Content deleted Content added
linking |
started on new article |
||
Line 4:
[[User:IntegralPython/sandbox/Fractal measure| Fractal Measure]]
==Open set condition==
In fractal geometry, the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an iterative function system of contractive mappings f<sub>i</sub>, the open set condition requires that there exists a nonempty, open set S satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.
The open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket.
https://www.jstor.org/stable/4097989?read-now=1&refreqid=excelsior%3A19b81930eca74e0a264d556ab56211ae&seq=1#page_scan_tab_contents
===Computing Hausdorff measure===
When the open set condition holds and each ψ<sub>''i''</sub> is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>
:<math> \sum_{i=1}^m r_i^s = 1. </math>
where r<sub>i</sub> is the magnitude of the dilation of the similitude.
With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub> in the plane '''R'''<sup>2</sup> and let ψ<sub>''i''</sub> be the dilation of ratio 1/2 around ''a<sub>i</sub>''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>
Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC. In general a set ''E'' which is a fixed point of a mapping
: <math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>
is self-similar if and only if the intersections
:<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, </math>
where ''s'' is the Hausdorff dimension of ''E'' and ''H<sup>s</sup>'' denotes [[Hausdorff measure]]. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally. Indeed, under the same conditions as above, the unique fixed point of ψ is self-similar.
==Hand-eye calibration problem==
| |||