rational Bézier surfaces, which were restricted to triangulations. In 2011, García-Puente et al. ²⁸ presented the toric degenerations of toric surface patches and explained the geometric meaning of regular control structures (surfaces) of toric surface patches while all of the weights tend to infinity, which is the generalization of the geometric meaning of single weight of a rational Bézier curve/surface in Piegl ²¹ and Piegl and Tiller. ²² Zhu ²⁹ gave the toric degenerations of toric varieties and toric ideals induced by the regular decomposition. Wang et al. ³⁰ presented an algorithm to compute the number of regular control surfaces of a toric surface patch. The injectivity of curve and surface, which means no self-intersection, plays an important role in image warping and morphing, 3D deformation, volume morphing, and mechanical design. Zhu and Zhao^31,32 presented the geometric conditions on the control points for checking the injectivity of rational Bézier curves and surfaces. These geometric conditions on the control points also depend on the given lifting functions.

Since a NURBS surface can be decomposed into piecewise Bézier surfaces by knot insertion, the toric degenerations of Bézier curves/surfaces can be applied to study the geometric properties of NURBS curves/surfaces. Zhang et al. ³³ studied the toric degenerations of NURBS curves and stated that the limit of a NURBS curve is exactly its regular control curve when all of the weights go to infinity. If a lifting function was given, then the corresponding regular control structure (curve) can be obtained. The result of Zhang et al. ³³ was applied for checking the injectivity of NURBS curves. ³⁴

In this article, we define the regular control surface, which is a kind of control structure of NURBS surface, and prove that it is exactly the limiting position of the NURBS surface when all of the weights tend to infinity. This result not only generalizes the geometric property of a single weight of NURBS surface and explains the geometric meaning of weights of NURBS surface but also provides the possible applications for the injectivity checking and shape modification by many weights of NURBS surface. Furthermore, this result provides a potential application for computer animation.

The body of this article begins with a brief review of rational Bézier surface and its toric degeneration in section “Rational Bézier surface and its toric degeneration.” In section “On the limits of NURBS surfaces with varying weights,” we define a kind of special control structure of NURBS surface, which is called regular control surface. We prove that regular control surface is the limit of the NURBS surface when all of the weights approach infinity. Furthermore, we demonstrate that if a point set is the limit of a NURBS surface for a sequence of weights, then this set must be a regular control surface induced by some regular decomposition. In section “Examples,” we give some examples to explain the meaning of the regular control surfaces and provide some examples of application for surface deformation. Finally, we conclude the whole article.

Rational Bézier surface and its toric degeneration

Definition 1

Given control points ${\mathbf{b}}_{i,j}\in {\mathbb{R}}^3$ and weights ${\omega }_{i,j}\ge 0$ , where $i=0,1,\dots ,m$ and $j=0,1,\dots ,n$ , a rational Bézier surface $\mathbf{F}$ of degree $m\times n$ is defined by^35,36

$$\begin{array}{l}F(u,v)=\frac{{\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{\omega }_{i,j}{b}_{i,j}{B}_i^m(u)B_j^n(v)}}}{{\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{\omega }_{i,j}{B}_i^m(u)B_j^n(v)}}},\\ (u,v)\in [0,1]\times [0,1]\end{array}$$

where $B_i^m(u)=C_m^i(1-u{)}^{m-i}{u}^i$ and $B_j^n(v)=C_n^j(1-v{)}^{n-j}{v}^j$ are Bernstein basis functions of degree $m$ and $n$ . The control net $\mathcal{N}$ of the surface is the $(m+1)\times (n+1)$ grid formed by line segments that connecting the neighboring control points in the same row and column.

Krasauskas ²⁶ presented a kind of multi-sided surface patch, named toric surface patch, which is a multi-sided generalization of classical triangular and tensor-product Bézier surfaces. Let $\mathcal{A}\subset {\mathbb{Z}}^2$ be a finite set of lattice points, ${\Delta }_{\mathcal{A}}=\mathrm{conv}(\mathcal{A})$ be the convex hull of the lattice points of $\mathcal{A}$ . The boundary lines of ${\Delta }_{\mathcal{A}}$ are $f_1,f_2,\dots ,f_r$ , which satisfy $f_i(x)\ge 0(x\in {\Delta }_{\mathcal{A}},i=1,2,\dots ,r)$ . Following Krasauskas’ definition, the toric surface patch can be defined as follows.

Definition 2

Given $\mathcal{A}\subset {\mathbb{Z}}^2$ , control points set $\mathcal{B}=\{{\mathit{b}}_{i,j}|(i,j)\in \mathcal{A}\}\subset {\mathbb{R}}^3$ , and weights set $\omega =\{{\omega }_{i,j}>0|(i,j)\in \mathcal{A}\}$ , the toric surface patch is defined by ²⁶

$${\mathit{F}}_{\mathcal{A},\omega ,\mathcal{B}}(u,v)=\frac{\sum _{(i,j)\in \mathcal{A}}{\omega }_{i,j}{\mathit{b}}_{i,j}{\beta }_{i,j}(u,v)}{\sum _{(i,j)\in \mathcal{A}}{\omega }_{i,j}{\beta }_{i,j}(u,v)},(u,v)\in {\Delta }_{\mathcal{A}}$$

where ${\beta }_{i,j}(u,v)=c_{i,j}{f}_1(u,v{)}^{f_1(i,j)}\cdots f_r(u,v{)}^{f_r(i,j)}$ are the toric basis functions with the positive coefficients $c_{i,j}>0$ .

Let coefficients $c_{i,j}=C_m^i{C}_n^j{m}^{-m}{n}^{-n}$ and $\mathcal{A}={\mathcal{A}}_{m,n}:=\{(i,j)|i=0,1,\dots ,m;j=0,1,\dots ,n\}$ , where $m\mathrm{and}n$ are two positive integers, the lattice polygon ${⊡}_{{\mathcal{A}}_{m,n}}:=\mathrm{conv}({\mathcal{A}}_{m,n})$ and the boundary lines of ${⊡}_{{\mathcal{A}}_{m,n}}$ are defined by $f_1(u,v)=u$ , $f_2(u,v)=v$ , $f_3(u,v)=m-u$ , and $f_4(u,v)=n-v$ . Thus, the image of ${⊡}_{{\mathcal{A}}_{m,n}}$ under the map ${\mathbf{F}}_{\mathcal{A},\omega ,\mathcal{B}}$ is a rational tensor-product Bézier surface of degree $m\times n$ , denoted by ${\mathbf{F}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ . The following definition of rational tensor-product Bézier surface refers to Krasauskas, ²⁶ which is equivalent to Definition 1.

Definition 3

Given ${\mathcal{A}}_{m,n}$ , control points set $\mathcal{B}=\{{\mathbf{b}}_{i,j}|(i,j)\in {\mathcal{A}}_{m,n}\}\subset {\mathbb{R}}^3$ , and weights set $\omega =\{{\omega }_{i,j}>0|(i,j)\in {\mathcal{A}}_{m,n}\}$ , the rational tensor-product Bézier surface of degree $m\times n$ associated with rectangle ${⊡}_{{\mathcal{A}}_{m,n}}$ is parameterized by the rational map ${\mathbf{F}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}:{⊡}_{{\mathcal{A}}_{m,n}}\to {\mathbb{R}}^3$ ²⁶

$${\mathbf{F}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}(u,v)=\frac{\sum _{i=0}^m\sum _{j=0}^n{\omega }_{i,j}{\mathbf{b}}_{i,j}{\beta }_{i,j}(u,v)}{\sum _{i=0}^m\sum _{j=0}^n{\omega }_{i,j}{\beta }_{i,j}(u,v)},(u,v)\in {⊡}_{{\mathcal{A}}_{m,n}}$$

where ${\beta }_{i,j}(u,v)=C_m^i{C}_n^j{m}^{-m}{n}^{-n}{f}_1(u,v{)}^{f_1(i,j)}\cdots f_4(u,v{)}^{f_4(i,j)}$ .

Therefore, we use Definition 3 to represent a rational tensor-product Bézier surface in the rest of this article. Let $\lambda :{\mathcal{A}}_{m,n}\to \mathbb{R}$ be a lifting function, and $P_{\lambda }$ be the convex hull of the lifted points associated with lattice points of ${\mathcal{A}}_{m,n}$ , defined by $P_{\lambda }=\mathrm{conv}\{(i,j,\lambda (i,j))|(i,j)\in {\mathcal{A}}_{m,n}\}\subset {\mathbb{R}}^3$ . The upper faces of $P_{\lambda }$ are those having an outward-pointing normal vector with positive last coordinate. The union of the upper faces is called the upper hull of $P_{\lambda }$ . If we project this upper hull back to ${\mathbb{R}}^2$ , then it covers ${⊡}_{{\mathcal{A}}_{m,n}}$ and induces a regular domain decomposition ${\mathcal{T}}_{\lambda }$ of ${⊡}_{{\mathcal{A}}_{m,n}}$ . Collect the lattice points of ${\mathcal{A}}_{m,n}$ whose lifted points lie on a common upper face of upper hull of $P_{\lambda }$ into a subset $I$ of ${\mathcal{A}}_{m,n}$ , and the union of these subsets is called a regular decomposition $S_{\lambda }$ of ${\mathcal{A}}_{m,n}$ induced by lifting function $\lambda$ . ²⁸

Suppose that we have a regular decomposition $S_{\lambda }$ of ${\mathcal{A}}_{m,n}$ induced by the lifting function $\lambda$ , and $I$ is a subset of $S_{\lambda }$ . We can use $\omega {|}_I=\{{\omega }_{i,j}|(i,j)\in I\}$ and $\mathcal{B}{|}_I=\{{\mathbf{b}}_{i,j}|(i,j)\in I\}$ indexed by elements of $I$ as the sets of weights and control points to obtain a toric surface patch by Definition 2, denoted by ${\mathbf{F}}_{I,\omega {|}_I,\mathcal{B}{|}_I}$ .

