Approximation of minimal toric Bézier patch

Abstract

Toric Bézier patches are a multi-sided generalization of classical rational Bézier patches which are widely used in free-form surface modeling. In this article, we study the problem of finding the patch with minimal area among all possible toric Bézier patches with the given boundaries, that is, the Plateau–toric Bézier problem, and obtain an approximation of minimal toric Bézier patch. Some representative examples are given to verify our results.

Introduction

The study of the minimal surfaces is a hot topic as it exploits a broad spectrum of mathematics and physics. In mathematics, the minimal surfaces are a kind of surfaces which locally minimize their area. These surfaces, characterized by the fact that the mean curvatures vanish, have wide applications in surface design. Physically, the minimal surfaces are familiar as soap films; ¹ they have close relationships with a lot of problems in materials science, civil engineering, ship manufacture, and so on. For instance, Wang ² proposed a surface modeling method of atoms and molecules based on the minimal surfaces. Moreover, Séquin ³ applied them in designing the interactive computer-aided design (CAD) tools for engineering designs.

One of the oldest surface design problems in computer-aided geometric design (CAGD) is the Plateau problem: ⁴ to find the surface of minimal area among all possible surfaces with given boundaries.

Bézier curves and patches play an important role in CAGD. Triangular and rectangular Bézier patches are mostly used to design freedom surfaces as the surface fitting tools. ⁵ The problem of Plateau–Bézier, that is, to find the surface of minimal area among all Bézier patches with given boundaries has been explored by Monterde with the Dirichlet energy functional (cf. Monterde^4,6). Nitsche ⁷ and Osserman ⁸ gave a comprehensive survey of the minimal surfaces. Cosín and Monterde ⁹ studied the properties of the control nets of the Bézier patches which are an approximation of the minimal surfaces. The Plateau–Bézier problem for triangular Bézier patches was studied by Arnal et al. ¹⁰ Hao et al. ¹¹ investigated the Plateau quasi–Bézier problem and the harmonic quasi-Bézier patches. Ahmad and Masud^12,13 presented a curvature algorithm to reduce the area of a surface with variational minimization. In this article, we construct the minimal surface spanned by finite number of boundary curves with the toric Bézier patches.

Toric Bézier patches, introduced into the geometric modeling by Krasauskas ¹⁴ from the theory of toric varieties in algebraic geometry and toric ideals in combinatorics (cf. Sturmfels ¹⁵ and Cox et al. ¹⁶ ), are a multi-sided generalization of classical Bézier patches. There are wide shape possibilities for toric Bézier patches and many well-known surfaces of lower degrees are found to be special cases of toric Bézier patches. ¹⁷ In 2002, Krasauskas and Goldman^18,19 presented the de Casteljau pyramid algorithm, elevation algorithm, and blossoming algorithm of toric Bézier patches based on the depth of the convex polygons. García-Puente et al. ²⁰ gave the geometric significance of the control structures of toric Bézier patches. The G¹ continuity of toric Bézier patches was discussed by Sun and Zhu. ²¹ Liu et al. ²² constructed toric Bézier patches to fill n-sided holes.

In this article, we study the Plateau problem with multi-sided boundaries and obtain an approximation of minimal surface with toric Bézier patches. The rest of the article is organized as follows. In section “Toric Bézier patches and the Dirichlet functional,” we recall the notations and definitions of the toric Bézier patches and present the Dirichlet functionals of toric Bézier patches. In section “Extremals of the Dirichlet functional,” we present the toric Bézier patches of minimal area with the Dirichlet functionals. Moreover, the Dirichlet extremals and some representative examples are given in section “Examples.”

Toric Bézier patches and the Dirichlet functional

The Minkowski sum and the discrete convolution

Let P and Q be two arbitrary sets of finite integers in Z². The Minkowski sum of P and Q is the set

$$P\oplus Q=\{\mathbf{p}+\mathbf{q}|\mathbf{p}\in P\mathrm{and}\mathbf{q}\in Q\}$$

where $\mathbf{p}+\mathbf{q}$ is obtained by the addition of the coordinates of p and q. Similarly, $P^d=P\oplus P\oplus \cdots \oplus P$ (d-fold) (cf. Sturmfels ¹⁵ ).

