Fractals, Vol. 5, No. 4 (1997) 625–634c© World Scientific Publishing Company

THE STUDY ON BIVARIATE FRACTALINTERPOLATION FUNCTIONS AND CREATION

OF FRACTAL INTERPOLATED SURFACES

HEPING XIE and HONGQUAN SUNBeijing Graduate School, China University of Mining and Technology,

Beijing, 100083, P. R . China

Received July 1, 1997; Accepted August 21, 1997

Abstract

In this paper, the methods of construction of a fractal surface are introduced, the principleof bivariate fractal interpolation functions is discussed. The theorem of the uniqueness of aniterated function system of bivariate fractal interpolation functions is proved. Moreover, thetheorem of fractal dimension of fractal interpolated surface is derived. Based on these theorems,the fractal interpolated surfaces are created by using practical data.

1. INTRODUCTION

Fractal interpolation was first put forward by anAmerican mathematician, M. F. Barnsley,1 in 1986.It gives a new methodology for data fitting, whichnot only opens up a new research field for functionapproaching theory, but also provides powerful toolsfor computer graphics. This tool’s applicability isnow fully appreciated.

The use of linear functions, polynomial functionsand surface spring functions to establish variousmaterial object models in real life from traditionalEuclidean geometry is now common practice. Theavailability of a set of good theories and methods to

interpolate curves and surfaces in Euclidean spaceis the basis for new advances. The fractal inter-polated curves have been discussed in detail in theliterature.1,2,4,10

Fractal surfaces are found in great abundance innature. The surface shapes of mountains, topogra-phies, rocks, clouds and fractures are all real casesof fractal surfaces. Methods for constructing frac-tal surfaces which show beautiful pictures and vividlandscapes are found in literature (Song and Yang,5

Xie,6 Mandelbrot,7 Feder8 and Qi9). This is a usefultool for showing graphs of fractal bodies on a com-puter directly. However, with this method we can-not obtain the fractal surfaces which pass through

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the given data points. So the fractal surfaces con-structed with random methods cannot meet theneeds of practical work. Sometimes we know someof the data points on the surface of a fractal body(for example, profile of fracture and some data onfracture surface) and we want to fit a fractal surfaceto these data points so that we can study the frac-tal body as a whole. For this, the method of fractalinterpolated surface must be used. In this paper wewill discuss the principles and mathematical mod-els of bivariate fractal interpolation functions, provethe theorem of the uniqueness of an iterated func-tion system of fractal interpolated surface and thetheorem of fractal dimensions of fractal interpolatedsurface, and give an example that a set of practi-cal data are used to interpolate a fractal surface, sothat we can study the fractal features of materialfracture patterns and landscapes effectively.

2. CONSTRUCTION OF FRACTALSURFACES

2.1 Koch Fractal Surface

The literature (Xie6) gives the method for con-structing a Koch fractal surface (Fig. 1): translatinga Koch fractal curve along a direction perpendicularto the plane of the vertical section (or the profile)for a distance l0, we obtain then an irregular sur-face which we call a Koch fractal surface, as shownin Fig. 1. In order to measure the area of such asurface, we cover it with rectangles of length l0 andwidth δ. With the grids of small yardstick areasα = δ ,2 we need

N(δ) =l0δ× 1

δD(1)

small squares to cover the surface. Where D is afractal dimension of the Koch fractal curve givenby D = lg4 / lg3 . The first factor in Eq. (1) is thenumber of small squares with the side δ along thedirection of l0, and the second indicates the num-ber of the intervals with length δ covering the Koch

Fig. 1 Koch fractal surface.

fractal curve. Let D̃ be the fractal dimension of aKoch fractal surface, according to B. B. Mandel-brot’s theory, then

D̃ = 1 +D = 2.2618 (2)

2.2 Brownian Fractal Surface

For 0 < H < 1, we define an index-H Brownianfunction ZH(x, y): R2 → R to be a random func-tion. Such that:

(1) P{ZH(0, 0) = 0} = 1, and ZH(x, y) is a con-tinuous function of (x, y);

(2) For (x, y), (∆x,∆y) ∈ R2, the height incre-ments ∆ZH(∆x,∆y) = ZH(x+ ∆x, y + ∆y)−ZH(x, y) have the normal distribution with zeromean and variance |∆x2 + ∆y2|H .

