Bivariate fractal interpolation functions on triangular domain for numerical integration and approximation

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Bivariate fractal interpolation functions on triangular domain for

numerical integration and approximation

Aparna M.Pa,1,P. Paramanathanb,∗,2

aDepartment of Mathematics, Amrita School of Physical Sciences, Coimbatore, Amrita Vishwa Vidyapeetham, India.

bDepartment of Mathematics, Amrita School of Physical Sciences, Coimbatore, Amrita Vishwa Vidyapeetham, India.

ARTICLE INFO

Keywords:

Bivariate fractal interpolation func-

tion (BFIF)

Chromatic number

Double integration

ABSTRACT

The primary objectives of this paper are to present the construction of bivariate fractal interpo-

lation functions over triangular interpolating domain using the concept of vertex coloring and

to propose a double integration formula for the constructed interpolation functions. Unlike the

conventional constructions, each vertex in the partition of the triangular region has been assigned

a color such that the chromatic number of the partition is 3. A new method for the partitioning

of the triangle is proposed with a result concerning the chromatic number of its graph. Fol-

lowing the construction, a formula determining the vertical scaling factor is provided. With the

newly deﬁned vertical scaling factor, it is clearly observed that the value of the double integral

coincides with the integral value calculated using fractal theory. Further, a relation connecting

the fractal interpolation function with the equation of the plane passing through the vertices of

the triangle is established. Convergence of the proposed method to the actual integral value is

proven with suﬃcient lemmas and theorems. Suﬃcient examples are also provided to illustrate

the method of construction and to verify the formula of double integration.

1. Introduction

Fractal interpolation functions (FIF) are constructed using iterated function system (IFS)[1]. The IFS consists of a

complete metric space together with a ﬁnite set of contraction mappings [11]. Normally, each contraction map in the

IFS is composed of two types of functions, the former one responsible for the contraction of the entire interpolating

domain and the latter one for deﬁning the Read-Bejraktarevic operator whose ﬁxed point is the required fractal inter-

polation function. The fundamental problem in the theory of fractal interpolation is to establish the well deﬁniteness of

this operator. In [2], L. Dalla imposed a restrictive condition on the interpolation points for tackling this problem, when

the interpolating domain is a rectangle. For the same purpose, a piecewise function, deﬁned in terms of the usual IFS,

is proposed in [3]. For the triangular interpolating domain, the problem of well deﬁniteness was dealt by Geronimo

and Hardin and they proposed the method of vertex coloring to solve the problem [4]. The present paper further con-

nects this approach to deﬁne bivariate fractal interpolation functions and to derive double integration for such functions.

Barnsley in [5] introduced the idea of fractal interpolation functions using iterated function systems for the ﬁrst

time where single variable interpolation functions were generated as the attractors of the IFS. The construction of

bivariate fractal interpolation functions considered by L Dalla required the interpolation points on the boundary of

the rectangle to be collinear [2]. Malysz used the fold-out technique for constructing bivariate fractal interpolation

function over rectangles by taking the same vertical scaling factor [6]. In [7], considering the vertical scaling factor as

a function, Metzler and Yun generalised this construction. In [8], Massopust constructed bivariate fractal interpolation

functions on a triangular region with a restrictive condition on the interpolation points. The construction proposed

in his work was further modiﬁed in [4]. [4] introduces the idea of the coloring of the vertices to solve the problem

of well deﬁniteness. The construction of the bivariate fractal interpolation function is again considered in [9]. The

paper, however, fails to establish the well deﬁniteness of the fractal interpolation operator. The numerical integration

of fractal interpolation functions was ﬁrst carried out by Navascues in [10]. The derived formula for integration is then

compared with the compound trapezoidal rule there.

∗Corresponding author

ORCID(s): 0000-0003-0688-4858 (P. Paramanathan)

1mp_aparna@cb.students.amrita.edu (Aparna M.P)

2p_paramanathan@cb.amrita.edu (P. Paramanathan)

First Author et al.: Preprint submitted to Elsevier Page 1 of 14

arXiv:2210.06435v1 [math.GM] 8 Aug 2022

Short Title of the Article

The present paper aims to deﬁne double integration for two variable fractal interpolation functions constructed over

a triangular domain. By proposing a method for the partition of the triangle and introducing a new vertical scaling

factor, this paper provides a detailed explanation for the construction of these functions. Following the derivation of

double integration, the constructed fractal interpolation function is approximated to the equation of the plane passing

through the vertices of the triangle. After proving the theorems in error analysis using this approximation, the paper

shows the attractors of the IFS’s and the double integral values obtained for some functions.