Let $A=\{a_{\mathbf{p}}|\mathbf{p}\in P\}$ and $B=\{b_{\mathbf{q}}|\mathbf{q}\in Q\}$ be two arrays, then the discrete convolution $A\otimes B$ is indexed by the Minkowski sum $P\oplus Q$ , that is

$$A\otimes B=\{{(A\otimes B)}_{\mathbf{k}}|\mathbf{k}\in P\oplus Q\}$$

(1)

where $(A\otimes B{)}_{\mathbf{k}}=\sum _{\mathbf{p}+\mathbf{q}=\mathbf{k}}{a}_{\mathbf{p}}{b}_{\mathbf{q}}$ (cf. Goldman and Krasauskas ¹⁸ ).

Toric Bézier patches and their derivatives

Let $\sigma =\{{\sigma }_1,{\sigma }_2,\dots ,{\sigma }_m\}\in Z^2$ be a set of finite integers, the lattice polygon $I_{\sigma }$ denotes the convex hull of $\sigma$ with vertices ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n\in \sigma$ , and the kth edge of $I_{\sigma }$ is defined by line $L_k(u,v)=a_{k}u+b_{k}v+c_k,i=1,\dots ,n$ . Normalize these line equations so that every normal vector $(a_k,b_k)$ satisfies the following two additional conditions:

1. The normal vector $(a_k,b_k)$ is inward oriented.

2. $(a_k,b_k)$ is the primitive lattice vector, that is, it is the shortest vector in this direction with integer coordinates.

For each lattice point ${\sigma }_i$ of $\sigma$ , Krasauskas ¹⁴ defined the toric Bernstein basis function on $I_{\sigma }$

$${\beta }_{{\sigma }_i}(u,v)=c_{{\sigma }_i}{L}_1(u,v{)}^{L_1({\sigma }_i)}\cdots L_n(u,v{)}^{L_n({\sigma }_i)},(u,v)\in I_{\sigma }$$

(2)

where ${\sigma }_i\in \sigma$ and the coefficients $c_{{\sigma }_i}>0$ .

Let ${\sigma }^d$ be the d-fold Minkowski sum of $\sigma$ and $I^d$ , the convex hull of ${\sigma }^d$ . It is obvious that $I^d$ is a lattice polygon. Then the toric Bernstein basis functions $\{{\mathcal{B}}_{\gamma }^d(u,v){\}}_{\gamma \in {\sigma }^d}$ on $I^d$ can be computed by convolving the toric Bernstein basis functions ${\beta }_{\sigma }(u,v)=\{{\beta }_{{\sigma }_i}(u,v){\}}_{{\sigma }_i\in \sigma }$ , that is

$$\{{\mathcal{B}}_{\gamma }^d(u,v)\}={\beta }_{\sigma }(u,v)\otimes \cdots \otimes {\beta }_{\sigma }(u,v),(d-\mathrm{fold})$$

(3)

Equation (1) gives the following recurrence relation: ${\mathcal{B}}_{\gamma }^d(u,v)={\beta }_{{\sigma }_1}(u,v){\mathcal{B}}_{\gamma -{\sigma }_1}^{d-1}(u,v)+{\beta }_{{\sigma }_2}(u,v){\mathcal{B}}_{\gamma -{\sigma }_2}^{d-1}(u,v)+\cdots +{\beta }_{{\sigma }_m}(u,v){\mathcal{B}}_{\gamma -{\sigma }_m}^{d-1}(u,v)$ and ${\mathcal{B}}_{\gamma -{\sigma }_i}^{d-1}(u,v)=0$ if $\gamma -{\sigma }_i\notin {\sigma }^{d-1}$ (cf. Goldman and Krasauskas ¹⁸ ).