Figure 2 is the Brownian fractal surface made bya computer (Xie6). In this figure, we find that tracesof Fractional Brownian Motion (FBM) are similarto real mountains, so we could use the traces ofFBM to construct fractal landscapes. The varietyof height and direction of fractal landscapes is animportant foundation in estimating the height.

Fig. 2 Brownian fractal surface.

2.3 Construction of a Fractal Surfacewith Random Grids

The literature (Hu et al4) gives a method for con-structing fractal surfaces with random grids. Thismethod uses a rectangle as the generator of the frac-tal surface. We obtain the displacements in the

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Fig. 3 Construction of a fractal surface with random grids.

middle of the four sides of the rectangle and in thecenter of the rectangle by interpolating the givendata on the vertices linearly and adding randomdisturbers. We divide then the rectangle into foursubrectangles equally. Iterating the process we havethe steps of this algorithm as follows (Hu et al4)(Fig. 3):

(1) giving initial values (height) on the vertices ofthe rectangle;

(2) finding the displacements in the middle of thefour sides of the rectangle by linear interpola-tion;

(3) calculating new displacements by adding ran-dom disturbers to the displacements (up if thedisplacement is positive; down if the displace-ment is negative);

(4) finding the displacements in the center of therectangle by linear interpolation. This is theheight of the center of the rectangular field.

We divide the rectangle into four small equalrectangles and iterate the steps from (1) to (4)above, we obtain then the displacement (height) ofthe center of each rectangle. This is one-level re-cursive. Using the analogous method, we obtain arecursive process and an algorithm: level 1, level2, . . . , and level n.

3. ESTABLISHING OF THEFRACTAL INTERPOLATEDSURFACES

The ways of interpolating fractal surfaces have beendiscussed in the literature (Qi9). We here discussthe principles and the methods of fractal interpo-lated surfaces in detail.

Let I = [a, b], J = [c, d], andD = I×J = {(x, y) :a ≤ x ≤ b, c ≤ y ≤ d}. We divide D into grids withthe steps ∆x and ∆y such that

a = x0 < x1 < · · · < xN = b ,

c = y0 < y1 < · · · < yM = d .(3)

With a data set {(xn, ym, zn,m), n = 0, 1, 2, . . . ,N, m = 0, 1, 2, . . . ,M} on the grids, we con-struct an interpolation function f : D → R, suchthat f(xn, ym) = zn,m, n = 0, 1, 2, . . . ,N, m =0, 1, 2, . . . ,M .

3.1 Principles of Bivariate FractalInterpolation Functions

We will restrict our attention to the field K =D × [h1, h2](− ∞ < h1 < h2 < + ∞). Letd((c1, d1, e1), (c2, d2, e2)) = max{|c1 − c2|, |d1 −d2|, |e1 − e2|} for (c1, d1, e1), (c2, d2, e2) ∈ K.

Let In = [xn−1, xn], Jm = [ym−1, ym],Dn,m =In × Jm, n ∈ {1, 2, . . . ,N}, m ∈ {1, 2, . . . ,M}.Also let φn : I → In, ψm : J → Jm be contrac-tion mapping, so that:

φn(x0) = xn−1, φn(xN ) = xn

ψm(y0) = ym−1, ψm(yM ) = ym

|φn(c1)− φn(c2)| < k1|c1 − c2||ψm(d1)− ψm(d2)| < k2|d1 − d2| .

(4)

where c1, c2 ∈ I, d1, d2 ∈ J , 0 ≤ k1 < 1, 0 ≤ k2 ≤ 1.

Let Ln,m : D → R2 be a contraction transfor-mation: Ln,m(x, y) = (φn(x), ψm(y)). Let Fn,m :K → [h1, h2] be continuous, which must obey fourequations:

Fn,m(x0, y0, z0,0) = zn−1,m−1

Fn,m(xN , y0, zN,0) = zn,m−1

Fn,m(x0, yM , z0,M ) = zn−1,m

Fn,m(xN , yM , zN,M ) = zn,m .

