Solving Cubic Equations and Animating along Polynomial Curves in Java

Using Polynomial equations in Java

In this Java polynomial equation tutorial, you'll learn how to model and animate a moving object along a cubic curve using mathematical formulas. This hands-on project demonstrates how to solve polynomial equations and translate them into visual motion-ideal for senior secondary students learning both math and coding.

Understanding Polynomial and Cubic Equations | Maths Explanation for Java Kids

A polynomial equation expresses a relationship involving powers of a variable. For a cubic curve, the general form is:
y = ax³ + bx² + cx + d;
Here, a, b, c, and d are constants. Every third-degree polynomial equation has both a maximum and a minimum point. These turning points are useful in generating smooth motion when graphing or animating curves with Java.

Figure: Graph of cubic equation and polynomial curve in Java

Deriving the Equation of a Cubic Curve | Maths Explanation for Java Kids

To generate a cubic equation, all we will need are the maximum and minimum points of the curve.

y = ax³ + bx² + cx + d ----- (eqn 0)

By differentiating y = ax³ + bx² + cx + d, we get dy/dx = 3ax² + 2bx + c. Setting the derivative equal to zero at both the maximum and minimum points allows us to calculate a, b, c, and d.

^dy/_dx = y^I = 3ax² + 2bx + c
At maximum point, y^I = 0
y^I|_{(x = x_max)} = 0
3ax_max² + 2bx_max + c = 0 ----- (eqn 1)
At minimum point, y^I = 0
y^I|_{(x = x_min)} = 0 ----- (eqn 2)
3ax_min² + 2bx_min + c = 0
Subtracting both derived equations
y^I|_{(x = x_max)} - y^I|_{(x = x_min)}
⇒ 3a(x_max² - x_min²) + 2b(x_max - x_min) = 0
2b(x_max - x_min) = -3a(x_max² - x_min²)

b =	-3a(x_max - x_min)(x_max + x_min)
	2(x_max - x_min)

b = -³/₂a(x_max + x_min)

Substituting b in (eqn 1)
3ax_max² + 2bx_max + c = 0
3ax_max² + 2(^-3a/₂)(x_max + x_min)x_max + c = 0
3ax_max² - 3ax_max(x_max + x_min) + c = 0
3ax_max² - 3ax_max² - 3ax_maxx_min + c = 0
c = 3ax_maxx_min

From the general equation(eqn 0)
y = ax³ + bx² + cx + d
y_max = ax_max³ + bx_max² + cx_max + d
Substituting for b & c
⇒ y_max = ax_max³ - ³/₂a(x_max + x_min)x_max² + 3ax_maxx_minx_max + d
y_max = ax_max³ - ³/₂ax_max³ - ³/₂ax_max²x_min + 3ax_max²x_min + d
y_max = ¹/₂[2ax_max³ - 3ax_max³ - 3ax_max²x_min + 6ax_max²x_min + 2d]
y_max = ¹/₂[ -ax_max³ + 3ax_max²x_min + 2d]
2y_max = -a(x_max - 3ax_min)x_max² + 2d
2d = 2y_max + a(x_max - 3ax_min)x_max²
d = y_max + ^a/₂(x_max - 3ax_min)x_max²

From the general equation(eqn 0)
y = ax³ + bx² + cx + d
y_max = ax_max³ + bx_max² + cx_max + d
y_min = ax_min³ + bx_min² + cx_min + d
Subtracting both derived equations
y_max - y_min = a(x_max³ - x_min³) + b(x_max² - x_min²) + c(x_max - x_min)
y_max - y_min = (x_max - x_min)[a(x_max² + x_maxx_min + x_min²) + b(x_max + x_min) + c]
Substituting for b & c
y_max - y_min = (x_max - x_min)[a(x_max² + x_maxx_min + x_min²) - ^3a/₂(x_max + x_min)² + 3ax_maxx_min]
y_max - y_min = a(x_max - x_min)[x_max² + x_maxx_min + x_min² - ³/₂(x_max² + 2x_maxx_min + x_min²) + 3x_maxx_min]
2(y_max - y_min) = a(x_max - x_min)[2x_max² + 2x_maxx_min + 2x_min² - 3(x_max² + 2x_maxx_min + x_min²) + 6x_maxx_min]
2(y_max - y_min) = a(x_max - x_min)(2x_max² + 2x_maxx_min + 2x_min² - 3x_max² - 6x_maxx_min - 6x_min² + 6x_maxx_min)
2(y_max - y_min) = a(x_max - x_min)(-x_max² + 2x_maxx_min - x_min²)
2(y_max - y_min) = -a(x_max - x_min)(x_max² - 2x_maxx_min + x_min²)
2(y_max - y_min) = -a(x_max - x_min)(x_max - x_min)²
2(y_max - y_min) = -a(x_max - x_min)³

Hence:

a =	-2(y_max - y_min)
	(x_max - x_min)³

b = -³/₂a(x_max + x_min)
c = 3ax_maxx_min
&
d = y_max + ^a/₂(x_max - 3ax_min)x_max²

These formulas form the mathematical basis of our Java polynomial solver.

Generating and Animating along a Cubic Polynomial Curve in Java

Once we determine the constants, we can implement a Java cubic equation solver to animate motion along the curve. The following example shows how to code a polynomial equation in Java using simple variables and Java canvas graphics.

To animate an object along a polynomial curve, increment x continuously and compute its corresponding y value using the cubic polynomial equation.

This Java code allows you to visualize the trajectory of a polynomial equation by plotting the curve dynamically on a Java canvas. The roots of the polynomial equation and the coefficients determine the shape and symmetry of the curve.

Create a new Java project; call it Dymetric.
Create 3 new Java classes; File, New.
Call them Facet, PanelsCubicPath and CubicPath.
Type out the adjoining Java code for animating an image body through the path of a cubic / polynomial curve.

