java递归画谢尔宾斯基三角形

什么是谢尔宾斯基三角形

谢尔宾斯基三角形（英语：Sierpinski triangle）是一种分形，由波兰数学家谢尔宾斯基在1915年提出。它是自相似集的例子。它的豪斯多夫维是log(3)/log(2) ≈ 1.585。

形状

构造

递归

1. 画一个三角形。（多数使用等边三角形）
2. 沿三边中点的连线，将它分成四个小三角形。
3. 对其余三个小三角形重复1。

若取一个正方形或其他形状开始，用类似的方法构作，形状也会和谢尔宾斯基三角形相近。

Chaos Game

1. 用随机的方法（Chaos Game），都可得到谢尔宾斯基三角形：
2. 任意取平面上三点A,B,C，组成一三角形
3. 任意取三角形ABC内的一点P，画出 该点
4. 画出 P和三角形其中一个顶点的中点
5. 重复1

L系统
这条曲线以L系统来记述为：
变量: A , B 常数: + , - 公理: A 规则: A → B-A-B B → A+B+A A,B ： 向前
-： 左转60°
+： 右转60°

这里我们使用递归方法画谢尔宾斯基三角形。

思路

1、递归

先根据自己选的三个点$(x_1, y_1),(x_2, y_2),(x_3, y_3)$(x1​,y1​),(x2​,y2​),(x3​,y3​)画三条线，然后分别取$(x_1, y_1),(x_3, y_3);(x_1, y_1),(x_2, y_2);(x_2, y_2),(x_3, y_3)$(x1​,y1​),(x3​,y3​);(x1​,y1​),(x2​,y2​);(x2​,y2​),(x3​,y3​)的中点为$(x_4,y_4),(x_5,y_5),(x_6,y_6)$(x4​,y4​),(x5​,y5​),(x6​,y6​)，连接$(x_4,y_4),(x_5,y_5),(x_6,y_6)$(x4​,y4​),(x5​,y5​),(x6​,y6​)三点。此时除了中间的三角形外形成了三个新的三角形，分别把它们的顶点看做$(x_1, y_1),(x_2, y_2),(x_3, y_3)$(x1​,y1​),(x2​,y2​),(x3​,y3​)，再重复上面的步骤。

public void x(int x1, int y1, int x2, int y2, int x3, int y3, int n) {
		index = 0;
		if (n > 0) {// 迭代条件
			n--;
			g.drawLine(x1, y1, x2, y2);
			g.drawLine(x3, y3, x2, y2);
			g.drawLine(x1, y1, x3, y3);

			int x4 = (int) ((x1 + x3) / 2);
			int y4 = (int) ((y1 + y3) / 2);
			int x5 = (int) ((x1 + x2) / 2);
			int y5 = (int) ((y2 + y1) / 2);
			int x6 = (int) ((x3 + x2) / 2);
			int y6 = (int) ((y3 + y2) / 2);

			g.drawLine(x4, y4, x5, y5);
			g.drawLine(x6, y6, x5, y5);
			g.drawLine(x4, y4, x6, y6);

			// 迭代
			x(x1, y1, x5, y5, x4, y4, n);
			x(x5, y5, x2, y2, x6, y6, n);
			x(x4, y4, x6, y6, x3, y3, n);
		}
	}

示意图：

2、画窗体

和普通的画窗体没什么区别。

package 谢尔宾斯基三角形;

import java.awt.BorderLayout;
import java.awt.Color;
import java.awt.Dimension;
import java.awt.FlowLayout;
import javax.swing.JButton;
import javax.swing.JFrame;
import javax.swing.JPanel;

public class DrawUI {
	public void intiui() {
		// 创建画图板窗口
		JFrame jf = new JFrame("谢尔宾斯基三角形");

		// 创建监听器对象
		Listener ls = new Listener();

		// 设置面板属性
		jf.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
		jf.setSize(800, 700); // 设置大小为800*700
		jf.setLocationRelativeTo(null);// 设置居中显示
		jf.setLayout(new BorderLayout());// 设置Border布局（JFrame默认为Border布局）

		// 创建Panel（且Panel为流式布局）
		JPanel jp1 = new JPanel();
		JPanel jp2 = new JPanel();
		JPanel jp3 = new JPanel();
		JPanel jp4 = new JPanel();
		jp1.setLayout(new FlowLayout());
		jp2.setLayout(new FlowLayout());
		jp1.setPreferredSize(new Dimension(70, 100));
		jp2.setPreferredSize(new Dimension(40, 100));
		jf.add(jp1, BorderLayout.WEST);
		jf.add(jp2, BorderLayout.EAST);
		jf.add(jp3, BorderLayout.CENTER);
		jf.add(jp4, BorderLayout.SOUTH);
		jp3.addMouseMotionListener(ls);
		jp3.addMouseListener(ls);