(5)

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For any (x1, y1;x2, y2) ∈ D and z1, z2 ∈ [h1, h2], we have

|Fn,m(x1, y1, z1)− Fn,m(x2, y2, z2)| ≤ k3|z1 − z2| ,

n ∈ {1, 2, . . . ,N}, m ∈ {1, 2, . . . ,M}, 0 ≤ k3 < 1 .(6)

3.2 Iterated Formulas

Let φn(x) = anx + bn. With the conditions [Eq. 4], we have anx0 + bn = xn−1, anxN + bn = xn andobtain {

an = (xn − xn−1)/(xN − x0)

bn = (xn−1xN − xnx0)/(xN − x0), n ∈ {1, 2, . . . ,N} . (7)

∴ φn(x) = xn−1 +xn − xn−1

xN − x0(x− x0) , n ∈ {1, 2, . . . ,N} . (8)

Let ψm(y) = cmy + dm. Similarly, with the conditions [Eq. 4], we have

{cm = (ym − ym−1)/(yM − y0)

dm = (ym−1yM − ymy0)/(yM − y0), m ∈ {1, 2, . . . ,M} . (9)

∴ ψm(y) = ym−1 +ym − ym−1

yM − y0(y − y0) , m ∈ {1, 2, . . . ,M} . (10)

LetFn,m(x, y, z) = en,mx+ fn,my + gn,mxy + αn,mz + kn,m ,

n ∈ {1, 2, . . . ,N},m ∈ {1, 2, . . . M} .(11)

According to Eq. (5), we have

zn−1,m−1 = en,mx0 + fn,my0 + gn,mx0y0 + αn,mz0,0 + kn,m

zn,m−1 = en,mxN + fn,my0 + gn,mxNy0 + αn,mzN,0 + kn,m

zn−1,m = en,mx0 + fn,myM + gn,mx0yM + αn,mz0,M + kn,m

zn,m = en,mxN + fn,myM + gn,mxNyM + αn,mzN,M + kn,m .

Let αn,m be any real number. We find that we can always solve the above equations for en,m, fn,m, gn,mand kn,m in terms of the interpolation data and αn,m. We obtain

gn,m =zn−1,m−1 − zn−1,m − zn,m−1 + zn,m − αn,m(z0,0 − zN,0 − z0,M + zN,M )

x0y0 − xNy0 − x0yM + xNyM

en,m =zn−1,m−1 − zn,m−1 − αn,m(z0,0 − zN,0)− gn,m(x0y0 − xNy0)

x0 − xNn ∈ {1, 2, . . . ,N} ,

fn,m =zn−1,m−1 − zn−1,m − αn,m(z0,0 − z0,M )− gn,m(x0y0 − x0yM )

y0 − yMm ∈ {1, 2, . . . ,M} .

kn,m = zn,m − en,mxN − fn,myM − αn,mzN,M − gn,mxNyM(12)

Now let 0 ≤ αn,m < 1(n ∈ {1, 2, . . . ,N}, m ∈ {1, 2, . . . ,M}). We call the αn,m a vertical scalingfactor.

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Now we construct a new function Gn,m(x, y, z), such that

Gn,m(x, y, z)=

1

2(Fn,m(x, y, z) + Fn+1,m(x0, y, z)) where x=xN ;n=1, 2, . . . ,N − 1;m=1, 2, . . . ,M .

1

2(Fn,m(x, y, z) + Fn,m+1(x, y0, z)) when x=yM ;n=1, 2, . . . ,N ;m=1, 2, . . . ,M − 1 .

Fn,m(x, y, z) others .

n ∈ {1, 2, . . . ,N} , m ∈ {1, 2, . . . ,M} . (13)

We define an Iterated Function System (IFS)Wn,m(x, y, z) on the field K:

Wn,m(x, y, z) = (φn(x), ψm(y), Gn,m(x, y, z)) ,

n = 1, 2, . . . ,N ; m = 1, 2, . . . ,M .(14)

We will prove that for such defined IFS, we havea unique attractor G which is the graph of a con-tinuous function f , such that

G = {(x, y, f(x, y)) : (x, y) ∈ D} (15)

andf(xn, ym) = zn,m;n = 0, 1, . . . ,N,

m = 0, 1, . . . ,M .(16)