Hence

$$\frac{\partial {\mathcal{B}}_{\gamma }^d(u,v)}{\partial u}=d\sum _{{\sigma }_i\in \sigma }\frac{\partial {\beta }_{{\sigma }_i}(u,v)}{\partial u}{\mathcal{B}}_{\gamma -{\sigma }_i}^{d-1}(u,v)$$

(4)

Definition 1

A toric Bézier patch defined on $I^d$ in the projective space is a piece of a surface parameterized by the map $\mathcal{B}:I^d\to R{P}^4$ ¹⁴

$$\mathcal{B}(u,v)=\sum _{\gamma \in {\sigma }^d}{\mathcal{B}}_{\gamma }^d(u,v)({\omega }_{\gamma }{p}_{\gamma },{\omega }_{\gamma }),(u,v)\in I^d$$

where $\{{\mathcal{B}}_{\gamma }^d(u,v){\}}_{\gamma \in {\sigma }^d}$ are the toric Bernstein basis functions, $\{p_{\gamma }{\}}_{\gamma \in {\sigma }^d}$ are the control points, and $\{{\omega }_{\gamma }\ge 0{\}}_{\gamma \in {\sigma }^d}$ are the corresponding weights.

In fact, if we set $d=1$ , the toric Bernstein basis functions $\{{\mathcal{B}}_{\gamma }^d(u,v){\}}_{\gamma \in {\sigma }^d}$ are actually the basis functions $\{{\beta }_{{\sigma }_i}(u,v){\}}_{{\sigma }_i\in \sigma }$ . In this case, we get the toric Bézier patches defined on an arbitrary convex lattice polygon. ¹⁹

Notice that if $\sum _{\gamma \in {\sigma }^d}{\mathcal{B}}_{\gamma }^d(u,v){\omega }_{\gamma }\ne 0$ , we can project this surface $\mathcal{B}(u,v)$ into the affine space by the rational map ${\mathcal{B}}^{\star }:I^d\to {\mathbb{R}}^3$ , given by

$${\mathcal{B}}^{\star }(u,v)=\frac{\sum _{\gamma \in {\sigma }^d}{\mathcal{B}}_{\gamma }^d(u,v){\omega }_{\gamma }{p}_{\gamma }}{\sum _{\gamma \in {\sigma }^d}{\mathcal{B}}_{\gamma }^d(u,v){\omega }_{\gamma }},(u,v)\in I^d$$

so that the parametric surface ${\mathcal{B}}^{\star }(u,v)$ is exactly the same toric Bézier patch as defined by Krasauskas. ¹⁴

Remark 1

Let k and l be two positive integers. If $\sigma =\{(i,j)|i,j\in Z,i,j\ge 0,i+j\le k\}$ or $\sigma =\{(i,j)|i,j\in Z,0\le i\le k,0\le j\le l\}$ , then $I_{\sigma }$ is a triangle or rectangle and the toric Bernstein basis functions are exactly the classical triangular Bernstein basis functions or tensor-product Bernstein basis functions. Therefore, the toric surface patch is degenerated to a classical triangular Bézier surface patch of degree k or a tensor-product Bézier surface patch of bi-degree $(k,l)$ . ¹⁴

The lattice polygon $I^d$ implies the geometry of the toric Bézier patches and the index set ${\sigma }^d$ describes the topology structure of the control nets. ¹⁸ In addition, the toric Bézier patches have many good geometrical properties analogous to the classical Bézier surface patches, such as affine invariance, the convex hull property, a partition of unity, and interpolating the corner control points. ¹⁴

The Dirichlet functional

Let $I^d$ be the convex hull of ${\sigma }^d$ , we discuss the toric Bézier patch $\mathcal{B}(u,v)$ defined on $I^d$ . The area functional of $\mathcal{B}(u,v)$ is

$$\mathcal{A}\left(ℬ(u,v)\right)={\displaystyle \underset{I^d}{\int }\left(\Vert \frac{\partial ℬ(u,v)}{\partial u}\times \frac{\partial ℬ(u,v)}{\partial v}\Vert \right)dudv}={\displaystyle \underset{I^d}{\int }{(EG-F^2)}^{1/2}}dudv$$

where E, F, and G are the coefficients of the first fundamental form of toric Bézier patch. ⁴

Because the area functional is high nonlinearity, Douglas used the Dirichlet functional instead of the area functional to resolve the minimal surface problem. ⁸ The Dirichlet functional is defined as

