What is a superellipse
A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape.
In the Cartesian coordinate system, the set of all points (x, y) on the curve satisfy the equation
where n, a and b are positive numbers, and the vertical bars | | around a number indicate the absolute value of the number.
a = 1 and b = 0.75
There are many number of specific cases of superellipse like given in the image down below:
These can be achieved by varying the value of n in the equation. So now we try to implement this in python and to do that we are require some libraries.
Modules Required:
- matplotlib: To plot the curve of the equation. Its an 3rd party library in python and if you want to install it click here.
- math : Its an built in library of python which have almost all the mathematical tools.
# 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, 2.5na = 2 / n # defining the accuracy step = 100 piece =(pi * 2)/step xp =[];yp =[] t = 0for 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() |
Output:
when n = 2.5
Now let’s see what happens when we changes the value of n to 0.5
# 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, 0.5na = 2 / n # defining the accuracy step = 100 piece =(pi * 2)/step xp =[];yp =[] t = 0for 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() |
Output:
Source Code of the program in Java.
// 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<Integer, JLabel> 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); }); } } |
Output:
Reference Links:
1. Wikipedia – Superellipse
2. WolframMathWorld – Superellipse
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