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

Guorong Zhou; Qing-Bo Cai

doi:10.1515/math-2019-0021

Article Open Access

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

Guorong Zhou and Qing-Bo Cai

Published/Copyright: April 29, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

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.

Keywords: Surface modeling; Meyer-König-Zeller operator; basis function; triangular surface

1 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 B_n [2] defined for any function f ∈ C[0, 1],

Bn(f;x)=∑k=0nnkxk(1−x)n−kf(kn),x∈[0,1]. (1)

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] defined 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 specifically 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 𝓑_n [11] defined for any function f ∈ C[0, ∞)

Bn(f;t)=∑k=0∞n+k−1kxk(1+x)−n−kf(kn),x∈[0,∞). (2)

The Poisson curve [12] constructed by Poisson basis functions is related to Szász-Mirakyan operators M_n [13] defined for any function f ∈ C[0, ∞)

Mn(f,x)=∑k=0∞(nx)kk!e−nx,x∈[0,∞). (3)

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

Mn(f;x)=f(1),x=1;∑k=0∞mn,k(x)f(kn+k),0≤x<1; (4)

where mn,k(x)=n+kkxk(1−x)n+1, f ∈ C[0, 1], called Meyer-König-Zeller operator. Xiong and Yang [20] introduced a kind of bivariate Meyer-König-Zeller operator which is defined over a triangular domain. For any function f ∈ C[Δ], Δ = {(x, y)∣0 ≤ y ≤ x ≤ 1},

Mn(f;x,y)=f(1,y)x=1;∑k=0∞∑l=0kPn,k,l(x,y)f(kn+k,ln+k),x≠1; (5)

where, (x, y) ∈ Δ, P_n,k,l(x, y) = n+kkkl(1−x)n+1yl(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 defined 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 finite, we introduce a truncated triangular Meyer-König-Zeller surface and a redistributed triangular Meyer-König-Zeller surface in Section 4.

2 Triangular Meyer-König-Zeller basis functions

In this section, the definition and several main properties of bivariate Meyer-König-Zeller basis functions over a triangular domain will be given.

Firstly, we given the barycentric coordinates μ = (μ₁, μ₂, μ₃), μ_i ≥ 0 (i = 1, 2, 3) and μ₁ + μ₂ + μ₃ = 1.

Definition 2.1

Bivariate Meyer-König-Zeller basis functions over a triangle called triangular Meyer-König-Zeller basis functions are defined by

Pn,k,l(μ)=n+kkklμ1n+1μ2lμ3k−l; (6)

where μ ∈ D = {μ₁ > 0, μ₂ ≥ 0, μ₃ ≥ 0, μ₁ + μ₂ + μ₃ = 1} and k, l are natural numbers.

Several illustrations of bivariate Meyer-König-Zeller basis functions with n = 0, 2, 3 and k = 3 are shown in Figure 1. Triangular Meyer-König-Zeller basis functions have the properties which are similar to the bivariate Bernstein basis functions over a triangle.

$Figure 1 Triangular Meyer-König-Zeller basis functions with n = 0, 2, 3 and k = 3 which are P0,3,0(μ) (a1), P0,3,1(μ) (a2), P0,3,2(μ) (a3), P0,3,3(μ) (a4), P2,3,0(μ) (b1), P2,3,1(μ) (b2), P2,3,2(μ) (b3), P2,3,3(μ) (b4), P3,3,0(μ) (c1), P3,3,1(μ) (c2), P3,3,2(μ) (c3), P3,3,3(μ) (c4).$

Figure 1

Triangular Meyer-König-Zeller basis functions with n = 0, 2, 3 and k = 3 which are P_0,3,0(μ) (a1), P_0,3,1(μ) (a2), P_0,3,2(μ) (a3), P_0,3,3(μ) (a4), P_2,3,0(μ) (b1), P_2,3,1(μ) (b2), P_2,3,2(μ) (b3), P_2,3,3(μ) (b4), P_3,3,0(μ) (c1), P_3,3,1(μ) (c2), P_3,3,2(μ) (c3), P_3,3,3(μ) (c4).

It is obvious from the definition of P_n,k,l(μ), that it has non-negative properties.

Proposition 2.1

(Non-negative).

Pn,k,l(μ)≥0. (7)

Proposition 2.2

(Partition of Unity).