Definition 4

The union of toric surface patches ²⁸

$${\mathbf{F}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}(S_{\lambda })=\bigcup _{I\in S_{\lambda }}{\mathbf{F}}_{I,\omega {|}_I,\mathcal{B}{|}_I}$$

is called the regular control surface of rational Bézier surface ${\mathbf{F}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ induced by the regular decomposition $S_{\lambda }$ .

Let $\lambda :{\mathcal{A}}_{m,n}\to \mathbb{R}$ be a lifting function and $S_{\lambda }$ be the corresponding regular decomposition of ${\mathcal{A}}_{m,n}$ . For a given set of weights $\omega =\{{\omega }_{i,j}>0|(i,j)\in {\mathcal{A}}_{m,n}\}$ , we define another set of weights depending upon parameter $t$ and $\lambda$ by ${\omega }_{\lambda }(t)=\{t^{\lambda (i,j)}{\omega }_{i,j}|(i,j)\in {\mathcal{A}}_{m,n}\}$ .

Definition 5

Given ${\mathcal{A}}_{m,n}$ , control points set $\mathcal{B}=\{{\mathbf{b}}_{i,j}|(i,j)\in {\mathcal{A}}_{m,n}\}\subset {\mathbb{R}}^3$ , and weights set $\omega =\{{\omega }_{i,j}>0|(i,j)\in {\mathcal{A}}_{m,n}\}$ , the rational Bézier surface of degree $m\times n$ parameterized by $t$ is defined by ²⁸

$$F_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),ℬ}(u,v;t)=\frac{{\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{t}^{\lambda (i,j)}{\omega }_{i,j}{b}_{i,j}{\beta }_{i,j}(u,v)}}}{{\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{t}^{\lambda (i,j)}{\omega }_{i,j}{\beta }_{i,j}(u,v)}}},(u,v)\in {\square }_{{\mathcal{A}}_{m,n}}$$

Write ${\mathbf{F}}_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),\mathcal{B}}$ to represent the image of ${\mathbf{F}}_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),\mathcal{B}}({⊡}_{{\mathcal{A}}_{m,n}})$ . García-Puente et al. proved that the regular control surface is exactly the limiting position of a rational Bézier surface when $t$ tends to infinity, which generalizes the geometric property of a single weight of rational Bézier surface and explains the geometric meaning of rational Bézier surface while all of weights tend to infinity.

Theorem 1

Theorem 1 is as follows ²⁸

$$\underset{t\to +\infty }{lim}{\mathit{F}}_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),\mathcal{B}}={\mathit{F}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}(S_{\lambda })$$

Theorem 2

Given ${\mathcal{A}}_{m,n}$ , control points set $\mathcal{B}=\{{\mathit{b}}_{i,j}|(i,j)\in {\mathcal{A}}_{m,n}\}\subset {\mathbb{R}}^3$ . If $\mathit{F}\subset {\mathbb{R}}^3$ is a set for which there is a sequence ${\omega }^{(1)},{\omega }^{(2)},\dots$ of weights so that ²⁸

$$\underset{\tau \to +\infty }{lim}{\mathit{F}}_{{\mathcal{A}}_{m,n},{\omega }^{(\tau )},\mathcal{B}}=\mathit{F}$$

then there exists a lifting function $\lambda :{\mathcal{A}}_{m,n}\to \mathbb{R}$ and a weights set $\omega$ , such that $\mathit{F}={\mathit{F}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}(S_{\lambda })$ is a regular control surface.

Example 1

Let ${\mathcal{A}}_{2,2}$ be a $3\times 3$ grid of nine points. Figure 1 shows the weights of a biquadratic rational Bézier surface ${\mathbf{F}}_{{\mathcal{A}}_{2,2},\omega ,\mathcal{B}}$ , the lifting function $\lambda$ , and the corresponding regular decomposition $S_{\lambda }$ of ${\mathcal{A}}_{2,2}$ . The center element (green point) in Figure 1(c) does not participate in regular decomposition, since it does not lie on any upper face. So, the regular control surface ${\mathbf{F}}_{{\mathcal{A}}_{2,2},\omega ,\mathcal{B}}(S_{\lambda })$ (shown in Figure 2(b)) of ${\mathbf{F}}_{{\mathcal{A}}_{2,2},\omega ,\mathcal{B}}$ (shown in Figure 2(a)) is the union of four triangles and one toric surface patch. The surface ${\mathbf{F}}_{{\mathcal{A}}_{2,2},{\omega }_{\lambda }(t),\mathcal{B}}$ approaches this regular control surface while $t\to +\infty$ .

Figure 1.

(a) Weights set $\omega$ , (b) lifting function $\lambda$ , and (c) regular decomposition $S_{\lambda }$ of ${\mathcal{A}}_{2,2}$ .

Figure 2.

The regular control surface of rational Bézier surface: (a) rational Bézier surface ${\mathbf{F}}_{{\mathcal{A}}_{2,2},\omega ,\mathcal{B}}$ and (b) Regular control surface ${\mathbf{F}}_{{\mathcal{A}}_{2,2},\omega ,\mathcal{B}}(S_{\lambda })$ of ${\mathbf{F}}_{{\mathcal{A}}_{2,2},\omega ,\mathcal{B}}$ .

On the limits of NURBS surfaces with varying weights

Regular control surface

Definition 6

A NURBS surface of degree $p$ in $u$ direction and degree $q$ in $v$ direction is defined by ²²

$$\begin{array}{l}R(u,v)=\frac{{\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{\omega }_{i,j}{P}_{i,j}{N}_{i,p}(u)N_{j,q}(v)}}}{{\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{\omega }_{i,j}{N}_{i,p}(u)N_{j,q}(v)}}},\\ (u,v)\in [a,b]\times [c,d]\end{array}$$

where ${\omega }_{i,j}$ are the weights, ${\mathbf{P}}_{i,j}$ are the control points, and $N_{i,p}(u)$ and $N_{j,q}(v)$ are the B-spline basis functions of degree $p$ and $q$ defined on knot vectors, respectively

$$\begin{array}{c}U=\{\underset{p+1}{\underbrace{a,\dots ,a}},u_{p+1},\dots ,u_m,\underset{p+1}{\underbrace{b,\dots ,b}}\}\hfill \\ V=\{\underset{q+1}{\underbrace{c,\dots ,c}},v_{q+1},\dots ,v_n,\underset{q+1}{\underbrace{d,\dots ,d}}\}\hfill \end{array}$$

The control net $\mathcal{N}$ of the surface is the $(m+1)\times (n+1)$ grid formed by line segments that connecting the neighboring control points in the same row and column.

Without loss of generality, we assume that $a=c=0\mathrm{and}b=d=1$ . It ensures that the NURBS surface $\mathbf{R}(u,v)$ can interpolate the corner control points ${\mathbf{P}}_{0,0},{\mathbf{P}}_{m,0},{\mathbf{P}}_{m,n},\mathrm{and}{\mathbf{P}}_{0,n}$ .

In this article, we use the following representation to represent a NURBS surface $\mathbf{R}(u,v)$ , which is similar to Definition 3 and can be proved to be an equivalent to the classical definition of NURBS surface from Piegl and Tiller, ²² Farin and colleagues^35,36 and Davis ³⁷ (see Definition 6) after a simple reparametrization.

