the axiom a string of characters (each one having a meaning) that is used to start the generation of the fractal and an angle θ,

Transcrição

1 UFMG/ICEx/DCC Algoritmos e Estruturas de Dados II Exercício de Programação 3 Bacharelado em Ciência da Computação 2 o semestre de 2005 Informações sobre o exercício de programação Data que o trabalho deve ser entregue: 30 de agosto de 2005, até às 7:30 horas Comece a fazer este exercício logo, enquanto o problema está fresco na memória e o prazo para terminá-lo está tão longe quanto jamais poderá estar Fractais Fractais são padrões geométricos recursivos que seguem certas regras de formação O objetivo deste trabalho é fazer um programa para ler as regras de formação dos fractais e gerar esses fractais até um certo estágio ou nível Além dos fractais abaixo, dois novos padrões devem ser definidos A saída deve ser um arquivo contendo os comandos para gerar o fractal Sugere-se fortemente que você tente exibir o fractal gerado utilizando um pacote gráfico A seguir, está um texto em inglês, que já havia escrito, e que serve de especificação deste problema Problem Description There are fractal algorithms that generate a simple polygon chain For example, algorithms in the so called L-systems or string rewriting systems (deterministic fractals), and Brownian motion fractals (random fractals) String rewriting systems can be used to generate classic fractal curves such as the von Koch snowflake and the space filling curves of Peano and Hilbert Those fractals can be generated by providing: the axiom a string of characters (each one having a meaning) that is used to start the generation of the fractal and an angle θ, the production rules indicate how each character of the axiom can be replaced, and the number of cycles or stages indicate the level of recursivity Basically, the characters that appear on the axiom and production rules are: F: draw a line forward +: turn right by θ degrees -: turn left by θ degrees The following fractals are shown below: von Koch snowflake classic von Koch snowflake square wave von Koch snowflake sine wave 1 von Koch snowflake sine wave 2 Hilbert space filling Peano space filling Koch island 1

2 von Koch Snowflake Classic Shown in Figure 1 Axiom: F θ = π 3 Rules: F F-F++F-F p = 4 n von Koch snowflake curve (classic) first stage von Koch snowflake curve (classic) first two stages Figura 1: Fractal: von Koch snowflake curve (classic) von Koch Snowflake Square Wave Shown in Figure 2 Axiom: F θ = π 2 Rules: F F-F+F+FF-F-F+F p = 8 n 2

3 von Koch snowflake curve (square wave) first stage von Koch snowflake curve (square wave) first two stages Figura 2: Fracal: von Koch snowflake curve (square wave) first two stages von Koch Snowflake Sine Wave 1 Shown in Figure 3 Axiom: F θ = π 3 Rules: F F-F++FF--F+F p = 6 n

4 von Koch snowflake curve (sine wave 1) first stage von Koch snowflake curve (sine wave 1) first two stages Figura 3: Fractal: von Koch snowflake curve (sine wave 1) von Koch Snowflake Sine Wave 2 Shown in Figure 4 Axiom: F θ = π 3 Rules: F F-F+F+FF-F-F+F p = 8 n Hilbert Space Filling Shown in Figure 5 Axiom: X θ = π 2 Rules: X -YF+XFX+FY- Y +XF-YFY-FX+ 4

5 von Koch snowflake curve (sine wave 2) first stage von Koch snowflake curve (sine wave 2) first two stages Figura 4: Fractal: von Koch snowflake curve (sine wave 2) p = 4 n Peano Space Filling Shown in Figure 6 Axiom: X θ = π 2 Rules: X XFYFX+F+YFXFY-F-XFYFX Y YFXFY-F-XFYFX+F+YFXFY p = 9 n 1 5

6 Hilbert filling curve first stage Hilbert filling curve first two stages Figura 5: Fractal: Hilbert filling curve first two stages Koch Island Shown in Figure 7 Axiom: F+F+F+F θ = π 2 Rules: F F+F-F-FFF+F+F-F p = 4 9 n 1 Note: a closed line will be produced if all p points are generated

7 Peano filling curve first stage Peano filling curve first two stages Figura 6: Peano filling curve first two stages 7

8 Koch island curve first stage Koch island curve first two stages Figura 7: Koch island curve first two stages 8