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{D}(\mathcal{B}(u,v))=\frac{1}{2}\underset{I^d}{\int }({\Vert \frac{\partial \mathcal{B}(u,v)}{\partial u}\Vert }^2+{\Vert \frac{\partial \mathcal{B}(u,v)}{\partial v}\Vert }^2)\mathrm{dudv}}\\ \multicolumn{1}{c}{=\frac{1}{2}\underset{I^d}{\int }(E+G)\mathrm{dudv}}\end{array}$$

Actually, $(\mathrm{EG}-F^2{)}^{1/2}\le (\mathrm{EG}{)}^{1/2}\le ((E+G)/2)$ ; therefore, for any toric Bézier patch $\mathcal{B}(u,v)$ , $\mathcal{A}(\mathcal{B}(u,v))\le \mathcal{D}(\mathcal{B}(u,v))$ . Moreover, equality holds only if $E=G$ and $F=0$ , that is, for isothermal surfaces. ⁶

Douglas proved an important property that both functionals have the same extremals among all functions. ⁸ For the toric Bézier patches, both the functionals exit the minimums, but the extremal of Dirichlet functional we obtain is an approximation of the extremal of the area functional. ⁴

Proposition 1

The functionals $\mathcal{A}(\mathcal{B}(u,v))$ and $\mathcal{D}(\mathcal{B}(u,v))$ exist the minimums.

In fact, we only need to prove that the functionals $\mathcal{A}(\mathcal{B}(u,v))$ and $\mathcal{D}(\mathcal{B}(u,v))$ are bounded continuous functions. It is obvious that the area functional $\mathcal{A}(\mathcal{B}(u,v))$ and the Dirichlet functional $\mathcal{D}(\mathcal{B}(u,v))$ are real continuous functions in ${\mathbb{R}}^3$ . Furthermore, since $E\ge 0,G\ge 0$ , and $\mathrm{EG}-F^2\ge 0$ , both functionals are bounded from blow. In addition, when looking for a minimum, we can restrict both functionals to a compact subset. If a control point goes away, the same happens with a portion of the toric Bézier patch $\mathcal{B}(u,v)$ , then the area and the sum of E and G increase. Hence, we can assume a compact subset such that if one of the inner control points goes out of the compact subset, then the area functional $\mathcal{A}(\mathcal{B}(u,v))$ and the Dirichlet function $\mathcal{D}(\mathcal{B}(u,v))$ are greater than some bound. Therefore, when we restrict the functionals to a compact subset, the minimums of $\mathcal{A}(\mathcal{B}(u,v))$ and $\mathcal{D}(\mathcal{B}(u,v))$ exist (cf. Monterde ⁴ ).

Since the boundaries of toric Bézier patch are determined by the control points on the boundaries, the Plateau–toric Bézier problem can be presented as follows: given the boundary control points indexed by the lattice points on the edges of $I^d$ , to find the inner ones such that the area of the resulted toric Bézier patch has the minimal area among all the toric Bézier patches with the same boundary control points.

Extremals of the Dirichlet functional

From the previous analysis, we investigate the Plateau–toric Bézier problem by minimizing the Dirichlet functional to get an approximation of minimal toric Bézier patch. Instead of computing the Euler–Lagrange equation of the Dirichlet functional, we simply compute the patch $\mathcal{B}(u,v)$ where only the gradient of $\mathcal{D}(\mathcal{B}(u,v))$ vanishes.