Definition 7

Given ${\mathcal{A}}_{m,n}=\{(i,j)|i=0,1,\dots ,m;j=0,1,\dots ,n\}$ , where $m\mathrm{and}n$ are two positive integers, ${⊡}_{{\mathcal{A}}_{m,n}}=\mathrm{conv}({\mathcal{A}}_{m,n})=[0,m]\times [0,n]$ , the control points set $\mathcal{B}=\{{\mathbf{P}}_{i,j}|(i,j)\in {\mathcal{A}}_{m,n}\}\subset {\mathbb{R}}^3$ , and weights set $\omega =\{{\omega }_{i,j}>0|(i,j)\in {\mathcal{A}}_{m,n}\}$ , the NURBS surface of degree $p\times q$ is defined by

$$\begin{array}{l}{R}_{{\mathcal{A}}_{m,n},\omega ,ℬ}(u,v)=\frac{{\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{\omega }_{i,j}{P}_{i,j}{N}_{i,p}(u)N_{j,q}(v)}}}{{\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{\omega }_{i,j}{N}_{i,p}(u)N_{j,q}(v)}}},\\ (u,v)\in [0,1]\times [0,1]\end{array}$$

where $N_{i,p}(u)$ and $N_{j,q}(v)$ are the B-spline basis functions of degree $p$ and $q$ defined on knot vectors $U$ and $V$ , respectively

$$U=\{\underset{p+1}{\underbrace{0,\dots ,0}},u_{p+1},u_{p+2}\dots ,u_m,\underset{p+1}{\underbrace{1,\dots ,1}}\}$$

(1)

$$V=\{\underset{q+1}{\underbrace{0,\dots ,0}},v_{q+1},v_{q+2}\dots ,v_n,\underset{q+1}{\underbrace{1,\dots ,1}}\}$$

(2)

Write ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ as the image of ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}(u,v)$ on $[0,1]\times [0,1]$ to represent the NURBS surface. Without loses of generality, we assume that the interior knots in $U$ and $V$ are with multiplicity 1 for the NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ of degree $p\times q$ in this article (i.e. $u_{p+1}<u_{p+2}<\cdots <u_m$ and $v_{q+1}<v_{q+2}<\cdots <v_n$ in equations (1) and (2)). A NURBS surface can be subdivided into piecewise rational Bézier surfaces after inserting each existing interior knot $p-1$ times until all of interior knots $u_i(i=p+1,p+2,\dots ,m)$ have multiplicity $p$ , and inserting each existing interior knot $q-1$ times until all of interior knots $v_j(j=q+1,q+2,\dots ,n)$ have multiplicity $q$ . Note that the resulting surface is independent of the order of inserting knots. The new weights and control points of NURBS surface can be obtained by knot-insertion algorithm when a knot is inserted in knot vectors. Here, we omit the detail process of knot-insertion algorithm since it is well known.^22,38Figure 3 shows that a biquadratic NURBS surface is subdivided into four pieces of rational Bézier surfaces by use of knot insertion. Counting rules are similar to matrix, from left to right and from bottom to up. The Bézier surface patches in Figure 3(b) at the lower left and right are defined as the $(1,1)\mathrm{th}$ patch and $(1,2)\mathrm{th}$ patch, while the patches at the upper left and right are defined as the $(2,1)\mathrm{th}$ patch and $(2,2)\mathrm{th}$ patch, respectively.

Figure 3.

NURBS surface is subdivided into piecewise Bézier surfaces: (a) the biquadratic NURBS surface and (b) Four pieces of rational Bézier surfaces after knot insertion.

Note that after all interior knots are inserted and the piecewise Bézier surfaces are generated, we can get all of the control points and weights for the NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ using knot-insertion algorithm, and the B-spline basis functions are defined on knot vectors

$$U^M=\{\underset{p+1}{\underbrace{0,\dots ,0}},\underset{p}{\underbrace{u_{p+1},\dots ,u_{p+1}}},\dots ,\underset{p}{\underbrace{u_m,\dots ,u_m}},\underset{p+1}{\underbrace{1,\dots ,1}}\}$$

and

$$V^N=\{\underset{q+1}{\underbrace{0,\dots ,0}},\underset{q}{\underbrace{v_{q+1},\dots ,v_{q+1}}},\dots ,\underset{q}{\underbrace{v_n,\dots ,v_n}},\underset{q+1}{\underbrace{1,\dots ,1}}\}$$

where $M=(p-1)(m-p)$ and $N=(q-1)(n-q)$ . Denote the original control points set of ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ by $\mathcal{B}=\{{\mathbf{P}}_{i,j}^{0,0}={\mathbf{P}}_{i,j}|(i,j)\in {\mathcal{A}}_{m,n}\}\subset {\mathbb{R}}^3$ , and the original weights set of ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ by $\omega =\{{\omega }_{i,j}^{0,0}={\omega }_{i,j}>0|(i,j)\in {\mathcal{A}}_{m,n}\}$ . Let the final finite set $\overline{\mathcal{A}}=\{(i,j)|i=0,1,\dots ,(m-p+1)p;j=0,1,\dots ,(n-q+1)q\}$ and ${⊡}_{\overline{\mathcal{A}}}=[0,(m-p+1)p]\times [0,(n-q+1)q]$ , the final weights set $\overline{\omega }=\{{\omega }_{i,j}^{M,N}|(i,j)\in \overline{\mathcal{A}}\}$ and control points set $\overline{\mathcal{B}}=\{{\mathbf{P}}_{i,j}^{M,N}|(i,j)\in \overline{\mathcal{A}}\}$ can be computed by the knot-insertion algorithm. Let $K=m-p+1\mathrm{and}L=n-q+1$ . Using knot insertion, we transform the surface ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ into the union of $K\times L$ pieces of rational Bézier surfaces, which is also a NURBS surface denoted by ${\mathbf{R}}_{\overline{\mathcal{A}},\overline{\omega },\overline{\mathcal{B}}}$ . Let ${\mathcal{A}}^{k,l}=\{(i,j)|i=(k-1)p,(k-1)p+1,\dots ,\mathrm{kp};j=(l-1)q,(l-1)q+1,\dots ,\mathrm{lq}\}$ be a subset of $\overline{\mathcal{A}}$ , and ${⊡}_{{\mathcal{A}}^{k,l}}=[(k-1)p,\mathrm{kp}]\times [(l-1)q,\mathrm{lq}]\subset {⊡}_{\overline{\mathcal{A}}}$ . Thus, $\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathcal{A}}^{k,l}=\overline{\mathcal{A}}$ . Using the weights set ${\omega }^{k,l}=\{{\omega }_{i,j}^{M,N}|(i,j)\in {\mathcal{A}}^{k,l}\}$ and control points set ${\mathcal{B}}^{k,l}=\{{\mathbf{P}}_{i,j}^{M,N}|(i,j)\in {\mathcal{A}}^{k,l}\}$ indexed by the elements of ${\mathcal{A}}^{k,l}$ as weights and control points, we can define a rational Bézier surface ${\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }^{k,l},{\mathcal{B}}^{k,l}}$ of degree $p\times q$ , which is the $(k,l)\mathrm{th}$ piece rational Bézier surface of ${\mathbf{R}}_{\overline{\mathcal{A}},\overline{\omega },\overline{\mathcal{B}}}$ . Therefore

$${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}={\mathbf{R}}_{\overline{\mathcal{A}},\overline{\omega },\overline{\mathcal{B}}}=\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }^{k,l},{\mathcal{B}}^{k,l}}$$

We define a NURBS surface parameterized by $t$ with a lifting function $\lambda$ as follows.

Definition 8

Given ${\mathcal{A}}_{m,n}$ , the control points set $\mathcal{B}=\{{\mathbf{P}}_{i,j}|(i,j)\in {\mathcal{A}}_{m,n}\}\subset {\mathbb{R}}^3$ , and weights set $\omega =\{{\omega }_{i,j}>0|(i,j)\in {\mathcal{A}}_{m,n}\}$ , the lifting function $\lambda :{\mathcal{A}}_{m,n}\to \mathbb{R}$ lifts the lattice point $(i,j)\in {\mathcal{A}}_{m,n}$ to $(i,j,\lambda (i,j))\in {\mathbb{R}}^3$ , then the surface

$$\begin{array}{l}{R}_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),ℬ}(u,v;t)=\frac{{\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{t}^{\lambda (i,j)}{\omega }_{i,j}{P}_{i,j}{N}_{i,p}(u)N_{j,q}(v)}}}{{\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{t}^{\lambda (i,j)}{\omega }_{i,j}{N}_{i,p}(u)N_{j,q}(v)}}},\\ (u,v)\in [0,1]\times [0,1]\end{array}$$

is called the NURBS surface ${\mathit{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ of degree $m\times n$ parameterized by $t$ , where ${\omega }_{\lambda }(t)=\{t^{\lambda (i,j)}{\omega }_{i,j}|(i,j)\in {\mathcal{A}}_{m,n}\}$ is a set of weights which depend upon $\lambda$ and $t$ .

