A bidimensional fern computed by means of an 'Iterated Function System' -IFS- [Une fougère bidimensionnelle obtenue à l'aide de la méthode des 'Iterated Function Systems' -IFS-].




This bidimensional fern was computed starting with the two following points:
                    A = {0,0,0} (displayed as a bigger Red sphere)
                    B = {1,0,0} (displayed as a bigger Green sphere)
(by the way, one point is enough for this iterative process...). Then, the coordinates of these two points are iteratively transformed using one of the four following linear transformations chosen randomly (each one with a different probability) at each step:
                    /        \   /                     \ /      \   /         \
                    | X(i+1) |   |  +0.00   +0.00   0  | | X(i) |   |  +0.00  |
                    |        |   |                     | |      |   |         |
                    | Y(i+1) | = |  +0.00   +0.16   0  |.| Y(i) | + |  +0.00  |     probability=0.01
                    |        |   |                     | |      |   |         |
                    | Z(i+1) |   |    0       0     0  | | Z(i) |   |    0    |
                    \        /   \                     / \      /   \         /
                    /        \   /                     \ /      \   /         \
                    | X(i+1) |   |  +0.20   -0.26   0  | | X(i) |   |  +0.00  |
                    |        |   |                     | |      |   |         |
                    | Y(i+1) | = |  +0.23   +0.22   0  |.| Y(i) | + |  +1.60  |     probability=0.07
                    |        |   |                     | |      |   |         |
                    | Z(i+1) |   |    0       0     0  | | Z(i) |   |    0    |
                    \        /   \                     / \      /   \         /
                    /        \   /                     \ /      \   /         \
                    | X(i+1) |   |  -0.15   +0.28   0  | | X(i) |   |  +0.00  |
                    |        |   |                     | |      |   |         |
                    | Y(i+1) | = |  +0.26   +0.24   0  |.| Y(i) | + |  +0.44  |     probability=0.07
                    |        |   |                     | |      |   |         |
                    | Z(i+1) |   |    0       0     0  | | Z(i) |   |    0    |
                    \        /   \                     / \      /   \         /
                    /        \   /                     \ /      \   /         \
                    | X(i+1) |   |  +0.85   +0.04   0  | | X(i) |   |  +0.00  |
                    |        |   |                     | |      |   |         |
                    | Y(i+1) | = |  -0.04   +0.85   0  |.| Y(i) | + |  +1.60  |     probability=0.85
                    |        |   |                     | |      |   |         |
                    | Z(i+1) |   |    0       0     0  | | Z(i) |   |    0    |
                    \        /   \                     / \      /   \         /
Each point {X(i+1),Y(i+1),Z(i+1)} is displayed as a little sphere having the color of the initial point {X(0),Y(0),Z(0)} (Red for A and Green for B).


See the bidimensional fern with one starting point and the display of the number of iteration:




(CMAP28 WWW site: this page was created on 12/04/2006 and last updated on 04/26/2015 11:40:56 -CEST-)



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