Theorem 1

The toric Bézier patch $\mathcal{B}(u,v)=\sum _{\gamma \in {\sigma }^d}{\mathcal{B}}_{\gamma }^d(u,v)({\omega }_{\gamma }{p}_{\gamma },{\omega }_{\gamma })$ , defined on the $I^d$ , is an extremal of the Dirichlet functional with given boundaries if and only if the following linear equations hold

$${\displaystyle \underset{I^d}{\int }\left({\delta }_{I^d}^{\lambda ,u}{\displaystyle \sum _{\gamma \in {\sigma }^d}{\delta }_{I^d}^{\gamma ,u}}({\omega }_{\gamma }{p}_{\gamma },{\omega }_{\gamma })+{\delta }_{I^d}^{\lambda ,v}{\displaystyle \sum _{\gamma \in {\sigma }^d}{\delta }_{I^d}^{\gamma ,v}}({\omega }_{\gamma }{p}_{\gamma },{\omega }_{\gamma })\right)}dudv=0,\lambda \in {\sigma }^d$$

where

$$\begin{array}{c}\multicolumn{1}{c}{{\delta }_{I^d}^{\lambda ,u}=\sum _{{\sigma }_i\in \sigma }\frac{\partial {\beta }_{{\sigma }_i}(u,v)}{\partial u}{\mathcal{B}}_{\lambda -{\sigma }_i\in I^{d-1}}^{d-1}(u,v),}\\ \multicolumn{1}{c}{{\delta }_{I^d}^{\gamma ,u}=\sum _{{\sigma }_i\in \sigma }\frac{\partial {\beta }_{{\sigma }_i}(u,v)}{\partial u}{\mathcal{B}}_{\gamma -{\sigma }_i\in I^{d-1}}^{d-1}(u,v)}\\ \multicolumn{1}{c}{{\delta }_{I^d}^{\lambda ,v}=\sum _{{\sigma }_i\in \sigma }\frac{\partial {\beta }_{{\sigma }_i}(u,v)}{\partial v}{\mathcal{B}}_{\lambda -{\sigma }_i\in I^{d-1}}^{d-1}(u,v),}\\ \multicolumn{1}{c}{{\delta }_{I^d}^{\gamma ,v}=\sum _{{\sigma }_i\in \sigma }\frac{\partial {\beta }_{{\sigma }_i}(u,v)}{\partial v}{\mathcal{B}}_{\gamma -{\sigma }_i\in I^{d-1}}^{d-1}(u,v)}\end{array}$$

Proof

For an inner control point $({\omega }_{\lambda }{p}_{\lambda },{\omega }_{\lambda }),\lambda \in {\sigma }^d$ , we compute the gradient of the Dirichlet functional with respect to the coordinates of $({\omega }_{\lambda }{p}_{\lambda },{\omega }_{\lambda })$ . For any $h\in \{1,2,3,4\}$

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial \mathcal{D}(\mathcal{B}(u,v))}{\partial {({\omega }_{\lambda }{p}_{\lambda },{\omega }_{\lambda })}^h}}\\ \multicolumn{1}{c}{=\frac{1}{2}\underset{I^d}{\int }(〈\frac{\partial \mathcal{B}{(u,v)}_u}{\partial {({\omega }_{\lambda }{p}_{\lambda },{\omega }_{\lambda })}^h},\mathcal{B}{(u,v)}_u〉+〈\frac{\partial \mathcal{B}{(u,v)}_v}{\partial {({\omega }_{\lambda }{p}_{\lambda },{\omega }_{\lambda })}^h},\mathcal{B}{(u,v)}_v〉)\mathrm{dudv}}\end{array}$$

(5)

By the definition of toric Bézier patch and equation (4), we get the partial derivatives of $\mathcal{B}(u,v)$ with respect to u

$$\begin{array}{c}\mathcal{B}{(u,v)}_u=d\sum _{\gamma \in {\sigma }^d}(\sum _{{\sigma }_i\in \sigma }\frac{\partial {\beta }_{{\sigma }_i}(u,v)}{\partial u}{\mathcal{B}}_{\gamma -{\sigma }_i}^{d-1}(u,v))({\omega }_{\gamma }{p}_{\gamma },{\omega }_{\gamma })\hfill \end{array}$$

Similarly

$$\begin{array}{c}\mathcal{B}{(u,v)}_v=d\sum _{\gamma \in {\sigma }^d}(\sum _{{\sigma }_i\in \sigma }\frac{\partial {\beta }_{{\sigma }_i}(u,v)}{\partial v}{\mathcal{B}}_{\gamma -{\sigma }_i}^{d-1}(u,v))({\omega }_{\gamma }{p}_{\gamma },{\omega }_{\gamma })\hfill \end{array}$$