Suppose that the original weights set $\omega =\{{\omega }_{i,j}^{0,0}|(i,j)\in {\mathcal{A}}_{m,n}\}$ is replaced by ${\omega }_{\lambda }(t)=\{t^{\lambda (i,j)}{\omega }_{i,j}^{0,0}|(i,j)\in {\mathcal{A}}_{m,n}\}$ . Take ${\omega }_{\lambda }(t)$ as the new weights set, and control points set $\mathcal{B}=\{{\mathbf{P}}_{i,j}^{0,0}|(i,j)\in {\mathcal{A}}_{m,n}\}$ remains unchanged. Using knot insertion, the surface ${\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),\mathcal{B}}$ is transformed into the union of $K\times L$ pieces of rational Bézier surfaces, which is also a NURBS surface denoted by ${\mathbf{R}}_{\overline{\mathcal{A}},\overline{{\omega }_{\lambda }(t)},\overline{{\mathcal{B}}_{\lambda }(t)}}$ , where the weights set $\overline{{\omega }_{\lambda }(t)}=\{{\omega }_{i,j}^{M,N}(t)|(i,j)\in \overline{\mathcal{A}}\}$ and control points set $\overline{{\mathcal{B}}_{\lambda }(t)}=\{{\mathbf{P}}_{i,j}^{M,N}(t)|(i,j)\in \overline{\mathcal{A}}\}$ can be obtained by the knot-insertion algorithm. We use the weights set ${\omega }_{\lambda }^{k,l}(t)=\{{\omega }_{i,j}^{M,N}(t)|(i,j)\in {\mathcal{A}}^{k,l}\}\subset \overline{{\omega }_{\lambda }(t)}$ and control points set ${\mathcal{B}}_{\lambda }^{k,l}(t)=\{{\mathbf{P}}_{i,j}^{M,N}(t)|(i,j)\in {\mathcal{A}}^{k,l}\}\subset \overline{{\mathcal{B}}_{\lambda }(t)}$ indexed by the elements of ${\mathcal{A}}^{k,l}$ as weights and control points to define a rational Bézier surface ${\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}$ of degree $p\times q$ , which is the $(k,l)\mathrm{th}$ piece rational Bézier surface of ${\mathbf{R}}_{\overline{\mathcal{A}},\overline{{\omega }_{\lambda }(t)},\overline{{\mathcal{B}}_{\lambda }(t)}}$ . Therefore

$${\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),\mathcal{B}}={\mathbf{R}}_{\overline{\mathcal{A}},\overline{{\omega }_{\lambda }(t)},\overline{{\mathcal{B}}_{\lambda }(t)}}=\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}$$

According to the regular decomposition $S_{\lambda }$ of ${\mathcal{A}}_{m,n}$ presented in section “Rational Bézier surface and its toric degeneration,” we can define a regular decomposition $S_{\lambda }^{k,l}$ of ${\mathcal{A}}^{k,l}$ . The lifting function $\lambda :{\mathcal{A}}_{m,n}\to \mathbb{R}$ induces the regular decomposition $S_{\lambda }^{k,l}$ of ${\mathcal{A}}^{k,l}$ as follows. Since the control point ${\mathbf{P}}_{i,j}^{M,N}(t)\in {\mathcal{B}}_{\lambda }^{k,l}(t)$ indexed by lattice point $(i,j)\in {\mathcal{A}}^{k,l}$ is formed by the convex combination of some original control points ${\mathbf{P}}_{a,b}^{0,0},\dots ,{\mathbf{P}}_{c,d}^{0,0}$ of $\mathcal{B}$ , we assume that $(i,j)$ of ${\mathcal{A}}^{k,l}$ is related to the lattice points $(a,b),\dots ,(c,d)$ of ${\mathcal{A}}_{m,n}$ , then the lifting value $\lambda (i,j)$ is the maximum value of lifting values $\lambda (a,b),\dots ,\lambda (c,d)$ . According to all of the lifted points associated with lattice points of ${\mathcal{A}}^{k,l}$ , we can get the convex hull $U_{\lambda }^{k,l}=\mathrm{conv}\{(i,j,\lambda (i,j))|(i,j)\in {\mathcal{A}}^{k,l}\}\subset {\mathbb{R}}^3$ of the lifted points associated with lattice points of ${\mathcal{A}}^{k,l}$ . It induces the regular decomposition $S_{\lambda }^{k,l}$ of ${\mathcal{A}}^{k,l}$ , and each subset of $S_{\lambda }^{k,l}$ consists of the lattice points of ${\mathcal{A}}^{k,l}$ whose lifted points lie on a common face of upper hull of $U_{\lambda }^{k,l}$ .

Definition 9

The union of regular decompositions $S_{\lambda }^{k,l}$ of ${\mathcal{A}}^{k,l}$ is called the regular decomposition of $\overline{\mathcal{A}}$ induced by lifting function $\lambda$ , denoted by $\overline{S_{\lambda }}=\bigcup _{k=1}^K\bigcup _{l=1}^L{S}_{\lambda }^{k,l}$ .

Example 2

Let $m=n=3$ , ${\mathcal{A}}_{3,3}$ be a $4\times 4$ grid of 16 points, and a biquadratic NURBS surface defined on knot vectors $U=V=\{0,0,0,\frac{1}{2},1,1,1\}$ . The lifting values of ${\mathcal{A}}_{3,3}$ and $\overline{\mathcal{A}}$ are shown in Figure 4(a) and (b), respectively. The blue numbers in Figure 4(b) and (c) are the lifting values of ${\mathcal{A}}^{2,1}$ and their corresponding regular decomposition is shown in Figure 4(d). The union of regular decompositions of ${\mathcal{A}}^{k,l}(k,l=1,2)$ is the regular decomposition of $\overline{\mathcal{A}}$ , which is shown in Figure 4(e).

Figure 4.

The lifting values and regular decomposition: (a) lifting values of ${\mathcal{A}}_{3,3}$ , (b) lifting values of $\overline{\mathcal{A}}$ , (c) lifting values of ${\mathcal{A}}^{2,1}$ , (d) regular decomposition of ${\mathcal{A}}^{2,1}$ , and (e) regular decomposition of $\overline{\mathcal{A}}$ .

Define the net formed by connecting the control points of NURBS surface ${\mathbf{R}}_{\overline{\mathcal{A}},\overline{{\omega }_{\lambda }(t)},\overline{{\mathcal{B}}_{\lambda }(t)}}$ according to the topological structure of regular decomposition $\overline{S_{\lambda }}$ of $\overline{\mathcal{A}}$ as the control net of NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),\mathcal{B}}$ (or ${\mathbf{R}}_{\overline{\mathcal{A}},\overline{{\omega }_{\lambda }(t)},\overline{{\mathcal{B}}_{\lambda }(t)}}$ ). Note that the control net we defined is different with the control net of classical NURBS surface, which is the quadrilateral net. Consider the $(k,l)\mathrm{th}$ piece rational Bézier surface ${\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}$ , note that ${\mathbf{P}}_{i,j}^{M,N}(t)$ of ${\mathcal{B}}_{\lambda }^{k,l}(t)$ and ${\omega }_{i,j}^{M,N}(t)$ of ${\omega }_{\lambda }^{k,l}(t)$ are parameterized by $t$ . It means that the positions of control points may change by altering of parameter $t$ , then the corresponding control net may also change. Hence, we just want to know the limit position of the control net while $t\to +\infty$ , which is called the regular control net of NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ induced by lifting function $\lambda$ . To solve this problem, the key is how the positions of control points and their corresponding weights change when $t\to +\infty$ .

Consider the control points ${\mathbf{P}}_{i,j}^{M,N}(t)$ and weights ${\omega }_{i,j}^{M,N}(t)$ of the $(k,l)\mathrm{th}$ piece rational Bézier surface ${\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}$ . Choose any of the finial control points ${\mathbf{P}}_{i,j}^{M,N}(t)\in {\mathcal{B}}_{\lambda }^{k,l}(t)$ , $(i,j)\in {\mathcal{A}}^{k,l}$ , suppose it is formed by the convex combination of some original control points ${\mathbf{P}}_{a,b}^{0,0},\dots ,{\mathbf{P}}_{c,d}^{0,0}$ of $\mathcal{B}$ , where $\{(a,b),\dots ,(c,d)\}\subset {\mathcal{A}}_{m,n}$

$${\mathbf{P}}_{i,j}^{M,N}(t)=\frac{\sum _{i=a}^c\sum _{j=b}^{d}f(i,j)t^{\lambda (i,j)}{\omega }_{i,j}^{0,0}{\mathbf{P}}_{i,j}^{0,0}}{\sum _{i=a}^c\sum _{j=b}^{d}f(i,j)t^{\lambda (i,j)}{\omega }_{i,j}^{0,0}},(i,j)\in {\mathcal{A}}^{k,l}$$

where coefficients $f(i,j)$ are obtained by knot-insertion algorithm. Suppose the set $\psi =\max \{\lambda (a,b),\dots ,\lambda (c,d)\}$ , then

$$\underset{t\to +\infty }{lim}{\mathbf{P}}_{i,j}^{M,N}(t)=\frac{\sum _{(\alpha ,\beta )\in \psi }f(\alpha ,\beta ){\omega }_{\alpha ,\beta }^{0,0}{\mathbf{P}}_{\alpha ,\beta }^{0,0}}{\sum _{(\alpha ,\beta )\in \psi }f(\alpha ,\beta ){\omega }_{\alpha ,\beta }^{0,0}}$$

Since ${\omega }_{i,j}^{M,N}(t)=\sum _{i=a}^c\sum _{j=b}^{d}f(i,j)t^{\lambda (i,j)}{\omega }_{i,j}^{0,0},(i,j)\in {\mathcal{A}}^{k,l}$ , then

$$\underset{t\to +\infty }{lim}\frac{\sum _{(\alpha ,\beta )\in \psi }{t}^{\lambda (\alpha ,\beta )}f(\alpha ,\beta ){\omega }_{\alpha ,\beta }^{0,0}}{{\omega }_{i,j}^{M,N}(t)}=1$$

We denote $\sum _{(\alpha ,\beta )\in \psi }{t}^{\lambda (\alpha ,\beta )}f(\alpha ,\beta ){\omega }_{\alpha ,\beta }^{0,0}$ by ${\Theta }_t(i,j)$ , which equals ${\omega }_{i,j}^{M,N}(t)$ asymptotically, and $\sum _{(\alpha ,\beta )\in \psi }f(\alpha ,\beta ){\omega }_{\alpha ,\beta }^{0,0}$ by $\Theta (i,j)$ .

