Mid-Point Circle Drawing Algorithm

The mid-point circle drawing algorithm is an algorithm used to determine the points needed for rasterizing a circle. 

We use the mid-point algorithm to calculate all the perimeter points of the circle in the first octant and then print them along with their mirror points in the other octants. This will work because a circle is symmetric about its centre.

The algorithm is very similar to the Mid-Point Line Generation Algorithm. Here, only the boundary condition is different.

For any given pixel (x, y), the next pixel to be plotted is either (x, y+1) or (x-1, y+1). This can be decided by following the steps below.

1. Find the mid-point p of the two possible pixels i.e (x-0.5, y+1)
2. If p lies inside or on the circle perimeter, we plot the pixel (x, y+1), otherwise if it’s outside we plot the pixel (x-1, y+1)

Boundary Condition : Whether the mid-point lies inside or outside the circle can be decided by using the formula:- 

Given a circle centered at (0,0) and radius r and a point p(x,y) 
F(p) = x² + y² – r²
if F(p)<0, the point is inside the circle
F(p)=0, the point is on the perimeter
F(p)>0, the point is outside the circle 

In our program, we denote F(p) with P. The value of P is calculated at the mid-point of the two contending pixels i.e. (x-0.5, y+1). Each pixel is described with a subscript k.

P_k = (X_k — 0.5)² + (y_k + 1)² – r²
Now, 
x_k+1 = x_k or x_k-1 , y_k+1= y_k +1
? P_k+1 = (x_k+1 – 0.5)² + (y_k+1 +1)² – r²
= (x_k+1 – 0.5)² + [(y_k +1) + 1]² – r²
= (x_k+1 – 0.5)² + (y_k +1)² + 2(y_k + 1) + 1 – r²
= (x_k+1 – 0.5)² + [ – (x_k – 0.5)² +(x_k – 0.5)² ] + (y_k + 1)² – r² + 2(y_k + 1) + 1
= P_k + (x_k+1 – 0.5)² – (x_k – 0.5)² + 2(y_k + 1) + 1 
= P_k + (x²_k+1 – x²_k) – (x_k+1 – x_k) + 2(y_k + 1) + 1 
= P_k + 2(y_k +1) + 1, when P_k <=0 i.e the midpoint is inside the circle 
(x_k+1 = x_k) 
P_k + 2(y_k +1) – 2(x_k – 1) + 1, when P_k>0 I.e the mid point is outside the circle(x_k+1 = x_k-1) 

The first point to be plotted is (r, 0) on the x-axis. The initial value of P is calculated as follows:-

P1 = (r – 0.5)² + (0+1)² – r²
= 1.25 – r 
= 1 -r (When rounded off)

Examples:

Input : Centre -> (0, 0), Radius -> 3
Output : (3, 0) (3, 0) (0, 3) (0, 3)
         (3, 1) (-3, 1) (3, -1) (-3, -1)
         (1, 3) (-1, 3) (1, -3) (-1, -3)
         (2, 2) (-2, 2) (2, -2) (-2, -2)

Input : Centre -> (4, 4), Radius -> 2
Output : (6, 4) (6, 4) (4, 6) (4, 6)
         (6, 5) (2, 5) (6, 3) (2, 3)
         (5, 6) (3, 6) (5, 2) (3, 2)

CPP

// C++ program for implementing
// Mid-Point Circle Drawing Algorithm
#include<iostream>
using namespace std;

// Implementing Mid-Point Circle Drawing Algorithm
void midPointCircleDraw(int x_centre, int y_centre, int r)
{
    int x = r, y = 0;

    // Printing the initial point on the axes
    // after translation
    cout << "(" << x + x_centre << ", " << y + y_centre << ") ";

    // When radius is zero only a single
    // point will be printed
    if (r > 0)
    {
        cout << "(" << x + x_centre << ", " << -y + y_centre << ") ";
        cout << "(" << y + x_centre << ", " << x + y_centre << ") ";
        cout << "(" << -y + x_centre << ", " << x + y_centre << ")\n";
    }