Then we compute the derivatives with respect to the coordinates of $({\omega }_{\lambda }{p}_{\lambda },{\omega }_{\lambda })$

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial \mathcal{B}{(u,v)}_u}{\partial {({\omega }_{\lambda }{p}_{\lambda },{\omega }_{\lambda })}^h}=d\frac{\partial }{\partial {({\omega }_{\lambda }{p}_{\lambda },{\omega }_{\lambda })}^h}\frac{\partial }{\partial u}\mathcal{B}(u,v{)}_u}\\ \multicolumn{1}{c}{=d(\sum _{{\sigma }_i\in \sigma }\frac{\partial {\beta }_{{\sigma }_i}(u,v)}{\partial u}{\mathcal{B}}_{\lambda -{\sigma }_i}^{d-1}(u,v))e^h}\end{array}$$

(6)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial \mathcal{B}{(u,v)}_v}{\partial {({\omega }_{\lambda }{p}_{\lambda },{\omega }_{\lambda })}^h}=d\frac{\partial }{\partial {({\omega }_{\lambda }{p}_{\lambda },{\omega }_{\lambda })}^h}\frac{\partial }{\partial v}\mathcal{B}(u,v{)}_v}\\ \multicolumn{1}{c}{=d(\sum _{{\sigma }_i\in \sigma }\frac{\partial {\beta }_{{\sigma }_i}(u,v)}{\partial v}{\mathcal{B}}_{\lambda -{\sigma }_i}^{d-1}(u,v))e^h}\end{array}$$

(7)

and ${\mathcal{B}}_{\lambda -{\sigma }_i}^{d-1}(u,v)=0$ , if $\lambda -{\sigma }_i\notin {\sigma }^{d-1}$ , where $e^h,h\in \{1,2,3,4\}$ denotes the hth vector of canonical basis, that is, $e^1=\{1,0,0,0\},e^2=\{0,1,0,0\},e^3=\{0,0,1,0\},e^4=\{0,0,0,1\}$ .

Then substituting equations (6) and (7), $\mathcal{B}(u,v{)}_u$ and $\mathcal{B}(u,v{)}_v$ , into equation (5), the linear equations hold.

Remark 2

The triangular Bézier patch of minimal area ¹⁰ and the Plateau–Bézier problem ⁶ are the special cases of the above Theorem 1.

Actually, as mentioned in Remark 1, when $I_{\sigma }$ is a triangle or rectangle, the toric Bernstein basis functions $\{{\beta }_{\sigma }(u,v){\}}_{{\sigma }_i\in \sigma }$ defined on $I_{\sigma }$ are exactly the classical Bernstein basis functions, then toric Bézier patch $\mathcal{B}(u,v)$ is the classical triangular or tensor-product Bézier patch. Therefore, the triangular Bézier patch of minimal area ¹⁰ and the Plateau–Bézier problem ⁶ are the special cases of Theorem 1.

The above theorem demonstrates that the inner control points and weights of the toric Bézier patch are totally determined by the boundary control points and weights. Assuming the inner control points and weights as unknown, the linear equation system is always compatible and can be solved in terms of the boundary control points and weights.

Examples

In this section, some examples will be discussed. In order to simplify the expression and calculation, we assume the weights all equal in the following examples.

Example 1

In this example, we give a minimal toric Bézier patch defined on the convex hull $I_{\sigma }$ , as shown in Figure 1(a), where $\sigma =\{(2,0),(1,1),(0,1),(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1),(3,-1),(0,0),(1,0)\}$ , and the given boundary control points are $(4,0,10),(3,1,3),(2,2,8),(1,1,4),(1,0,8),(1,-1,3),(2,-2,8),(3,-3,4),(4,-2,7),(5,-1,4)$ .

Figure 1.

The minimal toric Bézier patch defined on a tetragon: (a) the convex hull, (b) the boundary curves, and (c) the toric Bézier patch.