Using this method, the positions of control points of the $(k,l)\mathrm{th}$ piece rational Bézier surface while $t$ approaches infinity can be obtained. When $t\to +\infty$ , denote the collections of the control points of ${\mathcal{B}}_{\lambda }^{k,l}(t)$ by $\widetilde{{\mathcal{B}}^{k,l}}=\underset{t\to +\infty }{lim}{\mathcal{B}}_{\lambda }^{k,l}(t)$ . Let ${\omega }_{\lambda }^{k,l}(t)=\bigcup _{(i,j)\in {\mathcal{A}}^{k,l}}{\Theta }_t(i,j)$ and $\widetilde{{\omega }^{k,l}}=\bigcup _{(i,j)\in {\mathcal{A}}^{k,l}}\Theta (i,j)$ . In this article, $\widetilde{{\omega }^{k,l}}$ can be regarded as the limes of ${\omega }_{\lambda }^{k,l}(t)$ for parameter $t$ going to infinity. Let $\widetilde{{\omega }^{k,l}}{|}_{\mathcal{I}}$ and $\widetilde{{\mathcal{B}}^{k,l}}{|}_{\mathcal{I}}$ indexed by the elements of a subset $\mathcal{I}$ of $S_{\lambda }^{k,l}$ be the sets of weights and control points, respectively. Using these weights and control points, we can construct a toric surface patch ${\mathbf{F}}_{\mathcal{I},\widetilde{{\omega }^{k,l}}{|}_{\mathcal{I}},\widetilde{{\mathcal{B}}^{k,l}}{|}_{\mathcal{I}}}$ . Let $S_{\lambda }^{k,l}$ be the regular decomposition of ${\mathcal{A}}^{k,l}$ . Given weights set $\widetilde{{\omega }^{k,l}}$ and control points set $\widetilde{{\mathcal{B}}^{k,l}}$ , we get the regular control surface of $(k,l)\mathrm{th}$ piece rational Bézier surface ${\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }^{k,l},{\mathcal{B}}^{k,l}}$

$${\mathbf{F}}_{{\mathcal{A}}^{k,l},\widetilde{{\omega }^{k,l}},\widetilde{{\mathcal{B}}^{k,l}}}(S_{\lambda }^{k,l})=\bigcup _{\mathcal{I}\in S_{\lambda }^{k,l}}{\mathbf{F}}_{\mathcal{I},\widetilde{{\omega }^{k,l}}{|}_{\mathcal{I}},\widetilde{{\mathcal{B}}^{k,l}}{|}_{\mathcal{I}}}^{k,l}$$

which is the limit of ${\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}$ when $t\to +\infty$ .

Definition 10

Given ${\mathcal{A}}_{m,n}=\{(i,j)|i=0,1,\dots ,m;j=0,1,\dots ,n\}$ , the control points set $\mathcal{B}=\{{\mathbf{P}}_{i,j}|(i,j)\in {\mathcal{A}}_{m,n}\}\subset {\mathbb{R}}^3$ , and weights set $\omega =\{{\omega }_{i,j}>0|(i,j)\in {\mathcal{A}}_{m,n}\}$ , for the regular decomposition $\overline{S_{\lambda }}$ of $\overline{\mathcal{A}}$ induced by a lifting function $\lambda$ , the surface

$$\begin{array}{l}{R}_{{\mathcal{A}}_{m,n},\omega ,ℬ}(\overline{S_{\lambda }})=R_{\overline{\mathcal{A}},\overline{\omega },\overline{ℬ}}(\overline{S_{\lambda }})\\ ={\displaystyle \underset{k=1}{\overset{K}{\cup }}{\displaystyle \underset{l=1}{\overset{L}{\cup }}{F}_{{\mathcal{A}}^{k,l},\widetilde{{\omega }^{k,l}},\widetilde{{ℬ}^{k,l}}}^{k,l}(S_{\lambda }^{k,l})}}={\displaystyle \underset{k=1}{\overset{K}{\cup }}{\displaystyle \underset{l=1}{\overset{L}{\cup }}{\displaystyle \underset{ℐ\in S_{\lambda }^{k,l}}{\cup }{F}_{ℐ,\widetilde{{\omega }^{k,l}}{|}_{ℐ},\widetilde{{ℬ}^{k,l}}{|}_{ℐ}}^{k,l}}}}\end{array}$$

is called the regular control surface of NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ induced by the regular decomposition $\overline{S_{\lambda }}$ .

Example 3

Consider a biquadratic NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ (shown in Figure 5(c)) defined on knot vectors $U=V=\{0,0,0,\frac{1}{2},1,1,1\}$ in Example 2 with weights set (shown in Figure 5(a)). The lifting function $\lambda$ and the corresponding regular decomposition are shown in Figure 4. We show the control net of NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},{\omega }_{\lambda }(t),\mathcal{B}}$ while $t=1.4$ in Figure 5(d), which is different with the general tensor-product control net (shown in Figure 5(b)). The regular control net of NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ (the limit control net while $t\to +\infty )$ is shown in Figure 5(e). Specifically, the part with blue color is the regular control net of the $(2,1)\mathrm{th}$ piece rational Bézier surface ${\mathbf{F}}_{{\mathcal{A}}^{2,1},{\omega }^{2,1},{\mathcal{B}}^{2,1}}$ , and the corresponding regular control surface of ${\mathbf{F}}_{{\mathcal{A}}^{2,1},{\omega }^{2,1},{\mathcal{B}}^{2,1}}$ is the union of three bilinear rational Bézier surfaces and two triangles. But one of these three bilinear rational Bézier surfaces degenerates to a triangle while $t\to +\infty$ , it means that the regular control surface of ${\mathbf{F}}_{{\mathcal{A}}^{2,1},{\omega }^{2,1},{\mathcal{B}}^{2,1}}$ consists of two bilinear rational Bézier surfaces and three triangles. Finally, the regular control surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}(\overline{S_{\lambda }})$ of ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ is the union of four pieces of regular control surfaces each of which consists of five parts (shown in Figure 5(f)). For details about construction of regular control surface and toric degeneration procedure while $t$ approaches infinity, refer to Example 4.

Figure 5.

Regular control surface and regular control net of NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ : (a) weights set $\omega$ , (b) control net of ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ , (c) NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ , (d) control net of $R_{{\mathcal{A}}_{3,3,{\omega }_{\lambda }(1.4)}},ℬ$ , (e) regular control net of ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ , and (f) regular control surface of ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ .

Toric degenerations of NURBS surfaces

It is easy to see that the regular control surface ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}(\overline{S_{\lambda }})$ is a $C^0$ piecewise surface. The following theorem shows that the regular control surface is exactly the limiting position of a NURBS surface when the weights are allowed to vary, which develops the meaning of weights of NURBS surface.

Theorem 3

Let ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ be a NURBS surface of degree $p\times q$ with control points set $\mathcal{B}=\{{\mathbf{P}}_{i,j}|(i,j)\in {\mathcal{A}}_{m,n}\}\subset {\mathbb{R}}^3$ and weights set $\omega =\{{\omega }_{i,j}>0|(i,j)\in {\mathcal{A}}_{m,n}\}$ defined on knot vectors $U$ and $V$ . Suppose that ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}(\overline{S_{\lambda }})$ is the regular control surface of ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ induced by the regular decomposition $\overline{S_{\lambda }}$ , then

$$\underset{t\to +\infty }{lim}{\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),\mathcal{B}}={\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}(\overline{S_{\lambda }})$$

Proof

Using knot insertion, the NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),\mathcal{B}}$ is transformed into the union of $K\times L$ pieces of rational Bézier surfaces

$${\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),\mathcal{B}}={\mathbf{R}}_{\overline{\mathcal{A}},\overline{{\omega }_{\lambda }(t)},\overline{{\mathcal{B}}_{\lambda }(t)}}=\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}$$

(3)

For ${\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}$ , according to triangle inequality

$$\begin{array}{l}\Vert F_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{ℬ}_{\lambda }^{k,l}(t)}-F_{{\mathcal{A}}^{k,l},\widetilde{{\omega }^{k,l}},\widetilde{{ℬ}^{k,l}}}(S_{\lambda }^{k,l})\Vert \\ \le \Vert F_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{ℬ}_{\lambda }^{k,l}(t)}-F_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),\widetilde{{ℬ}^{k,l}}}\Vert \\ +\Vert F_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),\widetilde{{ℬ}^{k,l}}}-F_{{\mathcal{A}}^{k,l},\widetilde{{\omega }^{k,l}},\widetilde{{ℬ}^{k,l}}}(S_{\lambda }^{k,l})\Vert \end{array}$$

