The following theorem supplies the basic elements necessary for construction of generalized fractal interpolation functions. Theorem 1. Let the interpolating data set D = {P_{i} = [x_{i } y_{i } z_{i}]^{T }Î R^{3}, x_{i }< x_{i+1}, i = 0, 1,.., n}(n ³ 2) be given. Consider the IFS s(D) = {R ^{3}; w_{1},..., w_{n}}, where w_{i} are given by (1) with and the real numbers d_{i}, h_{i}, l_{i}, m_{i }are parameters chosen such that w_{i }(i = 1,..., n ) are contractive mappings. Then, the limiting set F_{D}= lim_{k ®¥}W ^{k}( G ), is the graph of a continuous vector valued function f : [x_{0}, x_{n}] ® R^{2} having the interpolation property f(x_{i}) = [ y_{i }z_{i}]^{T }, i = 0,1,..., n. Proof. It is straightforward
to see that, with the special choice of e_{i},
f_{i}, g_{i }given here, the transformations
in the IFS take the form
For i = 1,..,n, denote by M_{i }the
sub-matrix of A_{i } whose elements are d_{i},
h_{i}, l_{i}, m_{i}
The conditions that d_{i}, h_{i}, l_{i}, m_{i } must obey in order that a transformation of the form of w_{i }Îs(D) be a contraction, are given in the following lemma. Lemma 1.
A transformation w_{i }having
the form (1)with
a_{i }<
1
is a contraction of R^{3} if
and only if the elements d_{i}, h_{i},
l_{i}, m_{i}_{ }of A_{i
}satisfy
one of the following two systems of inequalities
(d_{i}-m_{i})^{2}
+ 4 h_{i }l_{i }³
0 and d_{i}m_{i}-
h_{i }l_{i }- | d_{i
}+
m_{i
}|
+ 1 > 0 , (7a)
(d_{i}-m_{i})^{2} + 4 h_{i }l_{i} < 0 and 1 -d_{i}m_{i} + h_{i }l_{i} > 0 . (7b) Proof. Denote
the
spectral radius of a matrix M by r(M).
Then
is a necessary and sufficient condition for the mapping w_{i }to be a contraction w.r.t. a feasible metric d in R ^{3}. Let Q_{i }= (d_{i}-m_{i})^{2} + 4 h_{i }l_{i} , and let l_{1}(i) , l_{2}(i) , l_{3}(i) denote the eigenvalues of the matrix A_{i } in (2), more precisely let l_{1 }(i) = a_{i}, l_{2}(i)= ^{1}/_{2}( d_{i}+ m_{i }- ÖQ_{i }) and l_{3}(i)= ^{1}/_{2}( d_{i}+ m_{i}+ ÖQ_{i}). Then, (8) is equivalent to | l_{k}(i)| < 1, k = 1, 2, 3 Let Q_{i } ³ 0 ( this is equivalent to the first inequality in the first system in the assertion ). Then, l_{2}(i) , l_{3}(i) Î R, and l_{2}(i) £ l_{3}(i).So the condition {|l_{2}(i)| < 1 Ù |l_{3}(i)| < 1 } is equivalent to { -1 < l_{2}(i) Ù l_{3}(i) < 1 }, or {ÖQ_{i } < d_{i} + m_{i }+ 2 Ù ÖQ_{i }< -d_{i}-m_{i }+ 2} which, in turn, is equivalent to the set of inequalities ÖQ_{i }< 2 - d_{i}- m_{i} , d_{i}+ m_{i }³ 0, ÖQ_{i }< 2 + d_{i}+ m_{i} , d_{i}+ m_{i }< 0. By squaring both sides of both left inequalities and replacing Q_{i }= (d_{i}-m_{i})^{2} + 4 h_{i }l_{i}, the above set reduces to the second inequality of (7a). If, instead, Q_{i } < 0, the first inequality in the second system is satisfied. Furthermore, in this case l_{2}(i) and l_{3}(i) are conjugate complex numbers having common modulus |l_{2}(i)| = |l_{3}(i)| = ^{1}/_{2}| d_{i} + m_{i }+ i Ö(-Q_{i })| = ^{1}/_{2} Ö[(d_{i}+m_{i})^{2}- Q_{i }], therefore the condition {|l_{2}(i)| < 1 Ù |l_{3}(i)| < 1} yields Ö[ (d_{i}+m_{i})^{2}- Q_{i }] < 2 which reduces to Ö(d_{i}m_{i}-h_{i }l_{i}) < 1, and to the second inequality of (7b), which is thereby completely proved. à The folowing corollary gives the condition on matrices M_{i }( i = 1,..,n) under which the IFS is hyperbolic. Corollary 