Triangular Surface Patch Based on Bivariate Meyer-König-Zeller Operator

Abstract Based on the relationship between probability operators and curve/surface modeling, a new kind of surface modeling method is introduced in this paper. According to a kind of bivariate Meyer-König-Zeller operator, we study the corresponding basis functions called triangular Meyer-König-Zeller basis functions which are defined over a triangular domain. The main properties of the basis functions are studied, which guarantee that the basis functions are suitable for surface modeling. Then, the corresponding triangular surface patch called a triangular Meyer-König-Zeller surface patch is constructed. We prove that the new surface patch has the important properties of surface modeling, such as affine invariance, convex hull property and so on. Finally, based on given control vertices, whose number is finite, a truncated triangular Meyer-König-Zeller surface and a redistributed triangular Meyer-König-Zeller surface are constructed and studied.


Introduction
In computer aided geometric design (CAGD), representing a parametric curve or surface with shape preserving is important. Essentially, the shape preserving is guaranteed by the partition of unity and non-negativity of the basis functions which are used to construct the parametric curve or surface. As we all know, shape preserving is the main property of probability operators. Thus, the shape preserving construction methods of parametric curves or surfaces have certain correlations with some probability operators. It is easy to realize that the classical Bézier curve [1] constructed by Bernstein basis functions is related to the Bernstein operator Bn [2] de ned for any function f ∈ C [ , ], In recent years, based on the Phillips q-Bernstein operator [3], which is a generalization of the Bernstein operator, generalized Bézier curves and surfaces have been introduced in [4][5][6]. In [4], Oruç and Phillips constructed q-Bézier curves by the basis functions of Phillips q-Bernstein operator. Dişibüyük and Oruç [5,6] de ned the q-generalization of rational Bernstein-Bézier curves and tensor product q-Bernstein-Bézier surfaces. Moreover, Simeonov et al. [7] introduced a new variant of the blossom, the q-blossom, which is speci cally adapted to developing identities and algorithms for q-Bernstein bases and q-Bézier curves. In 2014, Han et al. [8] constructed a new generalization of Bézier curves and its corresponding tensor product surfaces based on Lupaş q-analogue of the Bernstein operator [9].
We realized directly the relationship between several probability operators and some curve modeling methods. For example, the rational Bézier curve of negative degree [10] constructed by Bernstein basis functions of negative degree is related to Baskakov operators Bn [11] de ned for any function f ∈ C[ , ∞) The Poisson curve [12] constructed by Poisson basis functions is related to Szász-Mirakyan operators Mn [13] de ned for any function f ∈ C[ , ∞) Goldman introduced the connection between probability theory and computer-aided geometric design in [14][15][16]. Fan and Zeng [17] presented a class of discrete distributions called S-λ distributions and constructed the corresponding S-λ basis functions from these distributions. Zhou et al. [18] extended the work of Fan and Zeng to surface modeling and constructed the tensor product S-λ basis functions and triangular S-λ basis function.
Therefore, we can construct a new modeling method based on the connection between probability operators and computer-aided geometric design. In 1960, Meyer König and Zeller [19] presented a univariate operator , called Meyer-König-Zeller operator. Xiong and Yang [20] introduced a kind of bivariate Meyer-König-Zeller operator which is de ned over a triangular domain. For where, (x, y) ∈ ∆, P n,k,l (x, y) = n+k k k l ( − x) n+ y l (x − y) k−l . In this paper, we introduce a new surface modeling method by the bivariate Meyer-König-Zeller operator.
The remainder of this paper is organized as follows. In Section 2, we present triangular Meyer-König-Zeller basis functions by the bivariate Meyer-König-Zeller operator de ned over a triangular domain (5), and study their main properties. In Section 3, we construct a triangular Meyer-König-Zeller surface patch by the triangular Meyer-König-Zeller basis functions, and prove that the new surface has the main surface modeling properties. For given control vertices, whose number is nite, we introduce a truncated triangular Meyer-König-Zeller surface and a redistributed triangular Meyer-König-Zeller surface in Section 4.