(4)

where ∥·∥ is the Hausdorff distance between two subsets of ${\mathbb{R}}^3$ . ²⁸ Since $\underset{t\to +\infty }{lim}{\mathcal{B}}_{\lambda }^{k,l}(t)=\widetilde{{\mathcal{B}}^{k,l}}$ , it comes to the conclusion that

$$\underset{t\to +\infty }{lim}\Vert {\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}-{\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),\widetilde{{\mathcal{B}}^{k,l}}}\Vert =0$$

(5)

This limit is with respect to the Hausdorff distance. Since $\widetilde{{\omega }^{k,l}}$ can be regarded as the limes of ${\omega }_{\lambda }^{k,l}(t)$ for parameter $t$ going to infinity, considering Theorem 1, when the control point $\widetilde{{\mathcal{B}}^{k,l}}$ are fixed while the parameter $t$ approaches infinity, the regular control surface induced by the regular decomposition $S_{\lambda }^{k,l}$ of ${\mathcal{A}}^{k,l}$ is the limit of rational Bézier surface, then we have

$$\underset{t\to +\infty }{lim}{\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),\widetilde{{\mathcal{B}}^{k,l}}}={\mathbf{F}}_{{\mathcal{A}}^{k,l},\widetilde{{\omega }^{k,l}},\widetilde{{\mathcal{B}}^{k,l}}}(S_{\lambda }^{k,l})$$

thus

$$\underset{t\to +\infty }{lim}\Vert {\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),\widetilde{{\mathcal{B}}^{k,l}}}-{\mathbf{F}}_{{\mathcal{A}}^{k,l},\widetilde{{\omega }^{k,l}},\widetilde{{\mathcal{B}}^{k,l}}}(S_{\lambda }^{k,l})\Vert =0$$

(6)

According to equations (4)–(6), the following result can be obtained

$$\underset{t\to +\infty }{lim}{\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}={\mathbf{F}}_{{\mathcal{A}}^{k,l},\widetilde{{\omega }^{k,l}},\widetilde{{\mathcal{B}}^{k,l}}}(S_{\lambda }^{k,l})$$

(7)

Since ${\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}$ is $C^0$ continuous, according to equations (3) and (7), we have

$$\begin{array}{cc}\multicolumn{1}{c}{\underset{t\to +\infty }{lim}{\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),\mathcal{B}}}& =\underset{t\to +\infty }{lim}{\mathbf{R}}_{\overline{\mathcal{A}},\overline{{\omega }_{\lambda }(t)},\overline{{\mathcal{B}}_{\lambda }(t)}}\hfill \\ \multicolumn{1}{c}{}& =\underset{t\to +\infty }{lim}\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}\hfill \\ \multicolumn{1}{c}{}& =\bigcup _{k=1}^K\bigcup _{l=1}^L\underset{t\to +\infty }{lim}{\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}\hfill \\ \multicolumn{1}{c}{}& =\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathbf{F}}_{{\mathcal{A}}^{k,l},\widetilde{{\omega }^{k,l}},\widetilde{{\mathcal{B}}^{k,l}}}(S_{\lambda }^{k,l})\hfill \end{array}$$

By Definition 10,

$${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}(\overline{S_{\lambda }})={\mathbf{R}}_{\overline{\mathcal{A}},\overline{\omega },\overline{\mathcal{B}}}(\overline{S_{\lambda }})=\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathbf{F}}_{{\mathcal{A}}^{k,l},\widetilde{{\omega }^{k,l}},\widetilde{{\mathcal{B}}^{k,l}}}(S_{\lambda }^{k,l})$$

To sum up

$$\underset{t\to +\infty }{lim}{\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),\mathcal{B}}={\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}(\overline{S_{\lambda }})$$

The proof is completed.

Since this property is proved based on toric degeneration of Bézier surface, we call it is the toric degeneration of NURBS surface. The above result shows that the regular control surface is just the limit of a NURBS surface under the toric degeneration. By the same method, we can prove that the regular control net of ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ induced by the lifting function $\lambda$ is the union of the control nets of ${\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}$ (see equation (3)) while $t\to +\infty$ . The following theorem indicates that if a point set is the limit of NURBS surfaces defined on finite set ${\mathcal{A}}_{m,n}$ with settled control points $\mathcal{B}$ , but with differing weights, then the set is a regular control surface induced by some regular decomposition.

Theorem 4

Let ${\mathcal{A}}_{m,n}=\{(i,j)|i=0,1,\dots ,m;j=0,1,\dots ,n\}$ and $\mathcal{B}=\{{\mathbf{P}}_{i,j}|(i,j)\in {\mathcal{A}}_{m,n}\}\subset {\mathbb{R}}^3$ be control points set. If $\mathit{S}\subset {\mathbb{R}}^3$ is a set for which there is a sequence ${\omega }^{(1)},{\omega }^{(2)},\dots$ of weights so that

$$\underset{\tau \to +\infty }{lim}{\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }^{(\tau )},\mathcal{B}}=\mathbf{S}$$

then there is a regular decomposition $\overline{S_{\lambda }}$ of $\overline{\mathcal{A}}$ induced by a lifting function $\lambda$ and a weights set $\omega =\{{\omega }_{i,j}|(i,j)\in {\mathcal{A}}_{m,n}\}$ , such that $\mathbf{S}$ is a regular control surface of ${\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}$ , that is $\mathbf{S}={\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}(\overline{S_{\lambda }})$ .

Proof

Using knot insertion, NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }^{(\tau )},\mathcal{B}}$ is transformed into the union of $K\times L$ pieces of rational Bézier surfaces

$${\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }^{(\tau )},\mathcal{B}}={\mathbf{R}}_{\overline{\mathcal{A}},{\overline{\omega }}^{(\tau )},\overline{\mathcal{B}}}=\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }^{k,l^{(\tau )}},{\mathcal{B}}^{k,l}}$$

where ${\mathcal{A}}^{k,l}=\{(i,j)|i=(k-1)p,(k-1)p+1,\dots ,\mathrm{kp};j=(l-1)q,(l-1)q+1,\dots ,\mathrm{lq}\}$ , ${⊡}_{{\mathcal{A}}^{k,l}}=[(k-1)p,\mathrm{kp}]\times [(l-1)q,\mathrm{lq}]$ , ${\mathcal{B}}^{k,l}=\{{\mathbf{P}}_{i,j}^{M,N}|(i,j)\in {\mathcal{A}}^{k,l}\}$ be the control points set, and ${\omega }^{k,l^{(\tau )}}=\{{\omega }_{i,j}^{M,N^{(\tau )}}|(i,j)\in {\mathcal{A}}^{k,l}\}(\tau =1,2,\dots )$ be a sequence of weights for the $(k,l)\mathrm{th}$ piece rational Bézier surface ${\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }^{k,l^{(\tau )}},{\mathcal{B}}^{k,l}}$ .

If a set ${\mathbf{F}}^{k,l}\subset {\mathbb{R}}^3$ satisfies

$$\underset{\tau \to +\infty }{lim}{\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }^{k,l^{(\tau )}},{\mathcal{B}}^{k,l}}={\mathbf{F}}^{k,l}$$

By Theorem 2, there is regular decomposition $S_{\lambda }^{k,l}$ of ${\mathcal{A}}^{k,l}$ induced by a lifting function $\lambda$ , weights set ${\omega }_{*}^{k,l}$ , and control points set ${\mathcal{B}}_{*}^{k,l}$ , such that ${\mathbf{F}}^{k,l}={\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{*}^{k,l},{\mathcal{B}}_{*}^{k,l}}(S_{\lambda }^{k,l})$ is a regular control surface.

Let $\mathbf{S}=\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathbf{F}}^{k,l}$ , note that the NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }^{(\tau )},\mathcal{B}}$ is coincident with ${\mathbf{R}}_{\overline{\mathcal{A}},{\overline{\omega }}^{(\tau )},\overline{\mathcal{B}}}$ , then

$$\begin{array}{cc}\multicolumn{1}{c}{\underset{\tau \to +\infty }{lim}{\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }^{(\tau )},\mathcal{B}}}& =\underset{\tau \to +\infty }{lim}{\mathbf{R}}_{\overline{\mathcal{A}},{\overline{\omega }}^{(\tau )},\overline{\mathcal{B}}}\hfill \\ \multicolumn{1}{c}{}& =\underset{\tau \to +\infty }{lim}\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }^{k,l^{(\tau )}},{\mathcal{B}}^{k,l}}\hfill \\ \multicolumn{1}{c}{}& =\bigcup _{k=1}^K\bigcup _{l=1}^L\underset{\tau \to +\infty }{lim}{\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }^{k,l^{(\tau )}},{\mathcal{B}}^{k,l}}\hfill \\ \multicolumn{1}{c}{}& =\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathbf{F}}^{k,l}=\mathbf{S}\hfill \end{array}$$

Choose the finite set ${\mathcal{A}}_{m,n}$ , weights set $\omega =\{{\omega }_{i,j}^{0,0}|(i,j)\in {\mathcal{A}}_{m,n}\}$ , and control points set $\mathcal{B}=\{{\mathbf{P}}_{i,j}^{0,0}|(i,j)\in {\mathcal{A}}_{m,n}\}$ satisfying $\overline{\mathcal{A}}=\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathcal{A}}^{k,l}$ , $\overline{\omega }=\bigcup _{k=1}^K\bigcup _{l=1}^L{\omega }_{*}^{k,l}$ , and $\overline{\mathcal{B}}=\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathcal{B}}_{*}^{k,l}$ after knot insertion, respectively. Let $\overline{S_{\lambda }}$ be the regular decomposition of $\overline{\mathcal{A}}$ induced by the lifting function $\lambda$ which takes the same lifting values associated with lattice points of ${\mathcal{A}}^{k,l}$ . By Definition 10

$$\mathbf{S}=\bigcup _{k=1}^K\bigcup _{l=1}^L{\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{*}^{k,l},{\mathcal{B}}_{*}^{k,l}}(S_{\lambda }^{k,l})={\mathbf{R}}_{{\mathcal{A}}_{m,n},\omega ,\mathcal{B}}(\overline{S_{\lambda }})$$

is a regular control surface. The proof is completed.