    // Initialising the value of P
    int P = 1 - r;
    while (x > y)
    {
        y++;

        // Mid-point is inside or on the perimeter
        if (P <= 0)
            P = P + 2*y + 1;
        // Mid-point is outside the perimeter
        else
        {
            x--;
            P = P + 2*y - 2*x + 1;
        }

        // All the perimeter points have already been printed
        if (x < y)
            break;

        // Printing the generated point and its reflection
        // in the other octants after translation
        cout << "(" << x + x_centre << ", " << y + y_centre << ") ";
        cout << "(" << -x + x_centre << ", " << y + y_centre << ") ";
        cout << "(" << x + x_centre << ", " << -y + y_centre << ") ";
        cout << "(" << -x + x_centre << ", " << -y + y_centre << ")\n";

        // If the generated point is on the line x = y then
        // the perimeter points have already been printed
        if (x != y)
        {
            cout << "(" << y + x_centre << ", " << x + y_centre << ") ";
            cout << "(" << -y + x_centre << ", " << x + y_centre << ") ";
            cout << "(" << y + x_centre << ", " << -x + y_centre << ") ";
            cout << "(" << -y + x_centre << ", " << -x + y_centre << ")\n";
        }
    }
}

// Driver code
int main()
{
    // To draw a circle of radius 3 centered at (0, 0)
    midPointCircleDraw(0, 0, 3);
    return 0;
}

C

// C program for implementing
// Mid-Point Circle Drawing Algorithm
#include<stdio.h>

// Implementing Mid-Point Circle Drawing Algorithm
void midPointCircleDraw(int x_centre, int y_centre, int r)
{
    int x = r, y = 0;

    // Printing the initial point on the axes
    // after translation
    printf("(%d, %d) ", x + x_centre, y + y_centre);

    // When radius is zero only a single
    // point will be printed
    if (r > 0)
    {
        printf("(%d, %d) ", x + x_centre, -y + y_centre);
        printf("(%d, %d) ", y + x_centre, x + y_centre);
        printf("(%d, %d)\n", -y + x_centre, x + y_centre);
    }

    // Initialising the value of P
    int P = 1 - r;
    while (x > y)
    {
        y++;

        // Mid-point is inside or on the perimeter
        if (P <= 0)
            P = P + 2*y + 1;

        // Mid-point is outside the perimeter
        else
        {
            x--;
            P = P + 2*y - 2*x + 1;
        }

        // All the perimeter points have already been printed
        if (x < y)
            break;

        // Printing the generated point and its reflection
        // in the other octants after translation
        printf("(%d, %d) ", x + x_centre, y + y_centre);
        printf("(%d, %d) ", -x + x_centre, y + y_centre);
        printf("(%d, %d) ", x + x_centre, -y + y_centre);
        printf("(%d, %d)\n", -x + x_centre, -y + y_centre);

        // If the generated point is on the line x = y then
        // the perimeter points have already been printed
        if (x != y)
        {
            printf("(%d, %d) ", y + x_centre, x + y_centre);
            printf("(%d, %d) ", -y + x_centre, x + y_centre);
            printf("(%d, %d) ", y + x_centre, -x + y_centre);
            printf("(%d, %d)\n", -y + x_centre, -x + y_centre);
        }
    }
}

// Driver code
int main()
{
    // To draw a circle of radius 3 centered at (0, 0)
    midPointCircleDraw(0, 0, 3);
    return 0;
}

Java

// Java program for implementing
// Mid-Point Circle Drawing Algorithm
class GFG {

    // Implementing Mid-Point Circle
    // Drawing Algorithm
    static void midPointCircleDraw(int x_centre,
                            int y_centre, int r)
    {

        int x = r, y = 0;

        // Printing the initial point
        // on the axes after translation
        System.out.print("(" + (x + x_centre)
                + ", " + (y + y_centre) + ")");

