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
D_{yz}
= {Q_{i} = [ y_{i } z_{i}]^{T},
i
= 0, 1,..., n} (n ³
2)
is given, satisfying
Given a polygonal line L with vertices {[p_{j }
q_{j}]^{T}, j =
0, 1, ..., m} (m ³ 2), such
that [p_{0} q_{0}]^{T}
= [ y_{0 } z_{0}]^{T
}=
Q_{0}
and [p_{m }q_{m}]^{T}
= [ y_{n } z_{n}]^{T }=
Q_{n
},
one can build a set of affine mappings { v_{i }:
R
^{2}®
R
^{2},
i
= 1,..., n}, having the form
and satisfying the four interpolation conditions
Let t(D_{yz}) = {R^{2}; v_{1},..., v_{n}} be the IFS connected to the transformations (10). It follows from (11) that for fixed i , two out of six parameters d_{i}, h_{i}, l_{i}, m_{i}, f_{i}, g_{i }are free. For the reason of convenience, one may suppose that these free parameters are items of the matrix M_{i}. So, the set of matrices M = {M_{i}, i = 1,.., n} appears as the parameter of the IFS t(D_{yz}) and this fact will be stressed by the notation t(D_{yz},M). If the elements of M_{i} for all i = 1,.., n satisfy conditions (7a) or (7b), it follows from the proof of Lemma 1 that r(M_{i}) < 1, and the IFS t(D_{yz}) is hyperbolic. Definition 1. Given the mesh D_{x} = {x_{0}< x_{1}< ...< x_{n }} and the data set D_{yz }= {Q_{i} = [ y_{i } z_{i}]^{T}, i = 0,..,n}, we call D = { [x_{i } y_{i } z_{i}]^{T }, i = 0, 1,..., n} a lifting of D_{yz }on D_{x}. Let t(D_{yz},M) be an IFS built as above and let D be a lifting of D_{yz }on D_{x}. Then, we call the IFS s(D) = s(D,M) built according to Theorem 1, a lifting of t(D) = t(D_{yz},M) on D_{x}. The following theorem then holds. Theorem 2. If the IFS t(D_{yz},M) is hyperbolic, then its lifting s(D,M) is also hyperbolic. Moreover, the attractor of t(D_{yz},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 D_{yz }. Proof. For every i = 1,..., n, (10) and (11) above yield y_{i }- y_{i-}_{1}- d_{i}(y_{n }-y_{0}) - h_{i}(z_{n }-z_{0}) = 0 and z_{i }- z_{i-1}- l_{i}(y_{n }-y_{0}) - m_{i}(z_{n }-z_{0}) = 0 , which, in connection with the hypothesis that s(D,M) is the lifting of t(D_{yz},M), gives c_{i}= 0 and k_{i} = 0 according to the formulas in Theorem 1. Furthermore, under these hypotheses, the expressions for the known terms f_{i }and g_{i } from Theorem 1 are consistent with (11). This means that every transformation w_{i} splits into two independent components w_{i}(P) = [u_{i}(x) v_{i}(y, z)]^{T}, the first one being the contraction of the real line u_{i}(x) = a_{i} x + e_{i}, and the second one, acting in the (y, z)-plane, being just v_{i}(y, z). Now, if the IFS t(D_{yz},M) is hyperbolic, r(M_{i}) < 1 holds for every i, and therefore r(A_{i}) = max{|a_{i}|, r(M_{i})}< 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(D_{yz},M) respectively. Let F_{x }and F_{yz }respectively denote projections of F on x-axis and yz-plane. Let F_{i }= w_{i}(F) and let (F_{i})_{yz }be the yz-projection of F_{i} . Then, by the block diagonal structure of w_{i} it follows that w_{i}(F) = F_{i} = [u_{i}(F_{x}) v_{i}(F_{yz})]^{T }or, v_{i}(F_{yz}) = (F_{i})_{yz }. Also, È_{i}F_{i } = F implies È_{i}(F_{i})_{yz} = F_{yz}, which is equivalent to È_{i} v_{i}(F_{yz}) = F_{yz }. So, F_{yz } is the fixed point of the Hutchinson operator È_{i} v_{i}( . ) and the (unique) attractor of t(D_{yz},M). Accordingly, F = F_{yz }and F_{yz} is a self-affine set. It is straightforward to see, from (11), that D_{yz }Ì F_{yz }i.e., the attarctor of t(D_{yz},M) interpolates the data set D_{yz} . à As far as the matter of supplying two extra conditions to (11) that will define the remaining parameters of transformation v_{i}, according to the literature, there are two ways. The first one ([2]) uses n prescribed points in the (y, z)-plane, {R_{i}, i = 1,..., n} and demands that v_{i}(p_{j}, q_{j}) =R_{i}, (j¹0,m), i = 1,..., n. In this way, the configuration of the points {R_{i}} influences the shape of the sequence of preattractors and the attractor itself. The second method reduces the number of parameters of v_{i }to four by demanding that {v_{i}} 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 D_{yz} = {(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 v_{1}, v_{2}, v_{3} and v_{4 } determined by (11) and by v_{1,4}(1/2, 1)=(1/2, 1/4), v_{2}(1/2, 1)=(1/4, 1), v_{3}(1/2, 1)=(3/4, 1) define the hyperbolic IFS t(D_{yz}). Choice of the mesh {x_{i}}={0, 1/4, 1/2, 3/4, 1} leeds to the lifted IFS s(D) ={R^{3}; w_{1}, w_{2}, w_{3}, w_{4}} where the matrix coefficients of w_{1}, w_{2}, w_{3} and w_{4 }are 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 W^{1}(L), W ^{2}(L) and W ^{3}(L) are successive approximations of F_{yz }, 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]^{2 }.
Figure 3. The vector-valued fractal interpolation function containing a self-affine Peano curve as a component . |