迭代函数系统与仿射映射

几何画板及绘图内容加载中，请稍等……

2018-07-19 08:05

函数变换几何作图

所属专题 Fractals

①本范例演示了通过四个仿射映射构成的迭代函数系统（IFS）生成的分形图形；②涉及的核心数学概念包括仿射变换、迭代函数系统、自相似性、分形几何；③作图要点提炼自构造摘要，使用了Sequence命令生成序列点，涉及的对象有d_{23}, d_{13}, c_{23}, c_{13}, b_{13}, b_{23}, a_{23}, a_{13}, d_{21}, d_{22}, d_{12}, d_{11}, c_{22}, c_{21}, c_{12}, c_{11}, b_{21}, b_{22}, b_{12}, b_{11}, a_{22}, a_{21}, a_{12}, a_{11}, A, text1, text1_1, L_M, L_P, text1_2, Button1, Button2, text1_3, G_1, G_2, G_3, G_4, G_5；④适合高中及以上数学学习者，用于理解分形几何和迭代函数系统的应用。

GeoGebra 命令及参数

Sequence((Element[L_M, i] * (x(A), y(A))) + Element[L_P, i], i, 1, 4, 1) → G_1
Sequence(Sequence[(Element[L_M, i] * (x(Element[G_1, j]), y(Element[G_1, j]))) + Element[L_P, i], i, 1, 4, 1], j, 1, 4, 1) → G_2
Sequence(Sequence[Sequence[(Element[L_M, i] * (x(Element[Element[G_2, j], k]), y(Element[Element[G_2, j], k]))) + Element[L_P, i], i, 1, 4, 1], j, 1, 4, 1], k, 1, 4, 1) → G_3
Sequence(Sequence[Sequence[Sequence[(Element[L_M, i] * (x(Element[Element[Element[G_3, j], k], l]), y(Element[Element[Element[G_3, j], k], l]))) + Element[L_P, i], i, 1, 4, 1], j, 1, 4, 1], k, 1, 4, 1], l, 1, 4, 1) → G_4
Sequence(Sequence[Sequence[Sequence[Sequence[(Element[L_M, i] * (x(Element[Element[Element[Element[G_4, j], k], l], m]), y(Element[Element[Element[Element[G_4, j], k], l], m]))) + Element[L_P, i], i, 1, 4, 1], j, 1, 4, 1], k, 1, 4, 1], l, 1, 4, 1], m, 1, 4, 1) → G_5