Ellipse Region Detection in C# | Ellipse Boundary & Collision Detection

Understanding the Ellipse Region | Maths Explanation for C# Kids

In this tutorial, we'll learn how to use C# ellipse region detection to determine whether a point or object lies inside a defined ellipse boundary. This concept combines two important ideas: geometry (the equation of an ellipse) and programming logic (using conditions in C#).

Being able to test if a point lies within an ellipse is useful in many areas — such as collision detection, interactive graphics, and educational simulations. Let's break it down step by step.

Checking the Boundaries of an Ellipse in C# | The Mathematics Behind the Ellipse

We'll use the standard ellipse equation to calculate if an (x, y) coordinate is within the ellipse region in C#.

As explained in the Equation of an Ellipse in C# tutorial, an ellipse centered at (h, k) with semi-major axis a and semi-minor axis b is defined by:

(x - h)²	+	(y - k)²	= 1
a²		b²

Every point (x, y) that satisfies this equation lies on the boundary of the ellipse. If the sum on the left-hand side is less than 1, then the point is inside the ellipse. If it's greater than 1, the point lies outside.

It can be deduced that y = k ± ^b/_a√(a² - (x - h)²) ;
And conversely x = h ± ^a/_b√(b² - (y - k)²)
Hence, the boundaries of any ellipse lie in the range
y ≥ k - ^b/_a√(a² - (x_external - h)²);
y ≤ k + ^b/_a√(a² - (x_external - h)²)
and
x ≥ h - ^a/_b√(b² - (y_external - k)²);
x ≤ h + ^a/_b√(b² - (y_external - k)²)

Tip: The equation (x - h)² / a² + (y - k)² ≤ 1 defines the entire region of the ellipse, not just its outline.

Step-by-Step Explanation for C# Algorithm

Use the ellipse equation to test points.
Apply the test to detect when an object enters the ellipse region.
Visualize the region on the C# windows form.

Code to Detect Entrance into an Elliptical Region in C#

Any point (x, y) that satisfies (x - h)² / a² + (y - k)² ≤ 1 lies inside the ellipse region. We'll translate this into C# code to perform region detection.

To check for when a second body enters the ellipse, we will continually use the x position of this second body in the ellipse equation to detect when its y position lies between the top and bottom limits at the x position in question:
y_{2nd_img}(top) > k - ^b/_a√(a² - (x_{2nd_img} - h)²)
and y_{2nd_img}(bottom) < k + ^b/_a√(a² - (x_{2nd_img} - h)²) ;
At the same time, we will use the y position of the second body in the ellipse equation to detect when its x position lies between the left and right limits at the y position in question:
x_{2nd_img}(left) > h - ^a/_b√(b² - (y_{2nd_img} - k)²)
and x_{2nd_img}(right) < h + ^a/_b√(b² - (y_{2nd_img} - k)²)

Figure: C# elliptical region detection example on C# windows form

Here's a C# code for ellipse region check using the C# windows form element. This approach works well for ellipse region collision detection in graphics or games.

Create a new C# Windows Forms Application project ; call it Dymetric_CS.
Create 2 new C# classes;
Call them Dymetric and EllipticalRegion.
Type out the adjoining C# code for detecting the instance a travelling body crosses the boundary of an ellipse.

By The Way: Notice how the equations for a circle are similar to those of an ellipse;
No surprise there!
A circle is just an ellipse in its simplest form.

How the C# Elliptical Region Detection Code Works

The ellipse is centered at (h, k) with radii a (horizontal) and b (vertical).
For each point (x, y), we calculate ((x - h)² / a²) + ((y - k)² / b²).
If the result is less than or equal to 1, the point lies inside the elliptical region.
Otherwise, it is outside the region.

This same principle is used in collision detection algorithms for games and simulations, where objects have elliptical or circular boundaries.

Applications of Ellipse Region Logic

Graphics and Animation: Detecting when a sprite enters an elliptical area on the canvas.
Mathematics Education: Demonstrating geometric regions and inequalities involving ellipses.
Game Development: Checking collisions or hitboxes shaped like ellipses instead of rectangles.
Data Visualization: Highlighting focus zones or interactive selections shaped as ellipses.

In all these cases, C# ellipse detection helps make interfaces interactive and geometrically accurate.

Key Takeaways on Elliptical Region Detection in C#

In this tutorial, you've learned:

The equation of an ellipse and how to test point positions,
How to use C# and C# windows form to visualize the region,
Practical applications of ellipse region detection in programming and mathematics.

Using C# ellipse boundary code, we can check if a moving object or point enters the defined elliptical region. This method is useful in maths programming and interactive learning for senior secondary students.

This simple concept links algebra, geometry, and coding — showing how mathematics powers real programming!

Summary: Visualizing Elliptical Region in C#

In this tutorial, we learned how to perform ellipse boundary detection in C#. By using the standard ellipse equation, you can efficiently determine whether a point lies inside or outside the elliptical region. This logic is widely used in collision detection, interactive graphics, and data visualization.

So! C# Fun Practice Exercise - Detect Elliptical Region

As a fun practice exercise, try implementing the same C# code but using the parmetric equation of an ellipse this time. This will really validate your understanding of coordinate geometry interpretation and C# graphical programming for ellipse region detection and mathematics application.