∑k=0∞∑l=0kPn,k,l(μ)=1. (8)

Proof

From Binomial expansion, it is obvious that

∑k=0∞∑l=0kPn,k,l(μ)=∑k=0∞n+kkμ1n+1∑l=0kklμ2lμ3k−l=∑k=0∞n+kkμ1n+1(μ2+μ3)k=∑k=0∞n+kkμ1n+1(1−μ1)k.

Let μ1=11+t,t∈[0,∞), then

∑k=0∞n+kkμ1n+1(1−μ1)k=∑k=0∞n+kk(1+t)−n−k−1tk=∑k=0∞−n−1k(1+t)−n−1−k(−t)k=(1+t−t)−n−1=1,

where −n−1k=(−1)kn+kk. Thus, the sum of all triangular Meyer-König-Zeller basis functions equals to 1.□

Proposition 2.3

(Interpolation).

Pn,k,l(1,0,0)=δ0,kδk,l. (9)

where δi,j=1i=j,0i≠j, is the Kronecker delta.

Proof

Since

Pn,k,l(1,0,0)=n+kkkl1n+10l0k−l,

when l = 0 and k = l = 0, P_n,0,0(1, 0, 0) = 1. It is obvious that P_n,k,l(1, 0, 0) = 0 in other cases.□

Proposition 2.4

(Linear independence).

∑k=0∞∑l=0kck,lPn,k,l(μ)=0⟺ck,l=0. (10)

Proof

Since 0 < μ₁ ≤ 1 and

∑k=0∞∑l=0kck,lPn,k,l(μ)=∑k=0∞n+kkμ1n+1∑l=0kck,lklμ2lμ3k−l,

we get

∑k=0∞∑l=0kck,lPn,k,l(μ)=0⟺∑l=0kck,lklμ2lμ3k−l=0,

by fixing the value of μ₁. As we know, klμ2lμ3k−l is a univariate Bernstein polynomial over [0, 1 − μ₁]. Thus,

∑l=0kck,lklμ2lμ3k−l=0⟺ck,l=0.

□

Proposition 2.5

(Boundary). When μ₂ = 0 or μ₃ = 0, triangular Meyer-König-Zeller basis functions transform to corresponding univariate Bernstein basis functions of negative degree [10].

Proof

When μ₂ = 0,

Pn,k,0(μ1,0,μ3)=n+kkk0μ1n+1μ3k=n+kkμ1n+1(1−μ1)k.

Let μ1=11−t,t∈(−∞,0], then

Pn,k,0(μ1,0,μ3)=n+kk(1−t)−n−1−k(−t)k=−n−1k(1−t)−n−1−ktk=Bk−n−1(t),

where, Bk−n−1(t) is a univariate Bernstein basis function of negative degree [10]. Similarly, P_n,k,k(μ₁,μ₂, 0) is also a univariate Bernstein basis function of negative degree.□

Proposition 2.6

(Symmetry).

Pn,k,l(μ1,μ2,μ3)=Pn,k,k−l(μ1,μ3,μ2).

Proof

Since kl=kk−l,

Pn,k,l(μ)=n+kkklμ1n+1μ2lμ3k−l=n+kkkk−lμ1n+1μ3k−lμ2k−(k−l)=Pn,k,k−l(μ1,μ3,μ2).

□

Proposition 2.7

(Degree Reduction).

Pn,k,l(μ)=μ1Pn−1,k,l(μ)+μ2Pn,k−1,l−1(μ)+μ3Pn,k−1,l(μ), (11)

where P_n,k,l(μ) ≡ 0, if one of n, k, l is negative or k < l.

Proof

We observe that

n+kkkl=(n+k)!n!k!k!l!(k−l)!=(n+k)!n!l!(k−l)!.

By the definition of triangular Meyer-König-Zeller basis functions,

Pn,k,l(μ)=(n+k−1)!(n+l+k−l)n!l!(k−l)!μ1n+1μ2lμ3k−l=μ1(n−1+k)!(n−1)!l!(k−l)!μ1nμ2lμ3k−l+μ2(n+k−1)!n!(l−1)!(k−l)!μ1n+1μ2l−1μ3k−l+μ3(n+k−1)!n!l!(k−1−l)!μ1n+1μ2lμ3k−1−l=μ1Pn−1,k,l(μ)+μ2Pn,k−1,l−1(μ)+μ3Pn,k−1,l(μ).

Proposition 2.8

(Degree Elevation).