		// 用数组和循环创建按钮并添加监听器 格式：类型[] 数组名 = {"变量","变量"...};
		String[] ShapeBtn = { "生成", "清屏" };
		for (int i = 0; i < ShapeBtn.length; i++) {
			JButton shape = new JButton(ShapeBtn[i]);
			jp1.add(shape);
			shape.addActionListener(ls);
		}

		// 创建颜色按钮
		Color[] ColorBtn = { Color.BLUE, Color.GREEN, Color.RED, Color.YELLOW, Color.CYAN, Color.BLACK };
		Dimension colorBtnSize = new Dimension(30, 30); // 设置颜色方块的大小
		for (int i = 0; i < ColorBtn.length; i++) {
			JButton color = new JButton(); // 创建颜色按钮对象
			color.setBackground(ColorBtn[i]);// 设置按钮的背景颜色
			color.setPreferredSize(colorBtnSize);
			jp2.add(color);
			color.addActionListener(ls);
		}

		// 设置窗口可见
		jf.setVisible(true);

		// 获取窗体的画布，并传给监听器对象（必须在设置窗口可见之后）
		ls.g = jp3.getGraphics();

	}

	public static void main(String[] args) {
		DrawUI ui = new DrawUI();
		ui.intiui();
	}
}

3、监听器

package 谢尔宾斯基三角形;

import java.awt.Color;
import java.awt.Graphics;
import java.awt.event.ActionEvent;
import java.awt.event.ActionListener;
import java.awt.event.MouseEvent;
import java.awt.event.MouseListener;
import java.awt.event.MouseMotionListener;
import javax.swing.JButton;

public class Listener implements ActionListener, MouseListener, MouseMotionListener {

	int x1;
	int y1;
	int x2;
	int y2;
	int x3;
	int y3;
	String action;
	Graphics g;
	int[] X = new int[3];
	int[] Y = new int[3];
	int index = 0;
	int n = 3;

	public void x(int x1, int y1, int x2, int y2, int x3, int y3, int n) {
		index = 0;
		if (n > 0) {// 迭代条件
			n--;
			g.drawLine(x1, y1, x2, y2);
			g.drawLine(x3, y3, x2, y2);
			g.drawLine(x1, y1, x3, y3);

			int x4 = (int) ((x1 + x3) / 2);
			int y4 = (int) ((y1 + y3) / 2);
			int x5 = (int) ((x1 + x2) / 2);
			int y5 = (int) ((y2 + y1) / 2);
			int x6 = (int) ((x3 + x2) / 2);
			int y6 = (int) ((y3 + y2) / 2);

			g.drawLine(x4, y4, x5, y5);
			g.drawLine(x6, y6, x5, y5);
			g.drawLine(x4, y4, x6, y6);

			// 迭代
			x(x1, y1, x5, y5, x4, y4, n);
			x(x5, y5, x2, y2, x6, y6, n);
			x(x4, y4, x6, y6, x3, y3, n);
		}
	}

	@Override
	public void mouseClicked(MouseEvent e) {
		if (index < 3) {
			// 获取三个顶点的坐标
			X[index] = e.getX();
			Y[index] = e.getY();
			System.out.println("第" + (index + 1) + "个点");
			g.fillOval(X[index] - 3, Y[index] - 3, 6, 6);
			index++;
		}

		// 把坐标储存在x_i， y_i中
		x1 = X[0];
		x2 = X[1];
		x3 = X[2];
		y1 = Y[0];
		y2 = Y[1];
		y3 = Y[2];
	}

	public void actionPerformed(ActionEvent e) {
		// TODO Auto-generated method stub
		action = e.getActionCommand();
		if (action.equals("")) {// 颜色
			// 获取事件源
			Object srcObj = e.getSource();
			// 获取按钮
			JButton srcBtn = (JButton) srcObj;
			// 获取颜色
			Color color = srcBtn.getBackground();
			g.setColor(color);
		} else if ("清屏".equals(action)) {
			g.clearRect(0, 0, 2000, 2000);
		} else if (action.equals("生成")) {
			x(x1, y1, x2, y2, x3, y3, n);
		}
	}

