Understanding the Quadratic Region Concept | Maths Explanation for VB.Net Kids
In this tutorial, you'll learn how to detect the region under a quadratic curve using VB.Net. The curve is defined by the equation y = a x² + b x + c, and we'll use the discriminant method to find when a point or object lies within the quadratic region. This concept helps students connect algebraic reasoning with programming and visualization using the VB.Net windows form.
What is a Quadratic Region? | Maths Explanation for VB.Net Kids
A quadratic region in VB.Net represents the area bounded by a quadratic curve.
Every quadratic equation has two x-values (roots) for any given y - except at its turning point (maximum or minimum).
We can use these roots as boundaries for region detection.
More technically, a quadratic region is the area defined by a quadratic inequality such as
y ≤ ax² + bx + c.
This concept is useful in computer graphics, physics simulations, and
quadratic curve collision detection (JS) projects.
Checking the Boundaries of a Quadratic Curve in VB.Net
To visualize the region under a quadratic curve, we'll use VB.Net to calculate the upper and lower limits dynamically. This makes it possible to detect when an object (like a moving ball) enters or exits the quadratic region.
As discussed in the Animating along a Straight Line in VB.Net tutorial,
any quadratic equation always have two roots for any value of y (except at it's maximum or minimum point).
All we need to do is use these two roots (x values) as boundaries for our check.
y = ax2 + bx + c
ax2 + bx + (c-y) = 0
| x = | -b ± √(b2 - 4a(c-y)) |
| 2a |
Our range will then be:
| -b - √(b2 - 4a(c-y)) | ≤ x ≤ | -b + √(b2 - 4a(c-y)) |
| 2a | 2a |
where a, b, and c are constants.
We will reuse the moving ball graphic from the
Solving and Animating a Quadratic Equation in VB.Net
tutorial and check for when it enters the region of our curve.
VB.Net Code Example: Detecting Entrance into a Quadratic Region
To check for when our ball enters the quadratic curve, we will continually check the x position
of the ball against the x position gotten using the quadratic equation at the same y position
as that of the ball.
We'll designate the coordinates of the ball as (xb, yb),
and those of the curve as (xq, yq).
To detect a point inside a parabola using VB.Net,
you can compare its coordinates to the quadratic curve.
We'll determine whether a moving ball lies within this region by solving
for x using the quadratic formula.
If y is less than or equal to the value of the quadratic equation,
the point lies within the region.
Create a new Visual Basic Windows Forms Application
project
;
call it Dymetric_VB.
Create 3 new VB.Net classes;
Call them Facet, Dymetric and QuadraticRegion.
Type out the adjoining VB.Net code for detecting the instance a travelling
body crosses the boundary of a quadratic curve.
Summary: Detecting Quadratic Boundaries with VB.Net
In this senior secondary VB.Net math tutorial, you've learnt how to identify whether a moving point lies inside a quadratic region. We've used simple algebra and the VB.Net canvas to visualize and draw the quadratic region bounded by a parabolic curve.
Formula Recap:
The general form of a quadratic equation is y = a x² + b x + c. To find the region under the curve, we can rearrange this equation to get a x² + b x + (c - y) = 0 and use the discriminant D = b² - 4a(c - y).
For any given y-value, if D is positive, the quadratic crosses that y-level at two x-values. The region between these two x-values represents the quadratic region.
y = ax² + bx + c
⟹ ax² + bx + (c - y) = 0
⟹ x = (-b ± √(b² - 4a(c - y))) / 2a
Thus, the quadratic region boundaries are:
(-b - √(b² - 4a(c - y))) / 2a ≤ x ≤ (-b + √(b² - 4a(c - y))) / 2a
Understanding how to compute and visualize quadratic regions in VB.Net bridges mathematical theory and practical coding. It helps students apply concepts from coordinate geometry in a real-world programming context.
Applying the Line Region Detection Logic in VB.Net
This tutorial teaches you to:
- Compute the region under a quadratic function in VB.Net
- Use real-time region detection to track an object's position
- Apply mathematical concepts like discriminants and boundaries in interactive graphics
To determine if a point lies inside a quadratic region, we've used a VB.Net quadratic region detection function. This approach is often used in interactive canvas demos and collision detection algorithms.
So! VB.Net Fun Practice Exercise - Detect Quadratic curve Boundary
As a fun practice exercise, try experimenting with different coefficients (a, b, and c)
to see how the quadratic region changes shape.