Pn,k,l(μ)=n+1n+k+1Pn+1,k,l(μ)+l+1n+k+1Pn,k+1,l+1(μ)+k−l+1n+k+1Pn,k+1,l(μ) (12)

Proof

Since μ₁ + μ₂ + μ₃ = 1,

Pn,k,l(μ)=n+kkklμ1n+1μ2lμ3k−l(μ1+μ2+μ3)=n+kkklμ1n+2μ2lμ3k−l+μ1n+1μ2l+1μ3k−l+μ1n+1μ2lμ3k−l+1=n+1n+k+1n+1+kkklμ1n+2μ2lμ3k−l+l+1n+k+1n+k+1k+1k+1l+1μ1n+1μ2l+1μ3k−l+k+1−ln+k+1n+k+1k+1k+1lμ1n+1μ2lμ3k+1−l=n+1n+k+1Pn+1,k,l(μ)+l+1n+k+1Pn,k+1,l+1(μ)+k+1−ln+k+1Pn,k+1,l(μ).

□

Proposition 2.9

(Integration).

∬DPn,k,l(μ)dσ=n+1(n+k+1)(n+k+2)(n+k+3). (13)

Proof

The integral area D can be described as {(μ₁, μ₂) ∣ 0 < μ₁ ≤ 1, 0 ≤ μ₂ ≤ 1 − μ₁}, thus

∬DPn,k,l(μ)dσ=∫01n+kkμ1n+1dμ1∫01−μ1klμ2l(1−μ1−μ2)k−ldμ2

Let t=μ21−μ1, then

∫01−μ1klμ2l(1−μ1−μ2)k−ldμ2=(1−μ1)k+1∫01kltl(1−t)k−ldt.

Moreover, from the integration of a univariate Bernstein polynomial, we get

∫01kltl(1−t)k−ldt=1k+1.

Therefore,

∬DPn,k,l(μ)dσ=1k+1∫01n+kkμ1n+1(1−μ1)k+1dμ1=n+1(n+k+1)(n+k+2)∫01n+k+2k+1μ1n+1(1−μ1)k+1dμ1=n+1(n+k+1)(n+k+2)(n+k+3).

□

Proposition 2.10

(Differentiation).

∂∂μ1Pn,k,l(μ)=(n+1)(n+k)nPn−1,k,l(μ); (14)

∂∂μ2Pn,k,l(μ)=(n+k)Pn,k−1,l−1(μ); (15)

∂∂μ3Pn,k,l(μ)=(n+k)Pn,k−1,l(μ). (16)

Proof

∂∂μ1Pn,k,l(μ)=(n+1)n+kkklμ1nμ2lμ3k−l=(n+1)(n+k)nn−1+kkklμ1nμ2lμ3k−l=(n+1)(n+k)nPn−1,k,l(μ).

Similarly,

∂∂μ2Pn,k,l(μ)=(n+k)n+k−1k−1k−1l−1μ1n+1μ2l−1μ3k−l=(n+k)Pn,k−1,l−1(μ);∂∂μ3Pn,k,l(μ)=(n+k)n+k−1k−1k−1lμ1n+1μ2lμ3k−l−1=(n+k)Pn,k−1,l(μ).

□

3 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).

Definition 3.1

For given control vertices {V_k,l ∈ R³}, k ≥ 0, 0 ≤ l ≤ k, we construct a surface denoted by

S(μ)=∑k=0∞∑l=0kPn,k,l(μ)Vk,l,μ∈D, (17)

which is called a triangular Meyer-König-Zeller surface. The mesh constituted by the line segments of V_k,lV_k+1,l, V_k,lV_k+1,l+1, V_k+1,lV_k+1,l+1 is called a control mesh.

Theorem 3.1

From the properties of triangular Meyer-König-Zeller basis functions, we can derive the geometric properties of a triangular Meyer-König-Zeller surface as follows:

Affine Invariance
Convex Hull Property
Interpolative control vertex V_0,0
Non-Degenerate
The boundary curves are rational Bézier curves in terms of the Bernstein basis functions of negative degree

Proof

The affine invariance is one of the most important geometric properties for curves and surfaces modeling in Computer-aided Geometric Design. For a given triangular Meyer-König-Zeller surface S(μ), an affine transformation operator 𝓐 is acting on it, i.e.

A(S(μ))=A∑k=0∞∑l=0kPn,k,l(μ)Vk,l.