	public void mousePressed(MouseEvent e) {
		// TODO Auto-generated method stub

	}

	public void mouseReleased(MouseEvent e) {
		// TODO Auto-generated method stub

	}

	public void mouseEntered(MouseEvent e) {
		// TODO Auto-generated method stub

	}

	public void mouseExited(MouseEvent e) {
		// TODO Auto-generated method stub
	}

	public void mouseDragged(MouseEvent e) {
		// TODO Auto-generated method stub

	}

	public void mouseMoved(MouseEvent e) {
		// TODO Auto-generated method stub

	}
}

此时已经可以通过改变颜色和$n$n的值来画出不同颜色和迭代次数的谢尔宾斯基三角形，但是通过改变$n$n的值来改变迭代次数未免太没有交互性，因此在窗体和监听器中添加滑动条，我们就可以通过滑动滑动条的滑块来改变$n$n的值。

4、滑动条

在窗体中

// 创建滑动条
JSlider JS = new JSlider();
// 添加滑动条
jp4.add(JS);
JS.setValue(3);
JS.setSnapToTicks(true);
JS.setMaximum(6);
JS.setMinimum(0);
JS.addChangeListener(ls);

在监听器中

	public void stateChanged(ChangeEvent e) {
		// 获取滑动条对象
		Object Ob = e.getSource();
		JSlider Obj = (JSlider) Ob;
		n = Obj.getValue();
		// 清屏
		g.clearRect(0, 0, 2000, 2000);
		// 画顶点
		for (int i = 0; i < 3; i++) {
			g.fillOval(X[i] - 3, Y[i] - 3, 6, 6);
		}
		// 递归
		x(x1, y1, x2, y2, x3, y3, n);
	}

完整代码

DrawUI.java

package 谢尔宾斯基三角形;

import java.awt.BorderLayout;
import java.awt.Color;
import java.awt.Dimension;
import java.awt.FlowLayout;
import javax.swing.JButton;
import javax.swing.JFrame;
import javax.swing.JPanel;
import javax.swing.JSlider;

public class DrawUI {
	public void intiui() {
		// 创建画图板窗口
		JFrame jf = new JFrame("谢尔宾斯基三角形");

		// 创建监听器对象
		Listener ls = new Listener();

		// 设置面板属性
		jf.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
		jf.setSize(800, 700); // 设置大小为800*700
		jf.setLocationRelativeTo(null);// 设置居中显示
		jf.setLayout(new BorderLayout());// 设置Border布局（JFrame默认为Border布局）

		// 创建Panel（且Panel为流式布局）
		JPanel jp1 = new JPanel();
		JPanel jp2 = new JPanel();
		JPanel jp3 = new JPanel();
		JPanel jp4 = new JPanel();
		jp1.setLayout(new FlowLayout());
		jp2.setLayout(new FlowLayout());
		jp1.setPreferredSize(new Dimension(70, 100));
		jp2.setPreferredSize(new Dimension(40, 100));
		jf.add(jp1, BorderLayout.WEST);
		jf.add(jp2, BorderLayout.EAST);
		jf.add(jp3, BorderLayout.CENTER);
		jf.add(jp4, BorderLayout.SOUTH);
		jp3.addMouseMotionListener(ls);
		jp3.addMouseListener(ls);

		// 创建滑动条
		JSlider JS = new JSlider();
		// 添加滑动条
		jp4.add(JS);
		JS.setValue(3);
		JS.setSnapToTicks(true);
		JS.setMaximum(6);
		JS.setMinimum(0);
		JS.addChangeListener(ls);

		// 用数组和循环创建按钮并添加监听器 格式：类型[] 数组名 = {"变量","变量"...};
		String[] ShapeBtn = { "生成", "清屏" };
		for (int i = 0; i < ShapeBtn.length; i++) {
			JButton shape = new JButton(ShapeBtn[i]);
			jp1.add(shape);
			shape.addActionListener(ls);
		}

		// 创建颜色按钮
		Color[] ColorBtn = { Color.BLUE, Color.GREEN, Color.RED, Color.YELLOW, Color.CYAN, Color.BLACK };
		Dimension colorBtnSize = new Dimension(30, 30); // 设置颜色方块的大小
		for (int i = 0; i < ColorBtn.length; i++) {
			JButton color = new JButton(); // 创建颜色按钮对象
			color.setBackground(ColorBtn[i]);// 设置按钮的背景颜色
			color.setPreferredSize(colorBtnSize);
			jp2.add(color);
			color.addActionListener(ls);
		}

		// 设置窗口可见
		jf.setVisible(true);

