A contraction is a transformation T that reduces the distance between every pair of points. |
That is, there is a number |
dist(T(x, y), T(x', y')) ≤ r⋅dist((x, y), (x', y')) |
for all pairs of points (x, y) and (x', y'). |
Here dist denotes the Euclidean distance between points: |
dist((x, y), (x', y')) = ((x - x')2 + (y - y')2)1/2 |
To save space we write dist((x, y), (x', y')) = d((x, y), (x', y')). |
The contraction factor of T is the smallest r satisfying |
d(T(x, y), T(x', y')) ≤ r⋅d((x, y), (x', y')) |
for all pairs of points (x, y), (x', y'). |
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In general, contractions can reduce distances between points by different amounts, depending on the position of the points. |
Here are some special kinds of contractions. |
A similarity reduces all distances by the same number,
|
d(T(x, y), T(x', y')) = r⋅d((x, y), (x', y')) |
for all pairs of points (x, y), (x', y'). |
The transformation T(x, y) = (r⋅x, r⋅y) is an example; its contraction factor is r. |
An affinity reduces distances by different amounts in different directions. For example, |
T(x, y) = (r⋅x, s⋅y), |
where both r < 1 and s < 1, and r and s are different. |
What is the contraction factor for the affinity T(x, y) = (x/2, y/3)? here is the Answer. |
If all the transformations of an IFS are contractions, then iterating the IFS is guaranteed to converge to a unique shape. |
Return to IFS Convergence.