        // When radius is zero only a single
        // point will be printed
        if (r > 0) {

            System.out.print("(" + (x + x_centre)
                + ", " + (-y + y_centre) + ")");

            System.out.print("(" + (y + x_centre)
                 + ", " + (x + y_centre) + ")");

            System.out.println("(" + (-y + x_centre)
                   + ", " + (x + y_centre) + ")");
        }

        // Initialising the value of P
        int P = 1 - r;
        while (x > y) {

            y++;

            // Mid-point is inside or on the perimeter
            if (P <= 0)
                P = P + 2 * y + 1;

            // Mid-point is outside the perimeter
            else {
                x--;
                P = P + 2 * y - 2 * x + 1;
            }

            // All the perimeter points have already
            // been printed
            if (x < y)
                break;

            // Printing the generated point and its
            // reflection in the other octants after
            // translation
            System.out.print("(" + (x + x_centre)
                    + ", " + (y + y_centre) + ")");

            System.out.print("(" + (-x + x_centre)
                    + ", " + (y + y_centre) + ")");

            System.out.print("(" + (x + x_centre) +
                    ", " + (-y + y_centre) + ")");

            System.out.println("(" + (-x + x_centre)
                    + ", " + (-y + y_centre) + ")");

            // If the generated point is on the
            // line x = y then the perimeter points
            // have already been printed
            if (x != y) {

                System.out.print("(" + (y + x_centre)
                      + ", " + (x + y_centre) + ")");

                System.out.print("(" + (-y + x_centre)
                      + ", " + (x + y_centre) + ")");

                System.out.print("(" + (y + x_centre)
                      + ", " + (-x + y_centre) + ")");

                System.out.println("(" + (-y + x_centre)
                    + ", " + (-x + y_centre) +")");
            }
        }
    }

    // Driver code
    public static void main(String[] args) {

        // To draw a circle of radius
        // 3 centered at (0, 0)
        midPointCircleDraw(0, 0, 3);
    }
}

// This code is contributed by Anant Agarwal.

Python3

# Python3 program for implementing
# Mid-Point Circle Drawing Algorithm

def midPointCircleDraw(x_centre, y_centre, r):
    x = r
    y = 0

    # Printing the initial point the
    # axes after translation
    print("(", x + x_centre, ", ",
               y + y_centre, ")",
               sep = "", end = "")

    # When radius is zero only a single
    # point be printed
    if (r > 0) :

        print("(", x + x_centre, ", ",
                  -y + y_centre, ")",
                  sep = "", end = "")
        print("(", y + x_centre, ", ",
                   x + y_centre, ")",
                   sep = "", end = "")
        print("(", -y + x_centre, ", ",
                    x + y_centre, ")", sep = "")

    # Initialising the value of P
    P = 1 - r

    while x > y:

        y += 1

        # Mid-point inside or on the perimeter
        if P <= 0:
            P = P + 2 * y + 1

        # Mid-point outside the perimeter
        else:
            x -= 1
            P = P + 2 * y - 2 * x + 1

        # All the perimeter points have
        # already been printed
        if (x < y):
            break

        # Printing the generated point its reflection
        # in the other octants after translation
        print("(", x + x_centre, ", ", y + y_centre,
                            ")", sep = "", end = "")
        print("(", -x + x_centre, ", ", y + y_centre,
                             ")", sep = "", end = "")
        print("(", x + x_centre, ", ", -y + y_centre,
                             ")", sep = "", end = "")
        print("(", -x + x_centre, ", ", -y + y_centre,
                                        ")", sep = "")

        # If the generated point on the line x = y then
        # the perimeter points have already been printed
        if x != y:

            print("(", y + x_centre, ", ", x + y_centre,
                                ")", sep = "", end = "")
            print("(", -y + x_centre, ", ", x + y_centre,
                                 ")", sep = "", end = "")
            print("(", y + x_centre, ", ", -x + y_centre,
                                 ")", sep = "", end = "")
            print("(", -y + x_centre, ", ", -x + y_centre,
                                            ")", sep = "")

