Solving Cubic Equations and Animating along Polynomial Curves in VB.Net

Using Polynomial equations in VB.Net

In this VB.Net 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 VB.Net 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 VB.Net.

Figure: Graph of cubic equation and polynomial curve in VB.Net

Deriving the Equation of a Cubic Curve | Maths Explanation for VB.Net 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 VB.Net polynomial solver.

Generating and Animating along a Cubic Polynomial Curve in VB.Net

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

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

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

Create a new Visual Basic Windows Forms Application project ; call it Dymetric_VB.
Create 3 new VB.Net classes;
Call them Facet, Dymetric and CubicPath.
Type out the adjoining VB.Net code for animating an image body through the path of a cubic / polynomial curve.

Key Takeaways on Cubic Path Animation in VB.Net

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 VB.Net
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 VB.Net 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 VB.Net provides a strong foundation for both mathematics and computational thinking.

Summary: Visualizing Polynomial Equations in VB.Net

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 VB.Net, understand the mathematics of cubic curves, and create a simple animation that moves an image body along a polynomial equation path.

This interactive VB.Net 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! VB.Net 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 VB.Net.

VB.Net Cubic Path Window Display Code Stub

Public Class Form1

    Private form_details As New Facet
    Private action_class As New Dymetric

    Private Sub Form1_Load(sender As Object, e As EventArgs) Handles MyBase.Load
        ' Fill in Form - Put button on form
        form_details.formFeatures(sender)
    End Sub

    Private Sub Form1_Paint(sender As Object, e As PaintEventArgs) Handles Me.Paint
        ' Colour button area
        form_details.decorateButtonArea(sender, e)

        ' Call MovingBody class into action
        action_class.decideAction(sender, Me.CreateGraphics(), form_details.CLICK_OCCURRED)

        ' Reset click variable
        form_details.CLICK_OCCURRED = False
    End Sub
End Class

VB.Net Cubic Path Facet Window Code Stub

Public Class Facet

    Dim screen_rect As Rectangle
    Public CLICK_OCCURRED As Boolean = False

    Public Sub formFeatures(sender As Object)
        'Set window position, width and height
        screen_rect = Screen.PrimaryScreen.Bounds
        sender.SetDesktopBounds(0, 0, screen_rect.Width, screen_rect.Height)

        ' Set a display text
        sender.Text = "useOfMaths.com"

        ' Set a background colour
        sender.BackColor = System.Drawing.Color.LightGray

        ' Set an icon image
        Dim path = System.IO.Path.GetDirectoryName(System.Reflection.Assembly.GetExecutingAssembly().CodeBase)
        path = New Uri(path).LocalPath
        Try
            sender.Icon = New Icon(path & "\useOfMaths.ico")
        Catch ex As System.IO.FileNotFoundException
            ' Well, just go on and use default pic
        End Try

        '
        'create a button - response_btn
        '
        Dim response_btn As New Button()
        response_btn.BackColor = System.Drawing.Color.Magenta
        response_btn.ForeColor = System.Drawing.Color.Blue
        response_btn.Name = "response_btn"
        response_btn.SetBounds(CInt(Math.Round(screen_rect.Width / 2)) - 50, 5, 100, 40)
        response_btn.Text = "Move"
        sender.Controls.Add(response_btn)
        AddHandler response_btn.Click, AddressOf response_btn_Click
    End Sub

    Public Sub decorateButtonArea(sender As Object, e As PaintEventArgs)
        ' Draw a dotted line
        Dim pencil As New System.Drawing.Pen(System.Drawing.Color.Black)
        pencil.DashStyle = Drawing2D.DashStyle.DashDot
        pencil.Width = 5
        e.Graphics.DrawLine(pencil, 0, 50, sender.Width, 50)
        pencil.Dispose()

        ' Colour region
        Dim paint_brush As New System.Drawing.SolidBrush(System.Drawing.Color.Pink)
        e.Graphics.FillRectangle(paint_brush, 0, 0, sender.Width, 50)
        paint_brush.Dispose()
    End Sub

    Public Sub response_btn_Click(sender As Object, e As EventArgs)
        ' turn this on on every button click
        CLICK_OCCURRED = True
        sender.Refresh()
    End Sub
End Class

VB.Net Cubic Path Code for Dymetric Class

Public Class Dymetric
    Private cube_curve As New CubicPath
    Private do_simulation = False

    ' decide what course of action to take
    Public Sub decideAction(sender As Object, g As Graphics, click_check As Boolean)
        If do_simulation And click_check Then
            ' do animation
            cube_curve.play(sender, g)
            do_simulation = False
        Else
            ' Put ball on screen
            cube_curve.prep(sender, g)
            do_simulation = True
        End If
    End Sub
End Class

VB.Net Animation Code for Cubic Path Class

Public Class CubicPath

    Private x_max, y_max, x_min, y_min, x, y As Integer
    Private a, b, c, d As Double
    Private Const dotDIAMETER = 10

    Dim dot_colour As New System.Drawing.SolidBrush(System.Drawing.Color.Yellow)
    Dim bg_colour As New System.Drawing.SolidBrush(System.Drawing.Color.LightGray)

    ' draw first appearance of dot on the screen
    Public Sub prep(sender As Object, g As Graphics)
        x = 20
        x_max = Math.Round(sender.Width / 4) + 10
        y_max = 70
        x_min = Math.Round(3 * sender.Width / 4) - 10
        y_min = sender.Height - 70

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

        y = CInt(Math.Round(a * Math.Pow(x, 3) + b * Math.Pow(x, 2) + c * x + d))

        ' clear entire used canvas area
        g.FillRectangle(bg_colour, 0, 60, sender.Width, sender.Height)
        ' draw dot
        g.FillEllipse(dot_colour, x, y, dotDIAMETER, dotDIAMETER)
    End Sub

    ' repetitively clear and draw dot on the screen - Simulate motion
    Public Sub play(sender As Object, g As Graphics)
        ' condition for continuing motion
        Do While x < sender.Width - dotDIAMETER And y >= y_max
            ' redraw dot
            g.FillEllipse(dot_colour, x, y, dotDIAMETER, dotDIAMETER)

            x += 20
            y = CInt(Math.Round(a * Math.Pow(x, 3) + b * Math.Pow(x, 2) + c * x + d))
            ' take a time pause
            Threading.Thread.Sleep(50)
        Loop
    End Sub
End Class

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