You can also animate a point moving across the screen to test when it enters or exits the region on the VB.Net windows form.
Experiment with different equations and visualize how region boundaries change dynamically in VB.Net.
This is a great way to explore the relationship between algebra and geometry in senior secondary mathematics.
VB.Net Quadratic Curve Boundary Window Display Code Stub
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 Quadratic Curve Boundary Facet Window Code Stub
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 Quadratic Curve Boundary Code for Dymetric Class
Private quad_region As New QuadraticRegion
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
quad_region.play(sender, g)
do_simulation = False
Else
' Put ball on screen
quad_region.prep(sender, g)
do_simulation = True
End If
End Sub
End Class
VB.Net Animation Code for Quadratic Region Class
' ball coordinates
Private x_ball, y_ball As Integer
Private previous_x As Integer = 0
Private previous_y As Integer = 0
Private Const ballDIAMETER = 80
Dim ball_colour As New System.Drawing.SolidBrush(System.Drawing.Color.Yellow)
' quadratic variables
Private xq_start, yq_start, xq_min, yq_min, xq_stop, x, y As Integer
Private xq_lb, xq_ub As Double ' curve lower and upper boundary
Private a, b, c, discriminant As Double
Private Const dotDIAMETER = 5
Dim dot_colour As New System.Drawing.SolidBrush(System.Drawing.Color.Black)
Dim bg_colour As New System.Drawing.SolidBrush(System.Drawing.Color.LightGray)
' draw first appearance of ball on the screen
Public Sub prep(sender As Object, g As Graphics)
x_ball = 10
y_ball = Math.Round(sender.Height / 2)
ball_colour = New System.Drawing.SolidBrush(System.Drawing.Color.Yellow)
xq_start = Math.Round(sender.Width / 2) - 200
yq_start = 70
xq_min = Math.Round(sender.Width / 2)
yq_min = sender.Height - 70
xq_stop = Math.Round(sender.Width / 2) + 200
' constants
a = (yq_start - yq_min) / Math.Pow((xq_start - xq_min), 2)
b = -2 * a * xq_min
c = yq_min + a * Math.Pow(xq_min, 2)
discriminant = Math.Sqrt(b * b - 4 * a * (c - (y_ball - ballDIAMETER / 2)))
If a < 0 Then ' a is negative
xq_lb = (-b + discriminant) / (2 * a) ' lower boundary
xq_ub = (-b - discriminant) / (2 * a) ' upper boundary
Else
xq_lb = (-b - discriminant) / (2 * a) ' lower boundary
xq_ub = (-b + discriminant) / (2 * a) ' upper boundary
End If
' draw quadratic curve
For x = xq_start To xq_stop
y = CInt(Math.Round(a * x * x + b * x + c))
' redraw dot
g.FillEllipse(dot_colour, x, y, dotDIAMETER, dotDIAMETER)
Next x
If previous_x > 0 Then
' clear previous ball using background colour
g.FillEllipse(bg_colour, previous_x, previous_y, ballDIAMETER, ballDIAMETER)
End If
' draw ball
g.FillEllipse(ball_colour, x_ball, y_ball, ballDIAMETER, ballDIAMETER)
previous_x = x_ball
previous_y = y_ball
End Sub
' repetitively clear and draw ball on the screen - Simulate motion
Public Sub play(sender As Object, g As Graphics)
' condition for continuing motion
Do While x_ball < sender.Width - ballDIAMETER
' yellow outside the quadratic region
ball_colour = New System.Drawing.SolidBrush(System.Drawing.Color.Yellow)
If (x_ball <= xq_lb And x_ball + ballDIAMETER >= xq_lb) _
Or (x_ball <= xq_ub And x_ball + ballDIAMETER >= xq_ub) Then
' red on the quadratic outline
ball_colour = New System.Drawing.SolidBrush(System.Drawing.Color.Red)
ElseIf x_ball >= xq_lb And x_ball + ballDIAMETER <= xq_ub Then
' green inside the quadratic region
ball_colour = New System.Drawing.SolidBrush(System.Drawing.Color.Green)
End If
' clear previous ball using background colour
g.FillEllipse(bg_colour, previous_x, previous_y, ballDIAMETER, ballDIAMETER)
' redraw ball
g.FillEllipse(ball_colour, x_ball, y_ball, ballDIAMETER, ballDIAMETER)
previous_x = x_ball
x_ball += 5
' take a time pause
Threading.Thread.Sleep(50)
Loop
End Sub
End Class