# Driver Code
if __name__ == '__main__':

    # To draw a circle of radius 3
    # centered at (0, 0)
    midPointCircleDraw(0, 0, 3)


# Contributed by: SHUBHAMSINGH10
# Improved by: siddharthx_07

C#

// C# program for implementing Mid-Point
// Circle Drawing Algorithm
using System;

class GFG {

    // Implementing Mid-Point Circle
    // Drawing Algorithm
    static void midPointCircleDraw(int x_centre,
                            int y_centre, int r)
    {

        int x = r, y = 0;

        // Printing the initial point on the
        // axes after translation
        Console.Write("(" + (x + x_centre)
                + ", " + (y + y_centre) + ")");

        // When radius is zero only a single
        // point will be printed
        if (r > 0)
        {

            Console.Write("(" + (x + x_centre)
                + ", " + (-y + y_centre) + ")");

            Console.Write("(" + (y + x_centre)
                + ", " + (x + y_centre) + ")");

            Console.WriteLine("(" + (-y + x_centre)
                + ", " + (x + y_centre) + ")");
        }

        // Initialising the value of P
        int P = 1 - r;
        while (x > y)
        {

            y++;

            // Mid-point is inside or on the perimeter
            if (P <= 0)
                P = P + 2 * y + 1;

            // Mid-point is outside the perimeter
            else
            {
                x--;
                P = P + 2 * y - 2 * x + 1;
            }

            // All the perimeter points have already
            // been printed
            if (x < y)
                break;

            // Printing the generated point and its
            // reflection in the other octants after
            // translation
            Console.Write("(" + (x + x_centre)
                    + ", " + (y + y_centre) + ")");

            Console.Write("(" + (-x + x_centre)
                    + ", " + (y + y_centre) + ")");

            Console.Write("(" + (x + x_centre) +
                    ", " + (-y + y_centre) + ")");

            Console.WriteLine("(" + (-x + x_centre)
                    + ", " + (-y + y_centre) + ")");

            // If the generated point is on the
            // line x = y then the perimeter points
            // have already been printed
            if (x != y)
            {
                Console.Write("(" + (y + x_centre)
                    + ", " + (x + y_centre) + ")");

                Console.Write("(" + (-y + x_centre)
                    + ", " + (x + y_centre) + ")");

                Console.Write("(" + (y + x_centre)
                    + ", " + (-x + y_centre) + ")");

                Console.WriteLine("(" + (-y + x_centre)
                    + ", " + (-x + y_centre) +")");
            }
        }
    }

    // Driver code
    public static void Main()
    {

        // To draw a circle of radius
        // 3 centered at (0, 0)
        midPointCircleDraw(0, 0, 3);
    }
}

// This code is contributed by nitin mittal.

PHP

<?php
// PHP program for implementing
// Mid-Point Circle Drawing Algorithm

// Implementing Mid-Point
// Circle Drawing Algorithm
function midPointCircleDraw($x_centre,
                            $y_centre,
                            $r)
{
    $x = $r;
    $y = 0;

    // Printing the initial
    // point on the axes
    // after translation
    echo "(",$x + $x_centre,",", $y + $y_centre,")";

    // When radius is zero only a single
    // point will be printed
    if ($r > 0)
    {
        echo "(",$x + $x_centre,",", -$y + $y_centre,")";
        echo "(",$y + $x_centre,",", $x + $y_centre,")";
        echo "(",-$y + $x_centre,",", $x + $y_centre,")","\n";
    }

    // Initializing the value of P
    $P = 1 - $r;
    while ($x > $y)
    {
        $y++;

        // Mid-point is inside
        // or on the perimeter
        if ($P <= 0)
            $P = $P + 2 * $y + 1;

        // Mid-point is outside
        // the perimeter
        else
        {
            $x--;
            $P = $P + 2 * $y -
                  2 * $x + 1;
        }