Since the properties of non-negativity and partition of unity of triangular Meyer-König-Zeller basis functions hold, we have,

A(S(μ))=∑k=0∞∑l=0kPn,k,l(μ)A(Vk,l). (18)

Thus, the triangular Meyer-König-Zeller surface is affine invariant.
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}.
By the property of interpolation of triangular Meyer-König-Zeller basis functions, it is obvious that S(1, 0, 0) = V_0,0, i.e. the triangular Meyer-König-Zeller surface interpolates the corner control vertex.
Suppose that S(μ) collapses to a vertex Q ∈ R³, then

0=S(μ)−Q=∑k=0∞∑l=0kPn,k,l(μ)(Vk,l−Q).

For any v ∈ R³, we have

∑k=0∞∑l=0kPn,k,l(μ)(Vk,l−Q)∙v=0,

where • means the inner product. Since the triangular Meyer-König-Zeller basis functions are linearly independent,

(Vk,l−Q)∙v=0.

According to the arbitrariness of v, V_k,l = Q. Hence, the triangular Meyer-König-Zeller surface is non-degenerate, only if all control vertices were the same vertex.
The boundary curves of a triangular Meyer-König-Zeller surface are S(μ₁, 0, μ₃) and S(μ₁, μ₂, 0). By the boundary property of triangular Meyer-König-Zeller basis functions, we get

S(μ1,0,μ3)=∑k=0∞Pn,k,0(μ1,0,μ3)Vk,0,

which is a rational Bézier curve in terms of the Bernstein basis functions of negative degree with control vertices {V_k,0}. Similarly,

S(μ1,μ2,0)=∑k=0∞Pn,k,0(μ1,μ2,0)Vk,k

also is a rational Bézier curve in terms of the Bernstein basis functions of negative degree with control vertices {V_k,k}.□

4 Implementation

In this section, based on given control vertices some practical examples of triangular Meyer-König-Zeller surfaces will be shown.

Given control vertices {V_k,l}, 0 ≤ k ≤ m, 0 ≤ l ≤ k, where k, l and m are natural numbers. m is the number of layers of the triangular control mesh. k and l are the indices of the control vertices. In Figure 2, the topology of the triangular control mesh with m = 3 is shown.

$Figure 2 The topology of control mesh with m = 3.$

Figure 2

The topology of control mesh with m = 3.

Now, we face a difficulty in that the number of triangular Meyer-König-Zeller basis functions is infinite but the number of control vertices is finite. We solve this difficulty by discarding or redistributing redundant basis functions.

4.1 Truncated Triangular Meyer-König-Zeller Surface

Definition 4.1

For the given control vertices {V_k,l}, 0 ≤ k ≤ m, 0 ≤ l ≤ k, we have

S¯(μ)=∑k=0m∑l=0kPn,k,l(μ)Vk,l,μ∈D (19)

called a truncated triangular Meyer-König-Zeller surface.

Figure 3 shows the graphs of truncated triangular Meyer-König-Zeller surfaces with m = 2, n = 1 (the blue one) and m = 2, n = 3 (the red one). It is obvious that lim_μ₁→0S̄(μ) = 0. Therefore, we can define S̄(0, μ₂,μ₃) = 0, i.e. the truncated triangular Meyer-König-Zeller surface interpolates the coordinate origin point.

$Figure 3 The graphs of truncated triangular Meyer-König-Zeller surfaces: the green surface with m = 2, n = 1, the red surface with m = 2, n = 3.$

Figure 3

The graphs of truncated triangular Meyer-König-Zeller surfaces: the green surface with m = 2, n = 1, the red surface with m = 2, n = 3.

Remark 4.1

The truncated triangular Meyer-König-Zeller surface is equivalent to the triangular Meyer-König-Zeller surface with V_k,l = {0, 0, 0}, k > m, 0 ≤ l ≤ k, i.e.

S¯(μ)=∑k=0m∑l=0kPn,k,l(μ)Vk,l+P¯n,m(μ)V¯,

where P̄_n,m(μ) = 1−∑k=0m∑l=0kPn,k,l(μ) and V̄ = {0, 0, 0}. Therefore, S̄(μ) is a special kind of triangular Meyer-König-Zeller surface with the control mesh as shown in Figure 4.

$Figure 4 The actual control mesh of truncated triangular Meyer-König-Zeller surfaces with m = 2, n = 1 (the green one) and m = 2, n = 3 (the red one).$

Figure 4

The actual control mesh of truncated triangular Meyer-König-Zeller surfaces with m = 2, n = 1 (the green one) and m = 2, n = 3 (the red one).