The edges of $I_{\sigma }$ are defined by the following lines

$$\begin{array}{l}{L}_1(u,v)=u+1;L_2(u,v)=v+1;\\ L_3(u,v)=-u-v+1;\\ L_4(u,v)=-v+1\end{array}$$

Then we get the toric Bernstein basis function for every lattice point ${\sigma }_i\in \sigma$

$${\beta }_{{\sigma }_i}(u,v)=c_{{\sigma }_i}{(u+1)}^{L_1({\sigma }_i)}{(v+1)}^{L_2({\sigma }_i)}{(-u-v+2)}^{L_3({\sigma }_i)}{(-v+1)}^{L_4({\sigma }_i)},(u,v)\in I_{\sigma }$$

The corresponding coefficients $c_{{\sigma }_i}$ are taken as follows: $2,1,2,1,2,1,4,6,4,1,1,1$ .

By the basis functions on the boundary and the given control points, we get the boundary curves, which are shown in Figure 1(b).

Substituting ${\beta }_{{\sigma }_i}(u,v)$ into the linear equation system in Theorem 1, we can get the two unknown inner control points

$$p_{(0,0)}=(2,14.6,-11.7),p_{(1,0)}=(-2.96,10.1,4.79)$$

Figure 1(c) shows the corresponding toric Bézier patch.

Example 2

In this example, we give a minimal toric Bézier patch defined on the convex hull $I_{\sigma }^2$ , as shown in Figure 2(a), where $\sigma =\{(0,0),(0,1),(-1,0),(-1,-1),(0,-1),(1,0)\}$ , and the given boundary control points are $(2,2,3),(0,4,4),(-2,2,5),(-4,0,4),(-4,-2,3),(-4,-4,4),(-2,-4,5),(0,-4,3),(2,-2,5),(4,0,4)$ .

Figure 2.

The minimal toric Bézier patch defined on a pentagon: (a) the convex hull, (b) the boundary curves, and (c) the toric Bézier patch.

The edges of $I_{\sigma }$ are defined by the following lines

$$\begin{array}{l}{L}_1(u,v)=u-v+1;\\ L_2(u,v)=u+1;\\ L_3(u,v)=v+1;\\ L_4(u,v)=-u+v+1;\\ L_5(u,v)=-u-v+1\end{array}$$

Then we get the toric Bernstein basis function ${\beta }_{{\sigma }_i}(u,v)$ for every lattice point ${\sigma }_i$

$${\beta }_{{\sigma }_i}(u,v)={(u-v+1)}^{L_1({\sigma }_i)}{(u+1)}^{L_2({\sigma }_i)}{(v+1)}^{L_3({\sigma }_i)}{(-u+v+1)}^{L_4({\sigma }_i)}{(-u-v+1)}^{L_5({\sigma }_i)}$$

By equation (3), the toric Bernstein basis functions $\{{\mathcal{B}}_{\gamma }^2(u,v){\}}_{\gamma \in {\sigma }^2}$ defined on $I_{\sigma }^2$ are presented as

$${\left\{{ℬ}_{\gamma }^2(u,v)\right\}}_{\gamma \in {\sigma }^2}={\left\{{\beta }_{{\sigma }_i}(u,v)\right\}}_{{\sigma }_i\in \sigma }\otimes {\left\{{\beta }_{{\sigma }_i}(u,v)\right\}}_{{\sigma }_i\in \sigma },(u,v)\in I_{\sigma }^2$$

where ${\sigma }^2$ is the twofold Minkowski sum of $\sigma$ .

The boundary curves are shown in Figure 2(b). By Theorem 1, we get the inner control points

$$\begin{array}{c}\multicolumn{1}{c}{p_{(0,0)}=(47.1,86.61,2.05),}\\ \multicolumn{1}{c}{p_{(0,1)}=(-13.00,-15.06,5.72),}\\ \multicolumn{1}{c}{p_{(-1,0)}=(-1.63,21.06,-0.87)}\\ \multicolumn{1}{c}{p_{(-1,-1)}=(28.19,-9.61,8.20),}\\ \multicolumn{1}{c}{p_{(0,-1)}=(-22.34,-24.99,-4.26),}\\ \multicolumn{1}{c}{p_{(1,0)}=(-1.06,0.29,6.27)}\end{array}$$

Figure 2(c) shows the corresponding toric Bézier patch; notice that it is a pentagonal parametric surface.