        // All the perimeter points
        // have already been printed
        if ($x < $y)
            break;

        // Printing the generated
        // point and its reflection
        // in the other octants
        // after translation
        echo "(",$x + $x_centre,",", $y + $y_centre,")";
        echo "(",-$x + $x_centre,",", $y + $y_centre,")";
        echo "(",$x +$x_centre,",", -$y + $y_centre,")";
        echo "(",-$x + $x_centre,",", -$y + $y_centre,")","\n";

        // If the generated point is
        // on the line x = y then
        // the perimeter points have
        // already been printed
        if ($x != $y)
        {
            echo "(",$y + $x_centre,",", $x + $y_centre,")";
            echo "(",-$y + $x_centre,",", $x + $y_centre,")";
            echo "(",$y + $x_centre,",", -$x + $y_centre,")";
            echo "(",-$y + $x_centre,",", -$x + $y_centre,")","\n";
        }
    }
}

    // Driver code
    // To draw a circle of radius
    // 3 centered at (0, 0)
    midPointCircleDraw(0, 0, 3);

// This code is contributed by nitin mittal.
?>

Javascript

<script>
// javascript program for implementing
// Mid-Point Circle Drawing Algorithm
// Implementing Mid-Point Circle
    // Drawing Algorithm
    function midPointCircleDraw(x_centre , y_centre , r) {

        var x = r, y = 0;

        // Printing the initial point
        // on the axes after translation
        document.write("(" + (x + x_centre) + ", " + (y + y_centre) + ")");

        // When radius is zero only a single
        // point will be printed
        if (r > 0) {

            document.write("(" + (x + x_centre) + ", " + (-y + y_centre) + ")");

            document.write("(" + (y + x_centre) + ", " + (x + y_centre) + ")");

            document.write("(" + (-y + x_centre) + ", " + (x + y_centre) + ")<br/>");
        }

        // Initialising the value of P
        var P = 1 - r;
        while (x > y) {

            y++;

            // Mid-point is inside or on the perimeter
            if (P <= 0)
                P = P + 2 * y + 1;

            // Mid-point is outside the perimeter
            else {
                x--;
                P = P + 2 * y - 2 * x + 1;
            }

            // All the perimeter points have already
            // been printed
            if (x < y)
                break;

            // Printing the generated point and its
            // reflection in the other octants after
            // translation
            document.write("(" + (x + x_centre) + ", " + (y + y_centre) + ")");

            document.write("(" + (-x + x_centre) + ", " + (y + y_centre) + ")");

            document.write("(" + (x + x_centre) + ", " + (-y + y_centre) + ")");

            document.write("(" + (-x + x_centre) + ", " + (-y + y_centre) + ")<br/>");

            // If the generated point is on the
            // line x = y then the perimeter points
            // have already been printed
            if (x != y) {

                document.write("(" + (y + x_centre) + ", " + (x + y_centre) + ")");

                document.write("(" + (-y + x_centre) + ", " + (x + y_centre) + ")");

                document.write("(" + (y + x_centre) + ", " + (-x + y_centre) + ")");

                document.write("(" + (-y + x_centre) + ", " + (-x + y_centre) + ")<br/>");
            }
        }
    }

    // Driver code


        // To draw a circle of radius
        // 3 centered at (0, 0)
        midPointCircleDraw(0, 0, 3);

// This code is contributed by umadevi9616
</script>

Output: 

(3, 0) (3, 0) (0, 3) (0, 3)
(3, 1) (-3, 1) (3, -1) (-3, -1)
(1, 3) (-1, 3) (1, -3) (-1, -3)
(2, 2) (-2, 2) (2, -2) (-2, -2)

Time Complexity: O(x – y)
Auxiliary Space: O(1) 
References : Midpoint Circle Algorithm
Image References : Octants of a circle, Rasterised Circle, the other images were created for this article by the geek
Thanks Tuhina Singh and Teva Zanker for improving this article.