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10214 Hausdorff distance Let us consider the metric space (R2 , d) The symbol H(R2 ) indicates a space whose elements are the non-empty compact subsets of R2 The distance [TRI 93] from the point x element of R2 to the set B element of H(R2 ), noted d(x, B), is de ned by: d(x, B) = min{d(x, y) : y B} The distance from the set A element of H(R2 ) to the set B element of H(R2 ), noted d(A, B), is de ned by: d(A, B) = max{d(x, B) : x A}
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The Hausdorff distance between two sets A and B elements of H(R2 ), noted hd (A, B), is de ned by: hd (A, B) = max{d(A, B), d(B, A)} (104)
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Only when applied to closed and bounded sets also referred to as compacts does the Hausdorff distance verify all the properties of a distance (in particular, commutativity) Evidently, it is not to be confused with the concept of the Hausdorff dimension presented in this volume in 1 and 3 10215 Contracting transformation on the space H(R2 ) Let : R2 R2 be a contracting transformation de ned on the metric space (R , d) with the real s as contraction factor The transformation : H(R2 ) H(R2 ) de ned by:
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is contracting on (H(R2 ), hd ), with contraction factor s The symbol hd indicates the Hausdorff distance 10216 Iterated transformation system An IFS de ned on the complete metric space (R2 , d) is composed of a set of N transformations i : R2 R2 (i = 1, , N ), each of them associated with a Lipschitz factor si From now on, in this section, it will be considered that the N transformations are contracting: the transformation system is in this case called hyperbolic IFS The contraction factor of the hyperbolic IFS, noted s, is equal to max{si : i = 1, , N } 1022 Attractor of an iterated transformation system Let us consider an IFS {R2 ; i , i = 1, , N } It has been demonstrated [BARN 93] that the operator W : H(R2 ) H(R2 ) de ned by:
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is contracting and that its contraction factor corresponds with that of the IFS The operator W possesses a single xed point At H(R2 ) given by: At = W (At ) = lim W on (X),
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The object At is also called an IFS attractor It is invariant under the transformation W and is equal to the union of N copies of itself transformed by 1 , , N This invariant object is called self-similar or self-af ne when the elementary transformations i are af ne
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EXAMPLE 101 Let us consider the IFS {R2 ; i , i = 1, , 3} composed of the following af ne transformations: x y 1 2 = 0 1 2 x y = 2 0 1 3 x y = 2 0 0 x + y0 1 y 2 2 x0 0 x 2 y + y 0 1 2 2 x 0 0 x 2 y + y0 1 2 2 0
(108)
Its contraction factor is equal to 025 The attractor coded by the IFS, called the Sierpinski triangle, is represented in Figure 101
x y ( 20 , y 0 ) w1 (0,0) w2 (-x 0 , -y 0 ) w3
Figure 101 Attractor of the iterated transformation system of Example 101 The initial square, originally centered and with sides of length 2x0 , 2y0 , is transformed into three homothetic squares by contracting transformations 1 , 2 and 3 This process is then iterated
1023 Collage theorem This theorem, shown in [BARN 93], provides an upper bound to the Hausdorff distance hd between a point A included in H(R2 ) and the attractor At of an IFS THEOREM 101 We consider the complete metric space (R2 , d) Given a point A belonging to H(R2 ) and an IFS {R2 ; 1 , 2 , , n } with a real contraction factor