		// 获取窗体的画布，并传给监听器对象（必须在设置窗口可见之后）
		ls.g = jp3.getGraphics();

	}

	public static void main(String[] args) {
		DrawUI ui = new DrawUI();
		ui.intiui();
	}
}

Listener.java

package 谢尔宾斯基三角形;

import java.awt.Color;
import java.awt.Graphics;
import java.awt.event.ActionEvent;
import java.awt.event.ActionListener;
import java.awt.event.MouseEvent;
import java.awt.event.MouseListener;
import java.awt.event.MouseMotionListener;
import javax.swing.JButton;
import javax.swing.JSlider;
import javax.swing.event.ChangeEvent;
import javax.swing.event.ChangeListener;

public class Listener implements ActionListener, MouseListener, MouseMotionListener, ChangeListener {

	int x1;
	int y1;
	int x2;
	int y2;
	int x3;
	int y3;
	String action;
	Graphics g;
	int[] X = new int[3];
	int[] Y = new int[3];
	int index = 0;
	int n = 3;

	public void x(int x1, int y1, int x2, int y2, int x3, int y3, int n) {
		index = 0;
		if (n > 0) {
			n--;
			g.drawLine(x1, y1, x2, y2);
			g.drawLine(x3, y3, x2, y2);
			g.drawLine(x1, y1, x3, y3);

			int x4 = (int) ((x1 + x3) / 2);
			int y4 = (int) ((y1 + y3) / 2);
			int x5 = (int) ((x1 + x2) / 2);
			int y5 = (int) ((y2 + y1) / 2);
			int x6 = (int) ((x3 + x2) / 2);
			int y6 = (int) ((y3 + y2) / 2);

			g.drawLine(x4, y4, x5, y5);
			g.drawLine(x6, y6, x5, y5);
			g.drawLine(x4, y4, x6, y6);

			// 迭代
			x(x1, y1, x5, y5, x4, y4, n);
			x(x5, y5, x2, y2, x6, y6, n);
			x(x4, y4, x6, y6, x3, y3, n);
		}
	}

	@Override
	public void mouseClicked(MouseEvent e) {
		if (index < 3) {
			// 获取三个顶点的坐标
			X[index] = e.getX();
			Y[index] = e.getY();
			System.out.println("第" + (index + 1) + "个点");
			g.fillOval(X[index] - 3, Y[index] - 3, 6, 6);
			index++;
		}

		// 把坐标储存在x_i， y_i中
		x1 = X[0];
		x2 = X[1];
		x3 = X[2];
		y1 = Y[0];
		y2 = Y[1];
		y3 = Y[2];
	}

	public void actionPerformed(ActionEvent e) {
		// TODO Auto-generated method stub
		action = e.getActionCommand();
		if (action.equals("")) {// 颜色
			// 获取事件源
			Object srcObj = e.getSource();
			// 获取按钮
			JButton srcBtn = (JButton) srcObj;
			// 获取颜色
			Color color = srcBtn.getBackground();
			g.setColor(color);
		} else if ("清屏".equals(action)) {
			g.clearRect(0, 0, 2000, 2000);
		} else if (action.equals("生成")) {
			x(x1, y1, x2, y2, x3, y3, n);
		}
	}

	public void stateChanged(ChangeEvent e) {
		// 获取滑动条对象
		Object Ob = e.getSource();
		JSlider Obj = (JSlider) Ob;
		n = Obj.getValue();
		// 清屏
		g.clearRect(0, 0, 2000, 2000);
		// 画顶点
		for (int i = 0; i < 3; i++) {
			g.fillOval(X[i] - 3, Y[i] - 3, 6, 6);
		}
		// 递归
		x(x1, y1, x2, y2, x3, y3, n);
	}

	public void mousePressed(MouseEvent e) {
		// TODO Auto-generated method stub

	}

	public void mouseReleased(MouseEvent e) {
		// TODO Auto-generated method stub

	}

	public void mouseEntered(MouseEvent e) {
		// TODO Auto-generated method stub

	}

	public void mouseExited(MouseEvent e) {
		// TODO Auto-generated method stub
	}

	public void mouseDragged(MouseEvent e) {
		// TODO Auto-generated method stub

	}

	public void mouseMoved(MouseEvent e) {
		// TODO Auto-generated method stub

	}
}

运行

1. 在画板中随机选三个点。
2. 点击 “生成”。
3. 滑动底部的滑动条即可改变谢尔宾斯基三角形的迭代次数。

效果

滑动滑动条的滑块：