Figure 5 shows the graphs of the truncated triangular Meyer-König-Zeller surfaces with (m = 2, n = 1) Figure 5(a), (m = 2, n = 3) Figure 5(b), (m = 2, n = 6) Figure 5(c), (m = 2, n = 9) 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 define the extent of the surface approximating control mesh as

D(S¯(μ),L(μ))=maxμ{S¯(μ)}−maxμ{L(μ)} (20)

$Figure 5 The truncated triangular Meyer-König-Zeller surfaces with (m = 2, n = 1)(a), (m = 2, n = 3)(b), (m = 2, n = 6)(c), (m = 2, n = 9)(d).$

Figure 5

The truncated triangular Meyer-König-Zeller surfaces with (m = 2, n = 1)(a), (m = 2, n = 3)(b), (m = 2, n = 6)(c), (m = 2, n = 9)(d).

where L(μ) is the linear interpolation of the control mesh. Table 1 shows the values of D(S̄(μ), L(μ)) with m = 2 and different n.

Table 1

The values of D(S̄(μ), L(μ)) with m = 2 and different n.

n	1	3	6	9	30	40
D(S̄, L)	1.2086	1.1434	1.1076	1.0915	1.0635	1.0608

Figure 6 shows the graphs of the truncated triangular Meyer-König-Zeller surfaces with (m = 3, n = 1)(a) and (m = 3, n = 3)(b). Figure 7 shows the graphs of the truncated triangular Meyer-König-Zeller surfaces with (m = 4, n = 1)(a) and (m = 4, n = 3)(b).

$Figure 6 The truncated triangular Meyer-König-Zeller surfaces with (m = 3, n = 1)(a) and (m = 3, n = 3)(b).$

Figure 6

The truncated triangular Meyer-König-Zeller surfaces with (m = 3, n = 1)(a) and (m = 3, n = 3)(b).

$Figure 7 The truncated triangular Meyer-König-Zeller surfaces with (m = 4, n = 1)(a) and (m = 4, n = 3)(b).$

Figure 7

The truncated triangular Meyer-König-Zeller surfaces with (m = 4, n = 1)(a) and (m = 4, n = 3)(b).

4.2 Redistributed Triangular Meyer-König-Zeller Surface

We construct a kind of triangular Meyer-König-Zeller surface by redistributing redundant basis functions.

Definition 4.2

For given weight sequence {ωm,l}l=0m,ωm,l≥0 and ∑l=0mωm,l=1, we derive a triangular Meyer-König-Zeller surface

S^(μ)=∑k=0m∑l=0kPn,k,l(μ)Vk,l+∑l=0mωm,lP¯n,m(μ)Vm,l. (21)

called a redistributed triangular Meyer-König-Zeller surface.

It is obvious that

limμ1→0S(μ)=∑l=0mωm,llimμ1→0P¯n,m(μ)Vm,l=∑l=0mωm,lVm,l.

Thus, let Ŝ(0, μ₂, μ₃) = ∑l=0mωm,lVm,l. Figure 8 shows the graphs of the surface Ŝ(μ) with m = 2, n = 3 and (ω_2,l = 1/3, l = 0, 1, 2)(a), (ω_2,0 = ω_2,2 = 1/2, ω_2,1 = 0)(b), (ω_2,0 = 0.7, ω_2,1 = 0.2, ω_2,2 = 0.1)(c), (ω_2,0 = 0.1, ω_2,1 = 0.2, ω_2,2 = 0.7)(d).

$Figure 8 The graphs of redistributed triangular Meyer-König-Zeller surfaces with m = 2, n = 3 and different {ω2,l}.$

Figure 8

The graphs of redistributed triangular Meyer-König-Zeller surfaces with m = 2, n = 3 and different {ω_2,l}.

Theorem 4.1

Redistributed triangular Meyer-König-Zeller surfaces retain the properties of triangular Meyer-König-Zeller surfaces.

Affine Invariance.
Convex Hull Property.
Interpolative control vertex V_0,0.
Non-Degenerate.
The boundary curves are rational Bézier curves in terms of the Bernstein basis functions of negative degree.

Proof

Let 𝓐 be an affine transformation operator, then

A(S^(μ))=A∑k=0m∑l=0kPn,k,l(μ)Vk,l+∑l=0mωm,lP¯n,m(μ)Vm,l.