Examples

In this section, we give some representative examples to illustrate the meaning of regular control surface of NURBS surface. We give more details about the construction of the regular control surface and the procedure of the toric degenerations of NURBS surface in Example 4. In Examples 5 and 6, we will omit the details and illustrate only the final regular control surface and regular control net. Moreover, some examples for surface deformation using the toric degenerations of NURBS surfaces are also presented. This provides potential applications for computer animation.

Example 4

Let $m=n=3$ , ${\mathcal{A}}_{3,3}$ be a $4\times 4$ grid of 16 points. Consider a biquadratic NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ (shown in Figure 6(b)) defined on knot vectors $U=V=\{0,0,0,\frac{1}{2},1,1,1\}$ with weights set $\omega =\{{\omega }_{i,j}^{0,0}|i,j=0,1,2,3\}$ (shown in Figure 6(a)) and control points set $B=\{{\mathbf{P}}_{i,j}^{0,0}|i,j=0,1,2,3\}$ . Figure 6(c) shows the NURBS surface ${\mathbf{R}}_{\overline{\mathcal{A}},\overline{\omega },\overline{\mathcal{B}}}$ and the control net after inserting the knots $\frac{1}{2}$ in $U$ and $V$ , where the NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ is converted to the union of four pieces of biquadratic rational Bézier surfaces ${\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }^{k,l},{\mathcal{B}}^{k,l}}(k,l=1,2)$ .

Figure 6.

Regular control surface and regular control net of NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ : (a) weights set $\omega$ , (b) NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ , (c) NURBS surface ${\mathbf{R}}_{\overline{\mathcal{A}},\overline{\omega },\overline{\mathcal{B}}}$ , (d) control net of ${\mathbf{R}}_{{\mathcal{A}}_{3,3},{\omega }_{\lambda }(2),\mathcal{B}}$ , (e) regular control net of ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ , and (f) regular control surface of ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ .

Suppose that a lifting function $\lambda$ has the assignments shown in Figure 7(a) for lattice points of ${\mathcal{A}}_{3,3}$ . Figure 7(b) shows the lifting values associated with lattice points of $\overline{\mathcal{A}}$ . The red numbers shown in Figure 7(b) are the lifting values associated with new adding lattice points, and the parts in yellow, purple, cyan, and green colors are the lifting values of ${\mathcal{A}}^{1,1}$ , ${\mathcal{A}}^{1,2}$ , ${\mathcal{A}}^{2,1}$ , and ${\mathcal{A}}^{2,2}$ (shown in Figure 7(b)), respectively. The corresponding regular decompositions are shown in the yellow, purple, cyan, and green parts of Figure 7(c). The regular decomposition $\overline{S_{\lambda }}$ of $\overline{\mathcal{A}}$ (shown in Figure 7(c)) is the union of four pieces of regular decompositions of ${\mathcal{A}}^{k,l}$ . By the lifting function $\lambda$ , after inserting the knots, the NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},{\omega }_{\lambda }(t),\mathcal{B}}$ parameterized by $t$ is converted to the union of four pieces of biquadratic rational Bézier surfaces ${\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }_{\lambda }^{k,l}(t),{\mathcal{B}}_{\lambda }^{k,l}(t)}(k,l=1,2)$ , which is a NURBS surface ${\mathbf{R}}_{\overline{\mathcal{A}},\overline{{\omega }_{\lambda }(t)},\overline{{\mathcal{B}}_{\lambda }(t)}}$ with weights set $\overline{{\omega }_{\lambda }(t)}=\{{\omega }_{i,j}^{1,1}(t)|i,j=0,1,2,3,4\}$ and control points set $\overline{{\mathcal{B}}_{\lambda }(t)}=\{{\mathbf{P}}_{i,j}^{1,1}(t)|i,j=0,1,2,3,4\}$ .

Figure 7.

The lifting values and regular decomposition: (a) lifting values of ${\mathcal{A}}_{3,3}$ , (b) lifting values of $\overline{\mathcal{A}}$ , and (c) regular decomposition $\overline{S_{\lambda }}$ of $\overline{\mathcal{A}}$ .

Consider the rational Bézier surface ${\mathbf{F}}_{{\mathcal{A}}^{1,1},{\omega }_{\lambda }^{1,1}(t),{\mathcal{B}}_{\lambda }^{1,1}(t)}$ with weights set ${\omega }_{\lambda }^{1,1}(t)=\{{\omega }_{i,j}^{1,1}(t)|i,j=0,1,2\}$ and control points set ${\mathcal{B}}_{\lambda }^{1,1}(t)=\{{\mathbf{P}}_{i,j}^{1,1}(t)|i,j=0,1,2\}$ . The regular decomposition $S_{\lambda }^{1,1}$ of ${\mathcal{A}}^{1,1}$ are shown in the yellow part of Figure 7(c). Let $t\to +\infty$ , control points set and weights set

$$\begin{array}{l}\widetilde{{ℬ}^{1,1}}=\left\{\begin{array}{ccc}{P}_{0,2}^{0,0}& P_{1,2}^{0,0}& \frac{P_{1,2}^{0,0}+P_{2,2}^{0,0}}{2}\\ P_{0,1}^{0,0}& P_{1,1}^{0,0}& \frac{P_{1,1}^{0,0}+P_{2,1}^{0,0}}{2}\\ P_{0,0}^{0,0}& P_{1,0}^{0,0}& \frac{P_{1,0}^{0,0}+P_{2,0}^{0,0}}{2}\end{array}\right\},\\ \widetilde{{\omega }^{1,1}}=\left\{\begin{array}{ccc}\frac{{\omega }_{0,2}^{0,0}}{2}& \frac{{\omega }_{1,2}^{0,0}}{2}& \frac{{\omega }_{1,2}^{0,0}+{\omega }_{2,2}^{0,0}}{4}\\ {\omega }_{0,1}^{0,0}& {\omega }_{1,1}^{0,0}& \frac{{\omega }_{1,1}^{0,0}+{\omega }_{2,1}^{0,0}}{2}\\ {\omega }_{0,0}^{0,0}& {\omega }_{1,0}^{0,0}& \frac{{\omega }_{1,0}^{0,0}+{\omega }_{2,0}^{0,0}}{2}\end{array}\right\}\end{array}$$

are obtained. By the regular decomposition $S_{\lambda }^{1,1}$ of ${\mathcal{A}}^{1,1}$ , the control points set $\widetilde{{\mathcal{B}}^{1,1}}$ , and the weights set $\widetilde{{\omega }^{1,1}}$ , the regular control surface ${\mathbf{F}}_{{\mathcal{A}}^{1,1},\widetilde{{\omega }^{1,1}},\widetilde{{\mathcal{B}}^{1,1}}}(S_{\lambda }^{1,1})$ of ${\mathbf{F}}_{{\mathcal{A}}^{1,1},{\omega }^{1,1},{\mathcal{B}}^{1,1}}$ is the union of two rational Bézier surfaces of degree $2\times 1$ , whose weights of all of the control points are 0 except the four corner control points in each surface.

Similarly, the regular control surfaces of the other three pieces of rational Bézier surfaces can be obtained. Because the positions of control points change by altering of the parameter $t$ , so does the control net of NURBS surface. We show the control net of ${\mathbf{R}}_{{\mathcal{A}}_{3,3},{\omega }_{\lambda }(t),\mathcal{B}}$ while $t=2$ in Figure 6(d) and the regular control net of ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ (the limit control net while $t\to +\infty )$ in Figure 6(e). The regular control surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}(\overline{S_{\lambda }})$ (shown in Figure 6(f)) of ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}$ is the union of regular control surfaces of four pieces of rational Bézier surfaces. Specifically, ${\mathbf{R}}_{{\mathcal{A}}_{3,3},\omega ,\mathcal{B}}(\overline{S_{\lambda }})$ consists of four bilinear rational Bézier surfaces and four rational Bézier surfaces of degree $2\times 1$ , whose weights of all of the control points are 0 except the four corner control points in each surface. Figure 8 shows the NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},{\omega }_{\lambda }(t),\mathcal{B}}$ at $t=5,10,20,30,50$ and the regular control surface $(t\to +\infty )$ , respectively.

