实施超级椭圆

什么是超椭圆
超椭圆，也称为加布里埃尔·拉梅(GabrielLamé)后的拉梅曲线，是类似于椭圆的闭合曲线，保留了半长轴和半短轴的几何特征，以及它们的对称性，但总体形状不同。
在笛卡尔坐标系中，曲线上所有点(x，y)的集合满足方程

其中n，a和b是正数，竖线| |数字周围表示数字的绝对值。

a = 1和b = 0.75

超椭圆有很多特殊情况，如下图所示：

这些可以通过改变方程式中n的值来实现。因此，现在我们尝试在Python实现此功能，并且需要一些库。

所需模块：

• matplotlib ：绘制方程曲线。它是Python的第3方库，如果您要安装它，请单击此处。
• math ：内置的Python库，其中包含几乎所有数学工具。

# Python program to implement 
# Superellipse
  
# importing the required libraries
import matplotlib.pyplot as plt
from math import sin, cos, pi
   
def sgn(x):
    return ((x>0)-(x<0))*1
  
# parameter for marking the shape  
a, b, n = 200, 200, 2.5
na = 2 / n
# defining the accuracy
step = 100 
piece =(pi * 2)/step
xp =[];yp =[]
   
t = 0
for t1 in range(step + 1):
    # because sin ^ n(x) is mathematically the same as (sin(x))^n...
    x =(abs((cos(t)))**na)*a * sgn(cos(t))
    y =(abs((sin(t)))**na)*b * sgn(sin(t))
    xp.append(x);yp.append(y)
    t+= piece
   
plt.plot(xp, yp) # plotting all point from array xp, yp
plt.title("Superellipse with parameter "+str(n))
plt.show()

输出：

当n = 2.5时

现在让我们看看将n的值更改为0.5时会发生什么

# Python program to implement 
# Superellipse
# importing the required libraries
import matplotlib.pyplot as plt
from math import sin, cos, pi
   
def sgn(x):
    return ((x>0)-(x<0))*1
  
# parameter for marking the shape  
a, b, n = 200, 200, 0.5
na = 2 / n
# defining the accuracy
step = 100 
piece =(pi * 2)/step
xp =[];yp =[]
   
t = 0
for t1 in range(step + 1):
    # because sin ^ n(x) is mathematically the same as (sin(x))^n...
    x =(abs((cos(t)))**na)*a * sgn(cos(t))
    y =(abs((sin(t)))**na)*b * sgn(sin(t))
    xp.append(x);yp.append(y)
    t+= piece
   
plt.plot(xp, yp) # plotting all point from array xp, yp
plt.title("Superellipse with parameter "+str(n))
plt.show()

输出：

Java程序的源代码。

// Java program to implement
// Superellipse
import java.awt.*;
import java.awt.geom.Path2D;
import static java.lang.Math.pow;
import java.util.Hashtable;
import javax.swing.*;
import javax.swing.event.*;
  
public class SuperEllipse extends JPanel implements ChangeListener {
    private double exp = 2.5;
  
    public SuperEllipse()
    {
        setPreferredSize(new Dimension(650, 650));
        setBackground(Color.white);
        setFont(new Font("Serif", Font.PLAIN, 18));
    }
  
    void drawGrid(Graphics2D g)
    {
        g.setStroke(new BasicStroke(2));
        g.setColor(new Color(0xEEEEEE));
  
        int w = getWidth();
        int h = getHeight();
        int spacing = 25;
  
        for (int i = 0; i < w / spacing; i++) {
            g.drawLine(0, i * spacing, w, i * spacing);
            g.drawLine(i * spacing, 0, i * spacing, w);
        }
        g.drawLine(0, h - 1, w, h - 1);
  
        g.setColor(new Color(0xAAAAAA));
        g.drawLine(0, w / 2, w, w / 2);
        g.drawLine(w / 2, 0, w / 2, w);
    }
  
    void drawLegend(Graphics2D g)
    {
        g.setColor(Color.black);
        g.setFont(getFont());
        g.drawString("n = " + String.valueOf(exp), getWidth() - 150, 45);
        g.drawString("a = b = 200", getWidth() - 150, 75);
    }
  
    void drawEllipse(Graphics2D g)
    {
  
        final int a = 200; // a = b
        double[] points = new double[a + 1];
  
        Path2D p = new Path2D.Double();
        p.moveTo(a, 0);
  
        // calculate first quadrant
        for (int x = a; x >= 0; x--) {
            points[x] = pow(pow(a, exp) - pow(x, exp), 1 / exp); // solve for y
            p.lineTo(x, -points[x]);
        }
  
        // mirror to others
        for (int x = 0; x <= a; x++)
            p.lineTo(x, points[x]);
  
        for (int x = a; x >= 0; x--)
            p.lineTo(-x, points[x]);
  
        for (int x = 0; x <= a; x++)
            p.lineTo(-x, -points[x]);
  
        g.translate(getWidth() / 2, getHeight() / 2);
        g.setStroke(new BasicStroke(2));
  
        g.setColor(new Color(0x25B0C4DE, true));
        g.fill(p);
  
        g.setColor(new Color(0xB0C4DE)); // LightSteelBlue
        g.draw(p);
    }
  
    @Override
    public void paintComponent(Graphics gg)
    {
        super.paintComponent(gg);
        Graphics2D g = (Graphics2D)gg;
        g.setRenderingHint(RenderingHints.KEY_ANTIALIASING,
                           RenderingHints.VALUE_ANTIALIAS_ON);
        g.setRenderingHint(RenderingHints.KEY_TEXT_ANTIALIASING,
                           RenderingHints.VALUE_TEXT_ANTIALIAS_ON);
  
        drawGrid(g);
        drawLegend(g);
        drawEllipse(g);
    }
  
    @Override
    public void stateChanged(ChangeEvent e)
    {
        JSlider source = (JSlider)e.getSource();
        exp = source.getValue() / 2.0;
        repaint();
    }
  
    public static void main(String[] args)
    {
        SwingUtilities.invokeLater(() -> {
            JFrame f = new JFrame();
            f.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
            f.setTitle("Super Ellipse");
            f.setResizable(false);
            SuperEllipse panel = new SuperEllipse();
            f.add(panel, BorderLayout.CENTER);
  
            JSlider exponent = new JSlider(JSlider.HORIZONTAL, 1, 9, 5);
            exponent.addChangeListener(panel);
            exponent.setMajorTickSpacing(1);
            exponent.setPaintLabels(true);
            exponent.setBackground(Color.white);
            exponent.setBorder(BorderFactory.createEmptyBorder(20, 20, 20, 20));
  
            Hashtable labelTable = new Hashtable<>();
            for (int i = 1; i < 10; i++)
                labelTable.put(i, new JLabel(String.valueOf(i * 0.5)));
            exponent.setLabelTable(labelTable);
  
            f.add(exponent, BorderLayout.SOUTH);
  
            f.pack();
            f.setLocationRelativeTo(null);
            f.setVisible(true);
        });
    }
}

输出：

参考链接：

1.维基百科-超级椭圆
2. WolframMathWorld –超级椭圆