L-Systems and Affine Transformations

Size: px

Start display at page:

Download "L-Systems and Affine Transformations"

Morgan Mitchell
5 years ago
Views:

1 L-Systems and Affine Transformations Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna

2 Copyright 2014, Moreno Marzolla, Università di Bologna, Italy ( This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License (CC-BY-SA). To view a copy of this license, visit or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA. Image credits: the Web; The Computational Beauty of Nature website Complex Systems 2

$Introduction We have seen that fractals are very$ $fractals are used within living organisms for achieving$ special goals Human brain (maximize surface area)

special goals Human brain (maximize surface area)

3 Introduction We have seen that fractals are very effective for squeezing a great deal of length (perhaps infinite) into a finite area It is not surprising that fractals are used within living organisms for achieving special goals Human brain (maximize surface area) Respiratory system, circulatory system (tree-like branching) Complex Systems 3

4 Production systems The instructions for building such complex structures must be stored somewhere (e.g., within the DNA) It is therefore important to be able to store that information as compactly as possible L-Systems were proposed in 1968 by biologist Arstid Lindenmayer as a mathematical description of how plants grow Can be applied to other structures as well Complex Systems 4

5 L-Systems An L-System consists of a special seed cell (axiom), and a set of rules describing how new cells can be generated from old cells Example: expands to Axiom Rules B B F[-B] + B F FF B F[-B] + B FF[-F[-B] + B] + F[-B] + B FFFF[-FF[-F[-B]+B] + F[-B]+B] + FF[-F[-B]+B] + F[-B] + B... Complex Systems 5

6 Turtle graphics commands F Draw forward by a fixed length G Go forward by a fixed length (without drawing) + Turn right by a fixed angle (n+ turns right n times) - Turn left by a fixed angle (n- turns left n times) [ Save the current position in the stack ] Restore position from the top of the stack Draw forward by a length computed from the execution depth Complex Systems 6

7 Koch Curve Angle: 60 Axiom: F Rule: F F-F++F-F lsys -da 60 -axiom F -rule "F=F-F++F-F" Complex Systems 7

8 Koch Snowflake Angle: 60 Axiom: F++F++F Rule: F F-F++F-F lsys -da 60 -axiom F++F++F -rule "F=F-F++F-F" Complex Systems 8

9 Peano curve Angle: 90 Axiom: F Rule: F F-F+F+F+F-F-F-F+F lsys -da 90 -axiom "F" -rule "F=F-F+F+F+F-F-F-F+F" Complex Systems 9

10 Other examples Complex Systems 10

11 Affine Transformations Affine transformations can be used to easily describe fractal objects that are made of smaller copies of themselves Possibly scaled and/or translated and/or rotated Complex Systems 11

12 Translation ( x ' y ' ) = ( x y) + ( Δ x Δ y) Complex Systems 12

13 Scaling ( x ' y ' ) = ( sx 0 0 sy)( x y) Complex Systems 13

14 Reflection ( x ' y ' ) = ( )( x y) ( x ' y ' ) = ( )( x y) Complex Systems 14

15 Rotation ( x ' ) y ' = ( cos(θ) sin(θ) )( x y) sin(θ) cos(θ) Complex Systems 15

16 Composing linear transformations Rotation, scaling, flipping + translation ( x ' ) y ' = ( a b )( x y) c d + ( e ) f With a suitable choice for terms a through f, this can be rewritten using a single matrix product ( x ' )( 1) x y ' d e f y ) = ( a b c To compose multiple transformations: x '=C ( B( A x)) Complex Systems 16

17 The Multiple Copy Reduction Machine (MCRM) Seed Image Next Image A One or more affine linear transformations A A A Recursion Complex Systems 17

18 The Multiple Copy Reduction Machine (MCRM) Complex Systems 18 Source: Il linguaggio dei frattali, Le Scienze n. 266, october 1990

19 Example Complex Systems 19

20 Speeding up convergence Start with a random point Iterate the point for n steps (n large) At each step choose one transformation with probability proportional to the area of the transformed box, i.e., the determinant of matrix ( a b c d ) Skip the first iterations, plot the positions of all other points Complex Systems 20

21 The Multiple Copy Reduction Machine (MCRM) Single point One transformation Next point ( x y) ( x ' One transformation Random choice One transformation y ' ) Recursion Complex Systems 21

22 Examples Complex Systems 22

23 Examples Complex Systems 23

24 Examples Complex Systems 24

25 Examples Complex Systems 25

Parallelizing Loops. Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna.

Parallelizing Loops. Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna. Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna http://www.moreno.marzolla.name/ Copyright 2017, 2018 Moreno Marzolla, Università di Bologna, Italy (http://www.moreno.marzolla.name/teaching/hpc/)