C# Elliptical Boundary Window Display Code Stub

Window Display Class Files

C# Elliptical Boundary Code for Dymetric Class

using System.Windows.Forms;

namespace Dymetric
{
    class Dymetric
    {
        private EllipticalRegion ellipse_zone;
        private bool do_simulation;

        public Dymetric(int screen_width, int screen_height)
        {
            ellipse_zone = new EllipticalRegion(screen_width, screen_height);
            do_simulation = false;
        }

        // decide what course of action to take
        public void decideAction(PaintEventArgs e, bool click_check)
        {
            if (do_simulation && click_check)
            {
                // do animation
                ellipse_zone.inPlay(e);
                do_simulation = false;
            }
            else
            {
                // Put ball on screen
                ellipse_zone.clearAndDraw(e);
                do_simulation = true;
            }
        }
    }
}

C# Animation Code for Elliptical Region Class

using System;
using System.Threading;
using System.Drawing;
using System.Windows.Forms;

namespace Dymetric
{
    class EllipticalRegion
    {
        // square coordinates
        private int x_square, y_square, previous_x, previous_y;
        private const int squareLENGTH = 80;

        // circle variables
        private int h, k, a, b;
        private const int dotDIAMETER = 10;

        // we'll be drawing to and from a bitmap image
        private Bitmap offscreen_bitmap;
        Graphics offscreen_g;

        private Brush bg_colour;
        private Pen square_pen;

        public EllipticalRegion(int screen_width, int screen_height)
        {
            square_pen = new Pen(Color.Yellow);
            square_pen.Width = 5;
            bg_colour = new SolidBrush(Color.LightGray);

            // Create bitmap image
            offscreen_bitmap = new Bitmap(screen_width, screen_height - 55,
                System.Drawing.Imaging.PixelFormat.Format24bppRgb);

            // point graphic object to bitmap image
            offscreen_g = Graphics.FromImage(offscreen_bitmap);

            // Set background of bitmap graphic
            offscreen_g.Clear(Color.LightGray);

            previous_x = x_square = 10;
            previous_y = y_square = offscreen_bitmap.Height / 2 - squareLENGTH / 2;

            // ellipse centre coordinates
            h = offscreen_bitmap.Width / 2;
            k = offscreen_bitmap.Height / 2;
            // ellipse major and minor semi-axes
            a = offscreen_bitmap.Width / 3;
            b = offscreen_bitmap.Height / 3;
        }

        // draw first appearance of square on the screen
        public void clearAndDraw(PaintEventArgs e)
        {
            /*
             * draw to offscreen bitmap
             */
            // draw an ellipse
            offscreen_g.DrawEllipse(Pens.Black, h - a, k - b, 2 * a, 2 * b);

            // clear previous square
            offscreen_g.FillRectangle(bg_colour, previous_x - 5, previous_y - 5, squareLENGTH + 10, squareLENGTH + 10);
            // draw square
            offscreen_g.DrawRectangle(square_pen, x_square, y_square, squareLENGTH, squareLENGTH);
            previous_x = x_square;
            previous_y = y_square;

            // draw to screen
            Graphics gr = e.Graphics;
            gr.DrawImage(offscreen_bitmap, 0, 55, offscreen_bitmap.Width, offscreen_bitmap.Height);
        }

        // repetitively clear and draw square on the screen - Simulate motion
        public void inPlay(PaintEventArgs e)
        {
            // condition for continuing motion
            while (x_square < offscreen_bitmap.Width - squareLENGTH)
            {
                int square_left = x_square;
                int square_right = x_square + squareLENGTH;
                int square_top = y_square;
                int square_bottom = y_square + squareLENGTH;

                // determinants for each side of the square
                int x_left_det = (int)Math.Round(((double)b / a)
                        * Math.Sqrt(Math.Pow(a, 2) - Math.Pow((square_left - h), 2)));
                int x_right_det = (int)Math.Round(((double)b / a)
                        * Math.Sqrt(Math.Pow(a, 2) - Math.Pow((square_right - h), 2)));
                int y_up_det = (int)Math.Round(((double)a / b)
                        * Math.Sqrt(Math.Pow(b, 2) - Math.Pow((square_top - k), 2)));
                int y_down_det = (int)Math.Round(((double)a / b)
                        * Math.Sqrt(Math.Pow(b, 2) - Math.Pow((square_bottom - k), 2)));

                // check the bounds of the circle
                square_pen.Dispose();
                // yellow outside the circle
                square_pen = new Pen(Color.Yellow);
                if (square_top > k - x_left_det && square_bottom < k + x_left_det
                    && square_top > k - x_right_det && square_bottom < k + x_right_det
                    && square_left > h - y_up_det && square_right < h + y_up_det
                    && square_left > h - y_down_det && square_right < h + y_down_det)
                {
                    // green inside the circle
                    square_pen = new Pen(Color.Green);
                }
                square_pen.Width = 5;

                // redraw square
                clearAndDraw(e);

                x_square += 10;
                // take a time pause
                Thread.Sleep(50);
            }
            x_square = 10;
            y_square = offscreen_bitmap.Height / 2 - squareLENGTH / 2;
        }
    }
}

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