The organization of the paper is as follows: The second section presents the formulation of the IFS and proves

the corresponding theorems with the results concerning the partitioning of the triangle and its chromatic number. The

derivation of the double integration formula is proposed in the third section. In the fourth section, it is established that

the bivariate fractal interpolation functions deﬁned over triangular regions can be approximated to the equation of the

plane passing through the vertices of the triangle. The ﬁfth section provides the formula for the vertical scaling factor

and its upper bound. Using, the newly obtained approximating function, the propositions and theorems are proved for

the error analysis. The paper concludes by displaying the tables and graphs describing the results.

2. Construction of bivariate fractal interpolation function over triangular regions

2.1. Method of Partition

Consider the triangle 𝐷with vertices 𝐴(𝑥1, 𝑦1), 𝐵(𝑥2, 𝑦2), 𝐶(𝑥3, 𝑦3).The algorithm for the partition is as follows.

1. Divide the height of the triangle 𝐷into ′𝑑′number of equal parts, thereby getting ′𝑑+1′new points (𝑥𝑖, 𝑦1),(𝑥𝑖, 𝑦2), ..., (𝑥𝑖, 𝑦𝑑+1)

along the height of 𝐷, where 𝑦1=𝑦coordinate of the point 𝐴or 𝐵, 𝑦𝑑+1 =𝑦coordinate of the point 𝐶.

2. Draw lines 𝑦=𝑦𝑗, 𝑗 = 1,2, ..., 𝑑 parallel to 𝑋−axis from 𝐴𝐶 to 𝐵𝐶.

3. Divide each of the horizontal lines 𝑦=𝑦𝑗, 𝑗 = 1,2, ..., 𝑑 into ′𝑑′number of equal parts, generating ′𝑑+ 1′

new points denoted by (𝑥1, 𝑦𝑗),(𝑥2, 𝑦𝑗), ...(𝑥𝑑+1, 𝑦𝑗)along each horizontal line 𝑦=𝑦𝑗for 𝑗= 1,2, ..., 𝑑 where

(𝑥1, 𝑦𝑗),(𝑥𝑑+1, 𝑦𝑗)lies on the sides 𝐴𝐶 and 𝐵𝐶 respectively.

4. Then, join the new points as shown in Figure 1.

Figure 1: Partition of 𝐷when 𝑑= 4

In Figure 1, red corresponds to color 1, blue stands for color 2, and green for color 3.

First Author et al.: Preprint submitted to Elsevier Page 2 of 14

Short Title of the Article

Lemma 2.1. For the partition deﬁned above, if the height of the triangle 𝐷is divided into 𝑑number of equal parts,

then, there are

i) 2𝑑2− 2𝑑+ 3 subtriangles and

ii) 𝑑2+𝑑+ 1 vertices,

in the partition.

Proof. i) According to the partition deﬁned, the height of the triangle is divided into 𝑑number of equal parts,

resulting 𝑑+ 1 ′𝑦′values, where 𝑦𝑑+1 is the 𝑦coordinate of the top vertex of 𝐷. Along each horizontal line

𝑦=𝑦𝑗, 𝑗 = 1,2, ..., 𝑑 the line is divided into 𝑑number of equal parts. The coordinates of the newly obtained

points along the line 𝑦=𝑦𝑗are denoted by (𝑥1, 𝑦𝑗),(𝑥2, 𝑦𝑗), ..., (𝑥𝑑+1, 𝑦𝑗).It is observed from the partition that

between two consecutive points on the line 𝑦=𝑦𝑗,two subtriangles are obtained (a normal and an inverted

triangle). Hence, along each line 𝑦=𝑦𝑗,there are 2𝑑subtriangles for 𝑗= 2, ..., 𝑑. Considering the top most three

subtriangles, which are ﬁxed irrespective of 𝑑, there are (2𝑑)(𝑑− 1) + 3 subtriangles in the partition. 𝑖.𝑒, the total

number of subtriangles in the partition is 2𝑑2− 2𝑑+ 3.Hence the proof.

ii) Following the partition, since each horizontal line 𝑦=𝑦𝑗is divided into 𝑑equal parts, there are 𝑑+ 1 new

vertices along each line 𝑦=𝑦𝑗, 𝑗 = 1,2, ..., 𝑑. Now, including the top most vertex 𝐶, of the triangle 𝐷, there are

[(𝑑+ 1)𝑑] + 1 = 𝑑2+𝑑+ 1 vertices in the partition. Hence the proof.

Lemma 2.2. If 𝑑= 3𝑛+ 1, 𝑛 ∈𝑁, then the graph of the partition deﬁned above has chromatic number 3.

