Detect a Point Inside a Quadratic Region Using C# | Senior Secondary Maths Tutorial
Understanding the Quadratic Region Concept | Maths Explanation for C# Kids
In this tutorial, you'll learn how to detect the region under a quadratic curve using C#.
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 C# windows form.
What is a Quadratic Region? | Maths Explanation for C# Kids
A quadratic region in C# 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 C#
To visualize the region under a quadratic curve, we'll use C# 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.
Remember as discussed in the Animating along a Straight Line in C# tutorial,
that 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
Figure: Visualizing quadratic curve region in C# and an object trajectory passing through it on C# windows form.
C# 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).
Figure: Detecting and visualizing the quadratic region on a C# windows form using C#.
To detect a point inside a parabola using C#,
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 C# Windows Forms Application project
;
call it Dymetric_CS.
Create 2 new classes;
Call them Dymetric and QuadraticRegion.
Type out the adjoining C# code for detecting the instance a travelling
body crosses the boundary of a quadratic curve.
Summary: Detecting Quadratic Boundaries with C#
In this senior secondary C# math tutorial, you've learnt how to
identify whether a moving point lies inside a quadratic region.
We've used simple algebra and the C# 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 C#
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 C#
This tutorial teaches you to:
Compute the region under a quadratic function in C#
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 C# quadratic region detection function.
This approach is often used in interactive canvas demos and
collision detection algorithms.
So! C# 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 C# windows form.
Experiment with different equations and visualize how region boundaries change dynamically in C#.
This is a great way to explore the relationship between algebra and geometry in senior secondary mathematics.
public Dymetric(int screen_width, int screen_height)
{
quad_region = new QuadraticRegion(screen_width, screen_height);
do_simulation = false;
}
// decide what course of action to take publicvoid decideAction(PaintEventArgs e, bool click_check)
{ if (do_simulation && click_check)
{ // do animation
quad_region.inPlay(e);
do_simulation = false;
} else
{ // Put ball on screen
quad_region.clearAndDraw(e);
do_simulation = true;
}
}
}
}
C# Animation Code for Quadratic Region Class
using System; using System.Threading; using System.Drawing; using System.Windows.Forms;
// draw first appearance of ball on the screen publicvoid clearAndDraw(PaintEventArgs e)
{ /*
* draw to offscreen bitmap
*/ // draw quadratic curve for (x = xq_start; x <= xq_stop; x++)
{
y = (int)Math.Round(a * x * x + b * x + c); // redraw dot
offscreen_g.FillEllipse(dot_colour, x, y, dotDIAMETER, dotDIAMETER);
}