Since

∑k=0m∑l=0kPn,k,l(μ)+∑l=0mωm,lP¯n,m(μ)=1, (22)

A(S^(μ))=∑k=0m∑l=0kPn,k,l(μ)AVk,l+∑l=0mωm,lP¯n,m(μ)AVm,l. (23)

Thus, the redistributed triangular Meyer-König-Zeller surface is affine invariant.
It is obvious that ω_m,lP̄_n,m(μ) (l = 0, …, m) are non-negative. Moreover, according to equation (22), the redistributed triangular Meyer-König-Zeller surface that we obtain is included within the convex hull of the control mesh which is constituted by {V_k,l},(k = 0, …, m; l = 0, …, k).
It is obvious that Ŝ(1, 0, 0) = V_0,0, i.e. the redistributed triangular Meyer-König-Zeller surface interpolates the corner control vertex.
Suppose that Ŝ(μ) collapses to a vertex Q ∈ R³, then

0=S^(μ)−Q=∑k=0m∑l=0kPn,k,l(μ)(Vk,l−Q)+∑l=0mωm,lP¯n,m(μ)(Vm,l−Q).

For any v ∈ R³, we have

∑k=0m∑l=0kPn,k,l(μ)(Vk,l−Q)∙v+∑l=0mωm,lP¯(μ)(Vm,l−Q)∙v=0.

By linear independence of triangular Meyer-König-Zeller basis functions, we get

(Vk,l−Q)∙v=0,(k=0,…,m;l=0,…,k).

Hence, a redistributed triangular Meyer-König-Zeller surface is non-degenerate.
For boundary curves of a redistributed triangular Meyer-König-Zeller surface Ŝ(μ) are Ŝ(μ₁, 0, μ₃) and Ŝ(μ₁, μ₂, 0). By the boundary property of triangular Meyer-König-Zeller basis functions, we get

S^(μ1,0,μ3)=∑k=0mPn,k,0(μ1,0,μ3)Vk,0+∑l=0mωm,lP¯n,m(μ1,0,μ3)Vm,l=∑k=0mPn,k,0(μ1,0,μ3)Vk,0+∑l=0mωm,lVm,l1−∑k=0mPn,k,0(μ1,0,μ3), (24)

which is a rational Bézier curve in terms of the Bernstein basis functions of negative degree. The control vertices of Ŝ(μ₁, 0, μ₃) are {Vk,0}k=0∞. For all k ≥ m + 1, let V_k,0 = ∑l=0mωm,lVm,l. Similarly,

S^(μ1,μ2,0)=∑k=0mPn,k,k(μ1,μ2,0)Vk,k+∑l=0mωm,lVm,l1−∑k=0mPn,k,k(μ1,μ2,0)

is a rational Bézier curve in terms of the Bernstein basis functions of negative degree with control vertices {Vk,k}k=0∞. For all k ≥ m + 1, let Vk,k=∑l=0mωm,lVm,l.

□

Furthermore, we can let {ωm,l}l=0m be a series of basis functions with variable μ₂. Figure 9 shows the graph of a redistributed triangular Meyer-König-Zeller surface with ω_2,l(μ₂) = 2lμ2l(1−μ2)2−l which are the Bernstein basis functions of degree 2. It is obvious that

S^(0,μ2,μ3)=∑l=0mωm,l(μ2)P¯n,m(0,μ2,μ3)Vm,l=∑l=0mωm,l(μ2)Vm,l (25)

$Figure 9 The graph of a redistributed triangular Meyer-König-Zeller surface, where m = 2, n = 3 and ω2,l are the Bernstein basis functions of degree 2 with variable μ2.$
Figure 9
The graph of a redistributed triangular Meyer-König-Zeller surface, where m = 2, n = 3 and ω_2,l are the Bernstein basis functions of degree 2 with variable μ₂.

is a Bézier curve.

Remark 4.2

When {ωm,l}l=0m is a series of Bernstein basis functions, i.e. ω_m,l = mlμ2l(1−μ2)m−l, it is obvious that Ŝ(0, 0, 1) = V_m,0, Ŝ(0, 1, 0) = V_m,m.

5 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 affine invariance, possess the convex hull property, are non-degenerate, have interpolative control vertex V_0,0 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 finite, 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 = {0, 0, 0}, k > m, 0 ≤ 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.

Acknowledgement

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11626201, 11601266).

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Received: 2018-07-15

Accepted: 2019-02-05

Published Online: 2019-04-29

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0021

Keywords for this article

Surface modeling; Meyer-König-Zeller operator; basis function; triangular surface

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