4. THEOREM OF THE EXISTENCEAND UNIQUENESS OF AFRACTAL INTERPOLATEDSURFACE

Theorem 1 Let N and M be positive integers greaterthan one. Let {R3;Wn,m, n = 1, . . . ,N, m =1, . . . ,M} denote the IFS defined above, asso-ciated with the data set {(xn, ym, zn,m), n =

0, 1, 2, . . . ,N,m = 0, 1, 2, . . . ,M}. Let the verti-cal scaling factor αn,m obey 0 ≤ αn,m < 1 forn = 0, 1, . . . ,N,m = 0, 1, . . . ,M . There is thena metric ρ on R3, equivalent to the Euclideanmetric, such that the IFS is a contraction map-ping with respect to ρ. In particular, there is aunique nonempty compact set G ⊂ R3 such thatG =

⋃Nn=1

⋃Mm=1wn,m(G).

Proof. We define a metric ρ on R3 by

ρ((x1, y1, z1), (x2, y2, z2)) = |x1 − x2|+ |y1 − y2|

+ θ|z1 − z2| ,

where θ is a positive real number which we spec-ify in the process of the proof below. It is equiv-alent to the Euclidean metric on R3. Let n ∈{1, 2, . . . ,N},m ∈ {1, 2, . . . ,M}. For

Wn,m(x, y, z) = (φn(x), ψm(y), Gn,m(x, y, z)) ,

where φn(x) = anx + bn, ψm(y) = cmy + dm, andGn,m(x, y, z) is defined by Eq. (13).

For simplicity we restrict attention to the “oth-ers” case in Eq. (13). Then we have

ρ(Wn,m(x1, y1, z1),Wn,m(x2, y2, z2))

= ρ((φn(x1), ψm(y1), Gn,m(x1, y1, z1)), (φn(x2), ψm(y2), Gn,m(x2, y2, z2)))

= ρ((φn(x1), ψm(y1), Fn,m(x1, y1, z1)), (φn(x2), ψm(y2), Fn,m(x2, y2, z2)))

= ρ((anx1+bn, cmy1+dm, en,mx1+fn,my1+gn,mx1y1+αn,mz1,1+kn,m),

(anx2+bn, cmy2+dm, en,mx2+fn,my2+gn,mx2y2+αn,mz2,2+kn,m))

= |an||x1−x2|+|cm||y1−y2|+θ|en,m(x1−x2)+fn,m(y1−y2)+gn,m(x1y1−x2y2)+(z1 − z2)|

≤ (|an|+θ|en,m|)|x1−x2|+(|cm|+θ|fn,m|)|y1−y2|+θ|gn,m||x1(y1−y2)+y2(x1−x2)|+θ|αn,m||z1−z2|

≤ (|an|+θ(|en,m|+|gn,myM |))|x1−x2|+(|cm|+θ(|fn,m|+|gn,mxN |))|y1−y2|+θ|αn,m||z1−z2| .

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Now notice that |an| = |xn−xn−1|/|xN −x0| < 1and |cm| = |ym−ym−1|/|yM−y0| < 1 becauseN ≥ 2and M ≥ 2. We choose θ = min{θ1, θ2}, where

θ1 =min

1≤n≤N{1− |an|}

max1≤n≤N,1≤m≤M

{2(|en,m|+ gn,myM |)},

θ2 =min

1≤m≤M{1− |cm|}

max1≤n≤N,1≤m≤M

{2(|fn,m|+ gn,mxN |)}.

Let

a=1

2+

max1≤n≤N,1≤m≤M

{|an|, |cm|}

2< 1 .

∵ |an| < 1, |cm| < 1, (1 ≤ n ≤ N, 1 ≤ m ≤M)

δ= max1≤n≤N,1≤m≤M

{|αn,m|} < 1, α=max{a, δ} < 1 .

We have:

ρ(Wn,m(x1, y1, z1),Wn,m(x2, y2, z2))

≤ a|x1 − x2|+ a|y1 − y2|+ θδ|z1 − z2|

≤ α(|x1 − x2|+ |y1 − y2|+ θ|z1 − z2|)

= αρ((x1, y1, z1), (x2, y2, z2))

For the other two cases (x = xN ;n =1, 2, . . . ,N − 1;m = 1, 2, . . . ,M and y = yM ;n =1, 2, . . . ,N ;m = 1, 2, . . . ,M − 1) in Eq. (13), withthe similar proving method, we can obtain the sameconclusion.