Figure 8.

Toric degeneration of the biquadratic NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{3,3},{\omega }_{\lambda }(t),\mathcal{B}}$ : (a) t = 5, (b) t = 10, (c) t = 20, (d) t = 30, (e) t = 50, and (f) t → +∞.

Example 5

Let $m=n=4$ , ${\mathcal{A}}_{4,4}$ be a $5\times 5$ grid of 25 points. Consider a biquadratic NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ (shown in Figure 9(b)) defined on knot vectors $U=V=\{0,0,0,\frac{1}{4},\frac{3}{4},1,1,1\}$ with weights set (shown in Figure 9(a)). Figure 10(a) shows the lifting values of ${\mathcal{A}}_{4,4}$ by $\lambda$ . After inserting the knots $\frac{1}{4}$ and $\frac{3}{4}$ in $U$ and $V$ , ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ is converted to the union of nine pieces of biquadratic rational Bézier surfaces ${\mathbf{F}}_{{\mathcal{A}}^{k,l},{\omega }^{k,l},{\mathcal{B}}^{k,l}}(k,l=1,2,3)$ . Figure 10(b) shows the lifting values of $\overline{\mathcal{A}}$ and the red numbers are the lifting values of new adding lattice points. The nine parts in yellow, cyan, and purple colors in Figure 10(b) and (c) are the lifting values and corresponding regular decompositions of ${\mathcal{A}}^{k,l}(k,l=1,2,3)$ . The corresponding regular control net of ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ is shown in Figure 9(c). The regular control surface ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}(\overline{S_{\lambda }})$ (shown in Figure 9(d)) of ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ is the union of regular control surfaces of nine pieces of rational Bézier surfaces. In Figure 11, we show the toric degeneration procedure of the surface while $t$ increases.

Figure 9.

A biquadratic NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ and its regular control surface: (a) weights set $\omega$ , (b) NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ , (c) regular control net of ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ , and (d) regular control net of ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ .

Figure 10.

The lifting values and regular decomposition of $\overline{\mathcal{A}}$ : (a) lifting values of ${\mathcal{A}}_{4,4}$ , (b) lifting values of $\overline{\mathcal{A}}$ , and (c) regular decomposition of $\overline{\mathcal{A}}$ .

Figure 11.

Toric degeneration of the biquadratic NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ : (a) t = 1, (b) t = 3, (c) t = 6, (d) t = 10, and (e) t → + ∞.

Example 6

Consider a bicubic NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ (see Figure 12(c)) defined on $U=V=\{0,0,0,0,\frac{1}{3},1,1,1,1\}$ with weights set (see Figure 12(a)). Figure 12(b) shows the lifting function $\lambda$ . The regular control surface ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}(\overline{S_{\lambda }})$ (see Figure 12(d)) of ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ consists of one bilinear rational Bézier surface, two rational Bézier surfaces of degree $3\times 1$ , and one rational Bézier surface of degree $3\times 3$ , whose weights of all of the control points are 0 except the four corner control points in each surface. We show the toric degeneration procedure of the surface while $t$ increases in Figure 13.

Figure 12.

A bicubic NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ and its regular control surface: (a) weights set $\omega$ , (b) lifting function $\lambda$ , (c) NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ , and (d) regular control surface of ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ .

Figure 13.

Toric degeneration of the bicubic NURBS surface ${\mathbf{R}}_{{\mathcal{A}}_{4,4},\omega ,\mathcal{B}}$ : (a) t = 3, (b) t = 5, (c) t = 10, (d) t = 20, (e) t = 50 and (f) t → +∞.

In the next four examples, we indicate the application of our result for shape deformation. Through the toric degenerations of NURBS surface, if the regular decomposition of ${\mathcal{A}}_{m,n}$ is determined, then the limit of ${\mathbf{R}}_{{\mathcal{A}}_{m,n},{\omega }_{\lambda }(t),\mathcal{B}}$ is determined. This means, we can deform a NURBS surface to a special shape by a special lifting function. Furthermore, these results may apply for two- or three-dimensional (2D or 3D) computer animation.

Example 7

Given some data points (see Figure 14(a)), we use two pieces of NURBS surfaces to fit these data points, and these two surfaces are combined into a sphere (see Figure 14(b)). In this example, we aim to deform this sphere to a cube, which is the convex hull of these control points (see Figure 14(c)). The sphere is spliced by two pieces of NURBS surfaces of degree $2\times 2$ on knot vectors $U=V=\{0,0,0,\frac{1}{2},1,1,1\}$ , and the control net of upside piece of NURBS surface is shown in Figure 14(d). Using knot insertion, each NURBS surface is transformed into the union of four pieces of rational Bézier surfaces. It means that each piece of rational Bézier surface is 1/8 of the sphere, which is shown in Figure 15(a). Figure 15(b) shows a 1/8 of the target surface (the cube). So we need to deform each rational Bézier surface (i.e. 1/8 of the sphere, see Figure 15(a)) to 1/8 of the cube, which is the convex hull of the control points of the Bézier surface (see Figure 15(b)). We assume this 1/8 of the cube as the limit surface (i.e. regular control surface) of the Bézier surface, which leads to get the regular decomposition (see Figure 15(d)). The lifting functions which induce this regular decomposition are easy to obtain We select one of the lifting functions, whose lifting values are relatively small positive integers, as shown in Figure 15(d). Of course, we can also select the other lifting function, as long as it induces regular decomposition as shown in Figure 15(c). The union of four pieces of regular decompositions (or lifting functions) which corresponds to four pieces of rational Bézier surfaces is the regular decomposition (or lifting function) of NURBS surface after knot insertion. Then, the lifting function of each NURBS surface before knot insertion can be easily obtained (see Figure 14(e)). By this lifting function, while $t\to +\infty$ , the union of two pieces of NURBS surfaces (i.e. the sphere) degenerates to the cube shown in Figure 14(c). Figure 16 shows the shape deformation procedure of the sphere.

Figure 14.

The NURBS surface: (a) data points, (b) original surface, (c) target surface, (d) control net, and (e) lifting function.

Figure 15.

The 1/8 of the sphere: (a) rational Bézier surface, (b) target surface, (c) regular decomposition, and (d) lifting function.

Figure 16.

The shape deformation procedure of a sphere: (a) t = 1, (b) t = 3, (c) t = 7, and (d) t = 9.

Example 8

Figure 17 shows the shape deformation processes of a sphere using toric degenerations of NURBS surfaces. The lifting function and control net of each piece of NURBS surface are shown in Figure 17(a) and (b), respectively. The sphere is composed of two pieces of NURBS surfaces of degree $2\times 2$ on knot vectors $U=V=\{0,0,0,\frac{1}{2},1,1,1\}$ . The limit of the sphere is a diamond (see Figure 17(e)).

Figure 17.

The shape deformation procedure of a sphere: (a) lifting function of each piece of NURBS surface, (b) control net of each piece of NURBS surface, (c) t = 1, (d) t = 5, and (e) t = 40.

Example 9

Figure 18 shows the shape deformation procedure of a mold using toric degenerations of NURBS surfaces. The mold is composed of four pieces of NURBS surfaces of degree $2\times 1$ on knot vectors $U=\{0,0,0,\frac{2}{3},1,1,1\}$ and $V=\{0,0,1,1\}$ . The lifting function and control net of each piece of NURBS surface are shown in Figure 18(a) and (b), respectively. By these assumptions, while $t\to +\infty$ , Figure 18(c)–(e) show the shape deformation procedure of the mold.

Figure 18.

The shape deformation procedure of a mold: (a) lifting function of each piece of NURBS surface, (b) control net of each piece of NURBS surface, (c) t = 1, (d) t = 2, and (e) t = 30.

Example 10

Using toric degenerations of NURBS surfaces, the shape deformation procedure of the surface of an apple is shown in Figure 19. The surface of this apple is composed of four pieces of NURBS surfaces of degree $4\times 4$ on knot vectors $U=V=\{0,0,0,0,0,\frac{1}{2},1,1,1,1,1\}$ . Figure 19(a) and (b) shows the lifting function and control net of each piece NURBS surface, respectively, and Figure 19(c)–(e) show the deformation procedure of the apple.

Figure 19.

The shape deformation procedure of the surface of an apple: (a) lifting function of each piece NURBS surface, (b) control net of each piece NURBS surface, (c) t = 1, (d) t = 2, and (e) t = 13.

Conclusion

In this article, we define the regular control surface of NURBS surface by regular decomposition and develop the geometric meaning of weights of NURBS surface. The regular control surface is exactly the limit of the NURBS surface when the control points are fixed but all of the weights approach infinity. It means that when all of the weights tend to infinity, the NURBS surface approximates not all of control points simultaneously, but rather a part of control points. Moreover, a set is a regular control surface, if it is the limit of NURBS surfaces on finite set with settled control points, but with differing weights. Our work provides possible application for the injectivity of NURBS surfaces, which plays an important role in image warping and morphing, 3D deformation, and volume morphing. Moreover, we also provide an idea for shape modification and shape deformation of NURBS surface by altering many weights.