1. The IFS s(D) = {R^{3}; w_{1}, ..., w_{n}}, built according to the rules given in Theorem 1, is hyperbolic if and only if for all i = 1,..., n the elements of the parameter matrix M_{i }satisfy the conditions (7a) and (7b). Proof. It is straightforward since the hypotheses made in Theorem 1 about D yield a_{i }< 1 ( i = 1,.. ,n) and, therefore, Lemma 1 applies. à Let us denote by M the set of all the parameter matrices in the IFS s(D), M = {M_{i}, i = 1,..,n}. It is clear, now, that the uniqueness of the attractor of s(D) and, of course, the shape of the attractor itself depend on the parameter matrices M just as well as on the interpolation data D. Therefore, in the following, whenever this dependence will be relevant to our discussion we will introduce the set of parameter matrices in our notation, denoting by s(D,M) the corresponding IFS. Also we will denote by the same symbol the corresponding interpolation scheme, namely the rule by which the continuous vector valued function f , being the unique attractor of s(D,M), is associated to D. Example 1. Consider the data set D= {(0, 0, 0), (1/4, 0, 1/2), (1/2, 1/2, 1), (3/4, 1, 1/2), (1, 1, 0)}. By Theorem 1, a four term IFS s(D) = {R ^{3}; w_{1}, w_{2}, w_{3}, w_{4}}, such that its attractor is the graph of a vector valued fractal interpolation function f, can be associated to D. If the items of {M_{i }, i = 1,..,4} are chosen, for example, to be : d_{1 }= d_{2}= 1/2, d_{3 }= -1/4, d_{4 }= 0; h_{1 }= h_{4 }= 1/4, h_{2 }= -1/4, h_{3 }= 1/2; l_{1 }= l_{4 }= 1/4, l_{2 }= l_{3 }= -1/4; m_{1 }= 0, m_{2 }= 1/4, m_{3 }= -1/4, m_{4 }=1/2, then the affine mappings w_{1}, w_{2}, w_{3}, w_{4 } have the form (2), with Direct checking shows that this choice of d_{i}, h_{i}, l_{i}, m_{i} satisfies constraints in Lemma 1, so the coefficient matrices above define contractive linear mappings R ^{3}® R ^{3}. Indeed, one could also evaluate directly r( A_{1}) = r( A_{4}) = (1+ Ö2 )/4 » 0.603553, r(A_{2}) = ( 3 + Ö5 )/8 » 0.654508, and r(A_{3}) = Ö3 /4 » 0.433013, to discover that all are smaller than 1, which guarantees that the IFS s(D) is hyperbolic and its attractor F_{D} is unique. In Figure 1 (top) the preattractors W ^{1}(G), W ^{2}(G) and W ^{3}(G) are displayed, where G is the piecewise linear interpolant of D. The rest of Fig. 1 shows the projections of these preattractors on the coordinate planes. The interpolating points are marked in all projections.
Figure 1. Preattractors of F_{D} and their projections on the coordinate planes Figure 2 (left) shows the attractor of the orbit of G, this is F_{D} = {[x f(x) y(x)]^{T}, x ÎI= [0,1]} which is the graph of f (x) = [f(x) y(x)]^{T }. The projections of F_{D} on the coordinate planes, f ={ [x f(x)]^{T}, xÎI}, y={[x y(x)]^{T}, xÎI}and the parametric curve f = (f,y) are shown in Figure 2 (right). The functions f and y are called hidden variable fractal interpolation functions (Barnsley [2], [3], Massopust [14]) meaning that they depend on the "third variable" that is not present in the plane containing their graphs. In fact they do share the interpolation property of F_{D}but, while F_{D }is, by its own nature, a self affine set, in the general case the projections f and y are not self affine.
Figure 2. Components of f . |