From 021d83730d896b7cef1050fbdd4c4c766992a9b0 Mon Sep 17 00:00:00 2001 From: Alain Date: Sun, 6 Jun 2021 17:36:05 +0200 Subject: ifs work --- buch/papers/ifs/images/farn.eps | 2372 ++++++++++++++++++++++++++++++ buch/papers/ifs/images/farncolor.eps | 2666 ++++++++++++++++++++++++++++++++++ buch/papers/ifs/teil2.tex | 26 +- 3 files changed, 5063 insertions(+), 1 deletion(-) create mode 100644 buch/papers/ifs/images/farn.eps create mode 100644 buch/papers/ifs/images/farncolor.eps diff --git a/buch/papers/ifs/images/farn.eps b/buch/papers/ifs/images/farn.eps new file mode 100644 index 0000000..597745b --- /dev/null +++ b/buch/papers/ifs/images/farn.eps @@ -0,0 +1,2372 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: (MATLAB, The Mathworks, Inc. Version 9.7.0.1434023 \(R2019b\) Update 6. 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mit einem $c_i < 1$. Dann existiert eine eindeutige kompakte Menge \begin{equation} F = \bigcup\limits_{i = 1}^{m} S_i(F) \end{equation} -TODO Text +Weiter definieren wir die Transformation S auf kompakte Mengen ohne die leere Menge. +\begin{equation} + S(E) = \bigcup\limits_{i = 1}^m S_i(E) +\end{equation} +Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, und für jedes $i$ $S_i(E) \subset E$, gilt +\begin{equation} + F = \bigcap\limits_{k = 1}^{\infty} S^k(E). +\end{equation} +In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgeführt, gegen eine eindeutige Menge konvergiert. +Dies für jede Startmenge, solange diese ihre Transformierten wieder beinhaltet. +Auf den Beweis wird verzichtet. +\subsection{Beispiel: Barnsley-Farn} +\begin{figure} + \label{ifs:farn} + \centering + \makebox[\textwidth][c]{ + \includegraphics[width=1.4\textwidth]{papers/ifs/images/farn}} + \caption{Barnsley-Farn} +\end{figure} +\begin{figure} + \label{ifs:farncolor} + \centering + \includegraphics[width=0.7\textwidth]{papers/ifs/images/farncolor} + \caption{Vier Transformationen des Barnsley-Farn} +\end{figure} -- cgit v1.2.1