Example 3

This example shows the minimal toric Bézier patch defined on the convex hull $I_{\sigma }^2$ , as shown in Figure 3(a), where $\sigma =\{(0,1),(-1,0),(-1,-1),(0,-1),(1,0),(1,1),(0,0)\}$ , and the given boundary control points are $(-4,-1,2),(-5,-3,4),(-6,-6,2),(-3,2,6),(-4,-5,5),(1,5,3),(0,-4,2),(3,7,5),(2,-3,4),(6,4,2),(5,1,4),(4,-1,2)$ .

Figure 3.

The minimal toric Bézier patch defined on a hexagon: (a) the convex hull, (b) the boundary curves, and (c) the toric Bézier patch.

The lines of each edge of $I_{\sigma }$ are defined by

$$\begin{array}{c}\multicolumn{1}{c}{L_1(u,v)=u-v+1;L_2(u,v)=-v+1;}\\ \multicolumn{1}{c}{L_3(u,v)=-u+1}\\ \multicolumn{1}{c}{L_4(u,v)=-u+v+1;L_5(u,v)=u+1;}\\ \multicolumn{1}{c}{L_6(u,v)=v+1}\end{array}$$

The toric Bernstein basis function for every lattice point ${\sigma }_i$ is given by

$${\beta }_{{\sigma }_i}(u,v)={(u-v+1)}^{L_1({\sigma }_i)}{(-v+1)}^{L_2({\sigma }_i)}{(-u+1)}^{L_3({\sigma }_i)}{(-u+v+1)}^{L_4({\sigma }_i)}{(u+1)}^{L_5({\sigma }_i)}{(v+1)}^{L_6({\sigma }_i)}$$

Then by equation (3), the toric Bernstein basis functions $\{{\mathcal{B}}_{\gamma }^2(u,v){\}}_{\gamma \in {\sigma }^2}$ defined on $I_{\sigma }^2$ can be obtained by the discrete convolution of $\{{\beta }_{{\sigma }_i}(u,v){\}}_{{\sigma }_i\in \sigma }$ , where ${\sigma }^2$ is the twofold Minkowski sum of $\sigma$ .

The boundary curves are shown in Figure 3(b). By Theorem 1, we get the inner control points

$$\begin{array}{c}\multicolumn{1}{c}{p_{(-1,0)}=(0.99,-16.5,2.62),}\\ \multicolumn{1}{c}{p_{(-1,-1)}=(4.59,57.1,4.44),}\\ \multicolumn{1}{c}{p_{(0,1)}=(-20.0,19.8,0.53)}\\ \multicolumn{1}{c}{p_{(0,0)}=(-17.2,-36.3,-30.6),}\\ \multicolumn{1}{c}{p_{(0,-1)}=(36.3,-17.1,3.69),}\\ \multicolumn{1}{c}{p_{(1,1)}=(-0.25,-53.8,14.0)}\\ \multicolumn{1}{c}{p_{(1,0)}=(-5.25,22.5,9.48)}\end{array}$$

The corresponding toric Bézier patch is shown in Figure 3(c); notice that it is a hexagonal parametric surface patch.

Conclusion and future work

In this article, we study the Plateau–toric Bézier problem with the Dirichlet functional and present a method to construct the approximation of the minimal toric Bézier patch. Three examples with different types of parametric domains are illustrated to show the effectiveness of the presented method. Each example denotes a kind of minimal and multi-sided parametric surface patches which implies a lot of shape possibilities for engineering design and CAGD. The hexagonal toric Bézier patch constructed in Example 3 is the best solution in the examples for potential engineering problems, in which the domain of the patch is symmetric.

As well known, the minimal surfaces have close relationships with a lot of problems in biology, materials science, civil engineering, ship manufacture, and so on. Therefore, to study the potential application of the minimal surfaces in these fields is a very interesting topic. For future work, we will study not only the toric Bézier patches of minimal area in theory but also the minimal toric Bézier patches in engineering design and CAGD for applications.