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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

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 C# tutorial and check for when it enters the region of our curve.

visualizing quadratic curve region in C# and ball trajectory crossing into it on C# windows form
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).

Canvas quadratic region detection C# example
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.









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