So Wn,m (x, y, z) (n = 1, 2, . . . ,N ; m = 1,2, . . . ,M) is a contraction mapping.

Because Wn,m(x, y, z) is a contraction mapping,according to the IFS theory, there is a uniquenonempty compact set G ⊂ R3 such that G =⋃Nn=1

⋃Mm=1wn,m(G). This completes the proof.

Theorem 2 Let N and M be positive integersgreater than one. Let {R3;Wn,m, n = 1, 2, . . . ,N,m = 1, 2, . . . ,M} denote the IFS defined above,associated with the data set {(xn, ym, zn,m), n =0, 1, 2, . . . ,N,m = 0, 1, 2, . . . ,M}. Let the verti-cal scaling factor αn,m obey 0 ≤ αn,m < 1 forn = 1, 2, . . . ,N,m = 1, 2, . . . ,M, so that the IFS iscontraction mapping. Let G denote the attractor ofthe IFS. Then G is the graph of a continuous func-tion f : [x0, xN ] × [y0, yM ] → R which interpolates

the data {(xn, ym, zn,m), n = 0, 1, 2, . . . ,N,m =0, 1, 2, . . . ,M}. That is

G = {(x, y, f(x, y)); (x, y) ∈ [x0, xN ]× [y0, yM ]} .

Proof. Let F denote the set of continuousfunctions f : [x0, xN ] × [y0, yM ] → R such thatf(x0, y0) = z0,0, f(x0, yM ) = z0,M , f(xN , y0) = zN,0and f(xN , yM ) = zN,M . We define a metric ρ on F

by

ρ(f, g) = max{|f(x, y)− g(x, y)|} ,

(x, y) ∈ [x0, xN ]× [y0, yM ] for all f, g ∈ F .

Then (F, ρ) is a complete metric space.For simplicity we here also restrict attention

to the “others” case in the Eq. (13). So thatGn,m(x, y, z) = Fn,m(x, y, z).

Let the real numbers an, bn, cm, dm, en,m, fn,m,

gn,m and kn,m be defined by Eqs. (7), (9), and (12).Define a mapping T : F → F on the metric space(F, ρ) by

(Tf)(x, y) = en,mφ−1n (x) + fn,mψ

−1m (y)

+ gn,mφ−1n (x)ψ−1

m (y) + αn,mz + kn,m

∀(x, y) ∈ [x0, xN ]× [y0, yM ] ,

where φn(x) : [x0, xN ] → [xn−1, xn] and ψm(y) :[y0, yM ] → [ym−1, ym] are the invertible transfor-mations. See Eqs. (8) and (10).

(I) We verify that T does indeed take F intoitself.

Let f ∈ F . Then the function (Tf)(x, y) obeysendpoint conditions because

(Tf)(x0, y0) = e1,1φ−11 (x0)+f1,1ψ

−11 (y0)

+ g1,1φ−11 (x0)ψ−1

1 (y0)

+ α1,1f(φ−11 (x0), ψ−1

1 (y0))+k1,1

= e1,1x0+f1,1y0+g1,1x0y0

+ α1,1f(x0, y0)+k1,1

= e1,1x0+f1,1y0+g1,1x0y0+α1,1z0,0

+k1,1 = z0,0 .

Similarly, we can verify that (Tf)(x0, yM ) =z0,M , (Tf)(xN , y0) = zN,0, (Tf)(xN , yM) = zN,Mand (Tf)(x, y) is continuous on the field [xn−1, xn]×[ym−1, ym]. Then it remains to be demonstrated

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The Study on Bivariate Fractal Interpolation Functions and Creation of. . . 631

that (Tf)(x, y) is continuous at each of the points.So we conclude that T does indeed take F to F .

(II) Now we show that T is a contraction map-ping on the metric space (F, ρ).