Triangular Meyer-König-Zeller basis functions
In this section, the de nition and several main properties of bivariate Meyer-König-Zeller basis functions over a triangular domain will be given.
the bivariate Bernstein basis functions over a triangle. It is obvious from the de nition of P n,k,l (µ), that it has non-negative properties.
Thus, the sum of all triangular Meyer-König-Zeller basis functions equals to 1.
where P n,k,l (µ) ≡ , if one of n, k, l is negative or k < l.
Proof. We observe that n + k k By the de nition of triangular Meyer-König-Zeller basis functions,

Triangular Meyer-König-Zeller Surface
In this section, we will introduce a method that constructs a kind of surface called a triangular Meyer-König-Zeller surface by triangular Meyer-König-Zeller basis functions P n,k,l (µ).

De nition 3.1. For given control vertices
which is called a triangular Meyer-König-Zeller surface. The mesh constituted by the line segments of V k,l V k+ ,l , V k,l V k+ ,l+ , V k+ ,l V k+ ,l+ is called a control mesh.

) Non-Degenerate (5) The boundary curves are rational Bézier curves in terms of the Bernstein basis functions of negative degree
Proof.
Thus, the triangular Meyer-König-Zeller surface is a ne invariant.
(2) The properties of non-negativity and partition of unity of triangular Meyer-König-Zeller basis functions further guarantee that the triangular Meyer-König-Zeller surface is included within the convex hull of the control mesh which is constituted by {V k,l }.
(3) By the property of interpolation of triangular Meyer-König-Zeller basis functions, it is obvious that S( , , ) = V , , i.e. the triangular Meyer-König-Zeller surface interpolates the corner control vertex.
(4) Suppose that S(µ) collapses to a vertex Q ∈ R , then where • means the inner product. Since the triangular Meyer-König-Zeller basis functions are linearly independent, According to the arbitrariness of v, V k,l = Q. Hence, the triangular Meyer-König-Zeller surface is nondegenerate, only if all control vertices were the same vertex.  called a truncated triangular Meyer-König-Zeller surface.         Figure 5(d). These graphs indicate that the surfaces intuitively approximate the control mesh with increasing n. We conduct some numerical experiments to verify this conclusion. We de ne the extent of the surface approximating control mesh as where L(µ) is the linear interpolation of the control mesh. Table 1 shows the values of D(S(µ), L(µ)) with m = and di erent n.

. Redistributed Triangular Meyer-König-Zeller Surface
We construct a kind of triangular Meyer-König-Zeller surface by redistributing redundant basis functions.   Furthermore, we can let {ω m,l } m l= be a series of basis functions with variable µ . Figure 9 shows the graph of a redistributed triangular Meyer-König-Zeller surface with ω ,l (µ ) = l µ l ( − µ ) −l which are the Bernstein basis functions of degree 2. It is obvious that

Conclusion
We introduced a new kind of surface called a triangular Meyer-König-Zeller surface, which is constructed by triangular Meyer-König-Zeller basis functions. We have studied the main properties of triangular Meyer-König-Zeller basis functions which guarantee that triangular Meyer-König-Zeller surfaces have a ne invariance, possess the convex hull property, are non-degenerate, have interpolative control vertex V , and the boundary curves are rational Bezier curves in terms of the Bernstein functions of negative degree.
Moreover, for given control vertices, whose number is nite, we presented a truncated triangular Meyer-König-Zeller surface and a redistributed triangular Meyer-König-Zeller surface. We remarked that a truncated triangular Meyer-König-Zeller surface is a special kind of triangular Meyer-König-Zeller surface with special control vertices (V k,l = { , , }, k > m, ≤ l ≤ k). Theorem 4.1 shows that a redistributed a triangular Meyer-König-Zeller surface retained all the properties of triangular Meyer-König-Zeller surface.