$\boldsymbol {C^1 (\Omega )}$ NEM Interpolant

Farin [1] has proposed a $C^1(\Omega)$ interpolant based on Sibson's original $C^0 (\Omega)$ natural neighbor interpolant. By embedding Sibson's coordinate in the Bernstein-Bézier representation of a cubic simplex, a $C^1(\Omega)$ interpolant is realized. Bernstein-Bézier patches and related concepts are widely used in surface approximations and in the field of computer-aided geometric design [10]. A review article on triangular Bernstein-Bézier surfaces can be found in [11], while a general treatment of multivariate polynomials over multi-dimensional simplices is given by [12].

In what follows, multi-index notation denoted by the bold characters $\mathbf{i}$ and $\mathbf{j}$ is used. Multi-indexes are tuples of non-negative integers, the components of which are subscribed starting at zero; for instance, ${\mathbf{i}}= (i_1, i_2, \ldots , i_n)$. The norm of a multi-index ${\mathbf{i}}$, denoted by ${\lvert{\mathbf{i}}\rvert}$, is defined to be the sum of the components of ${\lvert{\mathbf{i}}\rvert}$, namely ${\lvert{\mathbf{i}}\rvert}= i_1 + i_2 + \ldots + i_n$ [10]. Let $\mathbf{u} = (u_1, u_2, \ldots ,u_n)$, with the property $\sum_I u_I = 1$, be the barycentric coordinate of a simplex $\delta \in \mathbf{R}^{n-1}$. A Bernstein-Bézier surface of degree $m$ over the simplex $\delta$ can be written in the form [12]

$$b(\mathbf{u}) = \sum_{{\lvert{\mathbf{i}}\rvert} = m} B_{ {\mathbf{i}}}^{m} (\mathbf{u}) b_{ {\mathbf{i}}} ,$$ (1)

where $b_{ {\mathbf{i}}}$ is known as the Bézier ordinate associated with the control point ${\mathbf{i}}/ m$. The control net of $b$ is the network of $(n+1)$-dimensional points ( ${\mathbf{i}}/m,b_{ {\mathbf{i}}}$). In Eq. (1), $B_{ {\mathbf{i}}}^{m}(\mathbf{u})$ are $m$-variate Bernstein polynomials in $n$ variables. To elaborate, they are the terms in the multinomial expansion of unity, i.e.

$$(u_1 + u_2 + \ldots + u_n)^m = \sum_{{\lvert{\mathbf{i}}\rvert} ... ...m}(\mathbf{u}) = \binom{m}{{\mathbf{i}}} u_1^{i_1} u_2^{i_2} \ldots u_n^{i_n} ,$$ (2)

where $\binom{m}{{\mathbf{i}}}$ is the multinomial coefficient which is defined as

$$\binom{m}{{\mathbf{i}}} = \frac{m!}{i_1! i_2! \ldots i_n!} .$$ (3)

The univariate linear Bernstein polynomials are {$1-\xi$, $\xi$}, while the cubic polynomials are {$(1-\xi)^3$, $3(1-\xi)^2 \xi$, $3(1-\xi)\xi^2$, $\xi^3$}, where $\xi \in [0, 1]$. Multivariate Bernstein polynomials have properties very much like their univariate counterparts. From the above equations, some of the important properties of Bernstein polynomials such as partition of unity, positivity, and cardinal interpolation, are easily inferred. The control points (circles) and associated Bézier ordinate values ( $b_{ {\mathbf{i}}}$) for a cubic Bernstein-Bézier triangular patch are shown in Fig. 2. The interested reader can refer to [13], [11], and [10] for further details on the properties and applications of Bernstein-Bézier patches.

Consider a point $\mathbf{x}\in \mathbf{R}^2$ which has $n$ natural neighbors. Let the Sibson coordinate of $\mathbf{x}$ be $\boldsymbol{\Phi} = (\phi_1(\mathbf{x}), \phi_2(\mathbf{x}) , \ldots , \phi_n(\mathbf{x}))$. Since $\sum_I \phi_I(\mathbf{x}) = 1$, we note that $\boldsymbol{\Phi}$ can be considered as a barycentric coordinate (non-unique) of the $n$-gon in the plane. The generalization of Bézier surfaces over a convex polygonal domain was proposed by [14]. By using $\boldsymbol{\Phi}$ instead of $\mathbf{u}$ in Eq. (1), we can construct the surface [1]

$$w^m(\boldsymbol{\Phi}) = \sum_{{\lvert{\mathbf{i}}\rvert} = m} B_{ {\mathbf{i}}}^{m} (\boldsymbol{\Phi}) b_{ {\mathbf{i}}} .$$ (4)

In the above equation, the Bézier ordinate $b_{ {\mathbf{i}}}$ are associated with the control point $\mathbf{q}_{ {\mathbf{i}}} \in \mathbf{R}^2$, where $\mathbf{q}_{ {\mathbf{i}}}$ are the projection of the control points of the $m$-variate Bézier polynomial over the $(n-1)$-dimensional simplex onto the plane [1]:

$$\mathbf{q}_{ {\mathbf{i}}} = \sum_{{\lvert{\mathbf{j}}\rvert} =... ...hbf{i}}/m) \mathbf{x}_{ {\mathbf{j}}} , \qquad {\lvert{\mathbf{i}}\rvert}= m.$$ (5)

On the basis of Eq. (5), one can infer that the components of the barycentric coordinate $\mathbf{u}$ of the $(n-1)$-dimensional simplex is identical to that of the Sibson coordinate $\boldsymbol{\Phi}$ of the mapped $n$-gon on the plane.

The connectivity rule for Bézier simplexes reflects the fact that the domain simplex has all vertices connected to all other vertices. If $\mathbf{q}_{ {\mathbf{i}}}$ and $\mathbf{q}_{ {\mathbf{j}}}$ are two Bézier points in the $n$-gon simplex, then the rule indicates that there must exist integers $r$ and $s$ such that the multi-indexes ${\mathbf{i}}$ and ${\mathbf{j}}$ satisfy

$${\mathbf{i}}- \mathbf{e}_r = {\mathbf{j}}- \mathbf{e}_s ,$$ (6)

where $\mathbf{e}_\alpha = (\delta_{1\alpha}, \delta_{2\alpha},\ldots,\delta_{k\alpha},\ldots,\delta_{n\alpha})$ denotes the multi-index having zero in all components except for the $\alpha$th component, which is one. The control points for the projection of a cubic tetrahedron ($m=3$, $n=4$) onto the plane is shown in Fig. 2.

If we choose $m = 1$ in Eq. (4) and let $w_I$ denote the nodal function values, we obtain

$$w^1(\mathbf{x}) = \sum_{I = 1}^n \phi_1^{\delta_{1I}}(\mathbf{x... ... \delta_{2I}, \ldots , \delta_{nI}} = \sum_{I = 1}^n \phi_I(\mathbf{x}) w_I$$ (7)

which is the original Sibson interpolant. Hence, Eq. (4) can be viewed as a generalized form of the Sibson interpolant.

For $m=3$, we arrive at the following $C^1(\Omega)$ surface representation [1]:

$$w^3(\boldsymbol{\Phi}) = \sum_{{\lvert{\mathbf{i}}\rvert} = 3} B_{ {\mathbf{i}}}^{m} (\boldsymbol{\Phi}) b_{ {\mathbf{i}}} .$$ (8)

The above surface form of a cubic Bézier $n$-gon in Sibson coordinates is used as the $C^1(\Omega)$ NEM trial function for the solution of fourth-order PDEs.