Key Takeaways on Cubic Path Animation in Java

In this tutorial, you learned how to:

Understand and derive cubic polynomial equations
Find coefficients from maximum and minimum points
Implement a polynomial equation solver using Java
Animate an object along a polynomial curve

By combining algebraic reasoning with code, senior secondary students can see how mathematics powers real-world applications like animation, computer graphics, and game design.

Applications of Polynomial Equations Java Programming and STEM Education

Polynomial equations are used in:

Data modeling and curve fitting
Graphics programming for drawing smooth curves
Physics simulations and motion paths
Machine learning and optimization problems

Learning how to solve polynomial equations in Java provides a strong foundation for both mathematics and computational thinking.

Summary: Visualizing Polynomial Equations in Java

Polynomial equations are powerful tools for generating smooth, curved motion in graphics and animations. In this tutorial, you've learnt how to solve polynomial equations in Java, understand the mathematics of cubic curves, and create a simple animation that moves an image body along a polynomial equation path.

This interactive Java polynomial solver visually demonstrates how mathematical equations can be represented as real motion on a graph. It's a simple yet powerful example of combining coding and mathematics for educational purposes.

So! Java Fun Practice Exercise - Animate along Cubic Path

As a fun practice exercise, try modifying the values of xmax, xmin, ymax, and ymin to observe how they affect the polynomial equation graph. You can also:

Write a function to calculate the roots of the polynomial.
Compare your results with a quadratic equation solver.
Build a reusable polynomial equation solver in Java.

Java Cubic Path Code - Main Class

package dymetric;

public class Dymetric {

    public static void main(String[] args) {
        Facet lcd = new Facet();
        lcd.setVisible(true);
    }

}

Java Cubic Path Window Setup - Facet Class

package dymetric;

import java.awt.*;
import javax.swing.*;

/**
* This class just creates a display window to attach our canvas to.
*/
public class Facet extends JFrame {

    public Container face;
    public ButtonandCanvasPanels components;
    public ImageIcon logo;

    public Facet() {
        // Give our window a title
        super("A window that will hold a Canvas and Button");
        // set window start points and dimensions
        setSize(780,400);
        // close the window when told to
        setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        setResizable(true); // should this window be resizable?

        // It'll be nice to have our own logo (feel free to use yours)
        logo = new ImageIcon(getClass().getResource("studyingPays.png"));
        this.setIconImage(logo.getImage());

        face = this.getContentPane();
        // background colour - may not be necessay since our canvas will have a colour of its own
        face.setBackground(Color.PINK);


        components = new ButtonandCanvasPanels();
        // using the default layout manager
        face.add(components.button_panel, BorderLayout.NORTH);
        // attach appropriate drawing component
        face.add(components.canvas_panel, BorderLayout.CENTER);
    }
}

Java Cubic Path Canvas and Button Controls Class

package dymetric;

import java.awt.event.*;
import java.awt.*;
import javax.swing.*;

public class ButtonandCanvasPanels implements ActionListener {
    public JPanel button_panel, canvas_panel;
    public JButton motion_bttn;
    public CubicPath cub_path;

    public ButtonandCanvasPanels() {
        button_panel = new JPanel();
        // pick a background colour
        button_panel.setBackground(Color.PINK);
        button_panel.setLayout(new FlowLayout(FlowLayout.CENTER, 0, 0));
        // O my; but for convenience sake let's add our control button here
        motion_bttn = new JButton("Move");
        motion_bttn.setBackground(new Color(255, 0, 255));
        motion_bttn.addActionListener(this);
        // using the default layout manager
        button_panel.add(motion_bttn);


        canvas_panel = new JPanel();
        canvas_panel.setLayout(new BorderLayout());
        cub_path = new CubicPath();
        // attach appropriate drawing component
        canvas_panel.add(cub_path, BorderLayout.CENTER);
    }

    /**
     * Respond to the button click event
     */
    public void actionPerformed(ActionEvent evt) {
        cub_path.moveCubic();
    }
}

Java Animation Code for Cubic Path Class

package dymetric;

import java.awt.*;

public class CubicPath extends Canvas {

    protected Color ball_colour;
    protected int x_start = 50;
    protected int x_max = 200;
    protected int y_max = 10;
    protected int x_min = 500;
    protected int y_min = 310;
    protected int x = x_start;
    protected int y;
    protected double a, b, c, d; // coefficients and constant
    protected final int aWIDTH, aHEIGHT;

    public CubicPath() {
        setBackground(Color.LIGHT_GRAY); // canvas color
        ball_colour = Color.RED;
        aWIDTH = aHEIGHT = 10;

        // constants
        a = (double) (-2 * (y_max - y_min)) / Math.pow((x_max - x_min), 3);
        b = -((double) 3 / 2) * a * (x_max + x_min);
        c = 3 * a * x_max * x_min;
        d = y_max + ((double) a / 2) * (x_max - 3 * x_min) * Math.pow(x_max, 2);

        y = (int)Math.round(a * Math.pow(x, 3) + b * Math.pow(x, 2) + c * x + d);
    }

    // Feel free to double buffer if flickering occurs
    public void paint(Graphics g) {
        g.setColor(ball_colour);
        // draw a dot
        g.fillOval(x, y, aWIDTH, aHEIGHT);
    }

    public void moveCubic() {
        // condition for continuing motion
        while (x <= 750 && y >= y_max) {
            paint(this.getGraphics());

            x += 20;
            y = (int)Math.round(a * Math.pow(x, 3) + b * Math.pow(x, 2) + c * x + d);
            // introduce a delay between renderings
            try {
                Thread.sleep(50);
            } catch (InterruptedException e) {
            }
        }
    }
}

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