Proof. According to the partition, each horizontal line 𝑦=𝑦𝑗is divided into 𝑑equal parts, generating 𝑑+ 1 points

along that line. Let the points be denoted by (𝑥1, 𝑦𝑗),(𝑥2, 𝑦𝑗), ..., (𝑥𝑑+1, 𝑦𝑗).Among these points, (𝑥1, 𝑦𝑗)and (𝑥𝑑+1, 𝑦𝑗)

are points on the two sides of the triangle 𝐷. The remaining points are intermediate points and there 𝑑− 1 such points

along that line, where 𝑑− 1 is a multiple of 3. Now, considering the coloring of theses points with the least number

of colors, since the points (𝑥1, 𝑦1)and (𝑥𝑑+1, 𝑦1)are adjacent with respect to the triangle 𝐷, they have to be colored

diﬀerently. Without loss of generality, let (𝑥1, 𝑦1)and (𝑥𝑑+1, 𝑦1)be colored with colors ’1’ and ’2’ respectively. Now,

since the point (𝑥2, 𝑦1)is adjacent to (𝑥1, 𝑦1),it should be colored ’2.’ Similarly, (𝑥3, 𝑦1)is colored ’1’. Proceeding

in this manner, the point (𝑥𝑑−1, 𝑦1)will be colored ’1.’ Then, the point (𝑥𝑑, 𝑦1)has to colored with a diﬀerent color

other than ’1’ and ’2’, since it is adjacent to both (𝑥𝑑−1, 𝑦1)and (𝑥𝑑+1, 𝑦1).Hence, a minimum of 3 colors are needed

to color the points along this line. The same reasoning can be applied along each of the horizontal lines 𝑦=𝑦𝑗and

along the slanting sides of 𝐷. Thus, it can be established that atleast 3 diﬀerent colors are needed to properly color the

graph. Therefore, the chromatic number of the graph is 3.

2.2. Construction

Consider a triangular domain 𝐷with vertices (𝑥1, 𝑦1),(𝑥2, 𝑦2),(𝑥3, 𝑦3),colored ’1’, ’2’ and ’3’ respectively. Let

the triangle be partitioned into 𝑁number of subtriangles 𝐷1, 𝐷2, ..., 𝐷𝑁such that 𝐷= ∪𝑁

𝑛=1𝐷𝑛and ∩𝑁

𝑛=1𝐷𝑛=𝜙. The

partitioning is done such that the chromatic number of their corresponding graph is 3. Each subtriangle be numbered

from 1 to 𝑁as shown in Figure 1. Set 𝑃= {(𝑥𝑛𝑗 , 𝑦𝑛𝑗 ) ∶ 𝑗= 1,2,3, 𝑛 = 1,2, ..., 𝑁}to be the set of all vertices of the

subtriangles 𝐷𝑛, 𝑛 = 1,2, ..., 𝑁 . Let 𝑧𝑛𝑗 =𝑓(𝑥𝑛𝑗 , 𝑦𝑛𝑗 )be the corresponding function values.

Let 𝑅= {(𝑥𝑛𝑗 , 𝑦𝑛𝑗 , 𝑧𝑛𝑗 ) ∶ 𝑗= 1,2,3, 𝑛 = 1,2, ..., 𝑁}be the data set. Without loss of generality, let (𝑥𝑛1, 𝑦𝑛1)denotes

the vertex colored ’1’, (𝑥𝑛2, 𝑦𝑛2)be the vertex with color ’2’, and (𝑥𝑛3, 𝑦𝑛3)be the vertex colored ’3’ in 𝐷𝑛.

Consider an invertible, aﬃne map 𝐿𝑛∶𝐷→𝐷𝑛such that

𝐿𝑛(𝑥𝑗, 𝑦𝑗)=(𝑥𝑛𝑗 , 𝑦𝑛𝑗 ),for 𝑗= 1,2,3.(1)

𝑖.𝑒, 𝐿𝑛maps (𝑥𝑗, 𝑦𝑗)to the vertex in 𝐷𝑛, which is colored 𝑗, for 𝑗= 1,2,3.The map 𝐿𝑛used here is given by,

𝐿𝑛(𝑥, 𝑦) = [𝛼𝑛1𝛼𝑛2

𝛼𝑛3𝛼𝑛4][𝑥

𝑦]+[𝛽𝑛1

𝛽𝑛2](2)

First Author et al.: Preprint submitted to Elsevier Page 3 of 14

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BivariatefractalinterpolationfunctionsontriangulardomainfornumericalintegrationandapproximationAparnaM.Pa,1,P.Paramanathanb,<,2aDepartmentofMathematics,AmritaSchoolofPhysicalSciences,Coimbatore,AmritaVishwaVidyapeetham,India.bDepartmentofMathematics,AmritaSchoolofPhysicalSciences,Coimbatore,AmritaVi...

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Bivariate fractal interpolation functions on triangular domain for numerical integration and approximation

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