Let f, g ∈ F . Let (x, y) ∈ [xn−1, xn] ×[ym−1, ym](n = 1, 2, . . . ,N,m = 1, 2, . . . ,M). Then

ρ(Tf, Tg) = |(Tf)(x, y)− (Tg)(x, y)|

= |αn,m‖f(φ−1n (x), ψ−1

m (y))

− g(φ−1n (x), ψ−1

m (y))|

≤ |αn,m|ρ(f, g) .

Let δ = max1≤n≤N,1≤m≤M

{|αn,m|} < 1, it follows that

ρ(Tf, Tg) ≤ δρ(f, g) .

We conclude that T : F → F is a contraction map-ping. The Contraction Mapping implies that T pos-sesses a unique fixed point in F . That is, thereexists a function f ∈ F such that

(Tf)(x, y) = f(x, y) for all (x, y)

∈ [x0, xN ]× [y0, yM ] .

If f ∈ F , then f is a function which passesthrough the interpolation points {(xn, ym, zn,m);n = 0, 1, 2, . . . ,N ;m = 0, 1, 2, . . . ,M}. ∀n ∈{1, 2, . . . ,N},m ∈ {1, 2, . . . ,M}, we have

(Tf)(xn−1, ym−1)

= en,mφ−1n (xn−1) + fn,mψ

−1m (ym−1)

+ gn,mφ−1n (xn−1)ψ−1

m (ym−1)

+ αn,mf(φ−1n (xn−1), ψ−1

m (ym−1)) + kn,m

= en,mx0 + fn,my0 + gn,mx0y0

+ αn,mf(x0, y0) + kn,m

= en,mx0 + fn,my0 + gn,mx0y0

+ αn,mz0,0 + kn,m = zn−1,m−1 .

We conclude that f is a function which passesthrough the interpolation points.

Finally, let G̃ denote the graph of f . Notice thatthe equations which define T can be rewritten

(Tf)(anx+ bn, cmy + dm)

= en,mx+ fn,my + gn,mxy + αn,mf(x, y)

+ kn,m, (x, y) ∈ [x0, xN ]× [y0, yM ],

n ∈ {1, 2, . . . ,N},m ∈ {1, 2, . . . ,M} .

So we have

Wn,m

x

y

f(x, y)

=

anx+ bn

cmy + dm

en,mx+ fn,my + gn,mxy + αn,mf(x, y) + kn,m

=

anx+ bn

cmy + dm

(Tf)(anx+ bn, cmy + dm)

=

anx+ bn

cmy + dm

f(anx+ bn, cmy + dm)

.

With (x, y) changing on the field [x0, xN ]×[y0, yM ],the right of this equation will form the graph off(x, y) on the field [x0, xN ]× [y0, yM ]. This impliesthat G̃ =

⋃Nn=1

⋃Mm=1Wn,m(G̃).

Because G̃ is a nonempty compact subset of R3,and for Theorem 1 there is only one nonemptycompact set G, the attractor of the IFS. It fol-lows that G̃ = G. So G is a graph of interpo-lation function f such that G = {(x, y, f(x, y));(x, y) ∈ [x0, xN ] × [y0, yM ]}.This completes theproof.

Definition 1. The function f(x, y) whose graphis the attractor of an IFS as described in The-orem 1 and 2 above, is called a fractal sur-face interpolation function corresponding to thedata {(xn, ym, zn,m);n = 0, 1, 2, . . . ,N,m = 0, 1,2, . . . ,M}.

5. THEOREM OF DIMENSIONSOF FRACTAL INTERPOLATEDSURFACES

Lemma 1 (Feder8) Let S1 and S2 belong respec-tively to E1-space and an E2-space, and denote byS the set in E-space, with E = E1+E2, which is ob-tained as the product of S1 and S2 (If E1 = E2 = 1,S is the set of points (x, y) in the plane, such thatx ∈ S1, y ∈ S2). If S1 and S2 are independent, the

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dimension of S is the sum of the dimension of S1

and S2, i.e.

D(S) = D(S1) +D(S2) (17)

For example, let S1 denote the set of a Koch frac-tal curve. The dimension D(S1) = 1.2618. LetS2 denote the line with a length l in Euclideanspace. The dimension D(S2) = 1. Let D denotethe dimension of a Koch fractal surface. We haveD = D(S1) +D(S2) = 1.2618 + 1 = 2.2618.

