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3. Block diagonal IFS

Nevertheless, under feasible conditions, by carefully  choosing the IFS the parametric curve { f(t) = [f(t) y(t)]T, t ÎI} can be built so to have the property of  self-affine planar interpolating curves, which is characteristic of most classical well known fractal curves such as the  Peano square (1890), the von Koch snowflake (1906) or the Sierpinski triangle (1915). Among these the Peano or space filling curves are especially interesting : these are curves  whose graph is identical to a nonempty pathwise connected compact subset of R 2 (Barnsley [3], p. 240), and find interesting applications  in computer graphics, see for example  [15]. The construction is as follows.

Suppose that  the data set  from  the (y, z)-plane Dyz = {Qi = [ y zi]T, i = 0, 1,..., n} (n ³ 2) is given, satisfying
 

                                   0  <   | Qi - Qi-1| <  |Qn - Q0 |, i = 1,..., n.                                    (9)

Given a polygonal line L with vertices {[p qj]T, j = 0, 1, ..., m} (m ³ 2), such that [p0  q0]T = [ y0  z0]T = Q0 and  [pm qm]T = [ y zn]T = Qn , one can  build  a set of affine mappings { vi : R 2® R 2, i =  1,..., n}, having the form
 

                 (10)

and satisfying  the four interpolation conditions
 

vi (y0 , z0) = Qi-1,     vi (yn , zn) = Qi,   i = 1, ..., n.                            (11)

Let  t(Dyz) = {R2; v1,..., vn} be the IFS connected to the transformations (10). It follows from (11) that for fixed  i , two out of six parameters di, hi, li, mi, fi, gi are free. For the reason of convenience, one may suppose that these free parameters are items of the matrix Mi. So, the set of matrices M = {Mi, i = 1,.., n} appears as the parameter of the IFS t(Dyz) and this fact will be stressed by the notation t(Dyz,M). If the elements of Mi for all  i = 1,.., satisfy conditions (7a) or (7b), it follows from the proof of Lemma 1 that r(Mi) < 1, and the IFS t(Dyz) is hyperbolic.

Definition 1. Given the mesh Dx = {x0< x1< ...< xn } and the data set  Dyz = {Qi = [ y zi]T, i = 0,..,n}, we call  D = { [x y zi]T , i = 0, 1,..., n} a  lifting  of  Dyz on Dx. Let  t(Dyz,M)  be an  IFS built as above and let  be a  lifting  of  Dyz on Dx. Then, we call the IFS s(D) = s(D,M)  built according to Theorem 1, a lifting of t(D) = t(Dyz,M) on Dx.

The following theorem then holds.

Theorem 2. If the  IFS  t(Dyz,M) is hyperbolic, then its lifting s(D,M) is also  hyperbolic. Moreover, the attractor of t(Dyz,M) is the orthogonal projection of  the attractor of s(D,M) onto the (y, z)-plane. It has self-affine structure and interpolates the data set Dyz .

Proof.    For every  i = 1,..., n, (10) and (11) above  yield yi - yi-1- di(yn -y0) - hi(zn -z0) = 0  and   zi - zi-1- li(yn -y0) - mi(zn -z0) = 0 , which,  in connection with the hypothesis that s(D,M) is the lifting of  t(Dyz,M),  gives ci= 0  and  ki = 0  according to the formulas in Theorem 1.  Furthermore, under these hypotheses, the expressions for the known terms fi and  g from Theorem 1 are consistent with (11). This means that every transformation wi splits  into two independent  components  wi(P) = [ui(x) vi(y, z)]T, the first one being  the contraction of  the real line ui(x) = ai x + ei, and the second one, acting in the (y, z)-plane, being just vi(y, z). Now, if the  IFS  t(Dyz,M) is hyperbolic,  r(Mi) < 1 holds for every i, and therefore  r(Ai) = max{|ai|, r(Mi)}< 1 also holds for i = 1,..., n, and  s(D,M) is  hyperbolic, too.

Let F and F denote the attractors of s(D,M) and t(Dyz,M)  respectively.  Let  Fx and Fyz respectively denote projections of  F on x-axis and yz-plane. Let  Fi = wi(F) and let  (Fi)yz be the yz-projection of Fi . Then, by  the block diagonal structure of wi it follows that wi(F) = Fi = [ui(Fx) vi(Fyz)]T or, vi(Fyz) = (Fi)yz . Also, ÈiF = F implies Èi(Fi)yz = Fyz, which is equivalent to  Èi vi(Fyz) = Fyz . So, Fyz  is the fixed point of the Hutchinson operator Èi vi( . ) and the (unique) attractor of  t(Dyz,M). Accordingly, F = Fyz and Fyz  is a self-affine set.

It is straightforward to see, from  (11), that  Dyz Ì Fyz i.e., the attarctor of t(Dyz,M) interpolates the data set Dyz . à

As far as the matter of supplying two extra conditions to (11) that will define the remaining parameters of transformation vi, according to the literature, there are two ways. The first one ([2]) uses n prescribed points in  the (y, z)-plane, {Ri, i = 1,..., n} and demands that vi(pj, qj) =Ri, (j¹0,m), i = 1,..., n. In this way, the configuration of the points {Ri} influences the shape of the sequence of preattractors and the attractor itself.

The second method reduces the number of parameters of vi to four by demanding that {vi} belong to some subset of the set of affine planar mappings like in the case of  Peano or space filling curves [12].

Example 2 (Space-filling curve). Let Dyz = {(0, 0), (0, 1/2), (1/2, 1/2), (1, 1/2), (1, 0)}, and let L= {(0, 0), (0, 1/2), (1/2, 1), (1, 1/2), (1, 0)}. The mappings v1, v2, v3  and v determined by (11) and  by  v1,4(1/2, 1)=(1/2, 1/4), v2(1/2, 1)=(1/4, 1), v3(1/2, 1)=(3/4, 1) define the hyperbolic IFS t(Dyz). Choice of the mesh {xi}={0, 1/4, 1/2, 3/4, 1} leeds to the lifted IFS s(D) ={R3; w1, w2, w3, w4} where  the  matrix coefficients of w1, w2, w3 and ware

The first three iterations W 1, W 2 and W 3 of the Hutchinson operator W associated  with s(D), applied to L, are shown in Figure 3 (top) along with the corresponding projections on the coordinate planes. The yz-projections of W1(L), W 2(L) and W 3(L) are successive approximations of  Fyz , the  yz-projection of the attractor F of s(D).  These self-affine continuous curves being contained in the unit square [0, 1]2 suggest that F is also a continuous curve, it has self-affine structure and it  "fills" [0, 1].

Figure 3. The vector-valued fractal interpolation function containing a self-affine Peano curve as a component .

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