According to the method for constructing a Kochfractal surface: translating a Koch fractal curvealong a direction perpendicular to the plane of thevertical section (or the profile) for a distance l0, weknow that “perpendicular direction” and “translat-ing a distance” may be an explanation of the con-dition of “independent” in Lemma 1.

Lemma 2 (Barnsley2) Let m be a positive integer;and consider the metric space (Rm, Euclidean). LetA and B be two compact sets in space Rm. Let Abe such that its fractal dimension is given by

D(A) = limε→0

ln(N(A, ε))

ln(1/ε). (18)

Let D(B) and D(A ∪ B) denote the fractal di-mensions of B and A ∪ B respectively. Supposethat D(B) ≤ D(A). Then D(A ∪B) = D(A).

According to Lemma 1 and Lemma 2, we have

Theorem 3 (Theorem of Fractal Dimensionof a Fractal Interpolated Surface) Let F

be a fractal interpolated surface corresponding tothe data {(xn, ym, zn,m);n = 0, 1, 2, . . . ,N,m =0, 1, 2, . . . ,M}.

Let Px denote a plane vertical to the X axis whichintersects the X axis at point x(x0 ≤ x ≤ xN ). LetPy denote a plane vertical to the Y axis which in-tersects the Y axis at point y(y0 ≤ y ≤ yM). LetSx = F ∩ Px, Sy = F ∩ Py. Then the dimension ofF is

dimH(F ) = D(F )

= maxx0≤x≤xN ,y0≤y≤yM

{D(Sx), D(Sy)}+ 1(19)

Proof. According to the definition of Sx, weknow that Sx is a plane curve. Translating Sx alonga direction perpendicular to the plane Px for a dis-tancex εx, we obtain an irregular surface which is

denoted by Vx. Obviously Vx is a compact set inspace R3. Let D(Sx) and D(Vx) denote the frac-tal dimensions of Sx and Vx, respectively. FromLemma 1 it follows that D(Vx) = D(Sx) + 1.

Because Sy is also a plane curve, with the simi-lar method, we translate Sy along a direction per-pendicular to the plane Py for a distance εy andwe obtain an irregular surface Vy. Let D(Sy) andD(Vy) denote the fractal dimensions of Sy and Vy,respectively. Then we have that D(Vy) = D(Sy)+1.

According to the construction of Vx and Vy , weknow that

F =

⋃x0≤x≤xN

Vx

⋃ ⋃y0≤y≤yM

Vy

. (20)

LetD(S∗x) = max

x0≤x≤xN{D(Sx)} ,

D(S∗y) = maxy0≤y≤yM

{D(Sy)} .(21)

Without loss of generality we can suppose thatD(S∗x) ≥ D(S∗y). It follows that

D(V ∗x ) = D(S∗x) + 1

≥ D(Vx), for all x ∈ [x0, xN ] . (22)

D(V ∗y ) = D(S∗y) + 1

≥ D(Vy), for all y ∈ [y0, ym] . (23)

Here V ∗x and V ∗y are constructed from S∗x and S∗yrespectively with the method above. We have

D(V ∗x ) ≥ D(V ∗y ) . (24)

From [Eq. (20)] we have that D(F ) =

D((⋃

x0≤x≤xN Vx)∪(⋃

y0≤y≤yM Vy))

.

(I) If D(⋃

x0≤x≤xN Vx)≥ D

(⋃y0≤y≤yM Vy

),

from Lemma 2 it follows that

D(F ) = D

⋃x0≤x≤xN

Vx

= D(V ∗x ) = D(S∗x) + 1 (25)

(II) If D(⋃

y0≤y≤yM Vy)≥ D

(⋃x0≤x≤xN Vx

),

from Lemma 2 it follows that

D(F ) = D

⋃y0≤y≤yM

Vy

= D(V ∗y ) = D(S∗y) + 1 . (26)

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The Study on Bivariate Fractal Interpolation Functions and Creation of. . . 633

Considering Eqs. (25) and (26), we conclude that

D(F ) = maxx0≤x≤xN ,y0≤y≤yM

{D(Sx),D(Sy)}+ 1 .

This completes the proof.

The practical significance of Theorem 3 is thatif we want to find the dimensions of the fractalinterpolated surface, we need only to find the di-mensions of fractal curves on the planes Sx and Sy,respectively. We can then find maximum of themand obtain the dimension of the fractal interpolatedsurface with [Eq. (19)]. The way to find the dimen-sions of the fractal interpolated curves is given inthe literature (Barnsley2).

6. EXAMPLES

Now we use a set of data to interpolate a fractalsurface. There are 4 data in the X direction andin the Y direction respectively on the interpola-tion field. The set of the data can be denoted by(xn, ym, zn,m)(n = 0, 1, . . . , 3; m = 0, 1, . . . , 3) (seeTable 1).

Figure 4 is the surface graph drawn with the orig-inal data. The dip is 40◦ and the direction 340◦.

Figure 5 gives a fractal interpolated surface fromthe fractal interpolation formula [Eq. (13)] with the

Table 1 The OriginalData for InterpolatingFractal Surface.

Y X

0 100 200 300

0 1 4 6 2

100 2 1 3 6

200 5 0 4 3

300 3 6 3 4

Fig. 4 The surface constructed with original data.

(a)

(b)

Fig. 5 Fractal interpolated surface (a) fractal dimensionD = 2.2675 (b) fractal dimension D = 2.5222.

original data in Table 1. The fractal dimension is2.2675 in Fig. 5(a), 2.5222 in Fig. 5(b). Obviously,with the same set of data, we can get different in-terpolated surfaces with different roughnesses.

7. CONCLUSIONS

With only a few data points, using the fractaltheory to interpolate a fractal surface that passesthrough the given points, it is possible to conductsignificant research while simulating some bodieswith complex geometrical shapes, such as materialfracture surfaces, fault surfaces and landscapes.

In this paper, we have discussed interpolationfunctions of fractal interpolated surface, proved thetheorem of uniqueness of an iterated function sys-tem of fractal interpolated surface and derived thetheorem of fractal dimensions of fractal interpolatedsurface. This lays a theoretical foundation for fur-ther research into the application and theory of frac-tal surfaces.

In formula [Eq. (11)], αn,m is an important pa-rameter showing the complex level of fractal inter-polated surfaces. The roughness of fractal interpo-lated surfaces can be adjusted with αn,m. Usingthe same interpolation data, we can obtain fractalinterpolated surfaces with different complex levelsfor different sets of data αn,m. Additional discus-sion and conclusions will be forthcoming in otherpapers.

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ACKNOWLEDGMENTS

The work was supported by the National Distin-guished Youth’s Science Foundation of China andthe Trans-Century Training Programme Founda-tion for the Talents by the State Education Com-mission and the National Natural Science Founda-tion of China.

REFERENCES

1. M. F. Barnsley, Fractal Functions and Interpola-tions, Constructive Approximation 2 (1986).

2. M. F. Barnsley, Fractals Everywhere (AcademicPress Orlando. FL.1988), pp. 172–247.

3. W. Q. Zeng et al., Fractal Theory and Its ComputerSimulation (Northeast University Press), 1993 (inChinese), pp. 74–105.

4. R. A. Hu et al., Fractal Computer Image and Its Ap-plications (China Railway Press),1995 (in Chinese),pp. 61–85.

5. W. S. Song and J. J. Yang, Modeling Methodfor The Earth’s Surface, (Mini-Micro-ComputerSystems), Vol. 17, No. 3, 1996 (in Chinese),pp. 32–36.

6. H. Xie, Fractals in Rock Mechanics (A. A. BalkemaPublishers, Netherlands, 1993), pp. 70–78.

7. B. B. Mandelbrot, The Fractal Geometry ofNature (W. H. Freeman, New York, 1982),pp. 361–366.

8. J. Feder, Fractals (Plenum press, New York, 1988),pp. 212–228.

9. X. D. Qi, Fractal and Computer Construction (Sci-ence Press), 1994 (in Chinese), pp. 70–72.

10. H. Xie and X. Q. Xue, Mathematical Foundationand Methods in Fractal Applications (Science Press),1997 (in Chinese).

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