Mid-Point Circle Drawing Algorithm

19 Mar, 2022
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.
 
Find the mid-point p of the two possible pixels i.e (x-0.5, y+1)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)
Find the mid-point p of the two possible pixels i.e (x-0.5, y+1)
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 circleF(p)=0, the point is on the perimeterF(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. 
Pk = (Xk — 0.5)2 + (yk + 1)2 – r2 Now, xk+1 = xk or xk-1 , yk+1= yk +1∴ Pk+1 = (xk+1 – 0.5)2 + (yk+1 +1)2 – r2 = (xk+1 – 0.5)2 + [(yk +1) + 1]2 – r2 = (xk+1 – 0.5)2 + (yk +1)2 + 2(yk + 1) + 1 – r2 = (xk+1 – 0.5)2 + [ – (xk – 0.5)2 +(xk – 0.5)2 ] + (yk + 1)2 – r2 + 2(yk + 1) + 1= Pk + (xk+1 – 0.5)2 – (xk – 0.5)2 + 2(yk + 1) + 1 = Pk + (x2k+1 – x2k) – (xk+1 – xk) + 2(yk + 1) + 1 = Pk + 2(yk +1) + 1, when Pk <=0 i.e the midpoint is inside the circle (xk+1 = xk) Pk + 2(yk +1) – 2(xk – 1) + 1, when Pk>0 I.e the mid point is outside the circle(xk+1 = xk-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
Java
Python3
C#
PHP
Javascript
// C++ program for implementing// Mid-Point Circle Drawing Algorithm#include<iostream>using namespace std; // Implementing Mid-Point Circle Drawing Algorithmvoid 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 codeint main(){    // To draw a circle of radius 3 centered at (0, 0)    midPointCircleDraw(0, 0, 3);    return 0;}
// C program for implementing// Mid-Point Circle Drawing Algorithm#include<stdio.h> // Implementing Mid-Point Circle Drawing Algorithmvoid 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 codeint main(){    // To draw a circle of radius 3 centered at (0, 0)    midPointCircleDraw(0, 0, 3);    return 0;}
// Java program for implementing// Mid-Point Circle Drawing Algorithmclass 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 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 Codeif __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# program for implementing Mid-Point// Circle Drawing Algorithmusing 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 program for implementing// Mid-Point Circle Drawing Algorithm // Implementing Mid-Point// Circle Drawing Algorithmfunction 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.?>
<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 geekThanks Tuhina Singh and Teva Zanker for improving this article. This article is contributed by Nabaneet Roy. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. 
nitin mittal
SHUBHAMSINGH10
siddharthx_07
pankajsharmagfg
umadevi9616
simranarora5sos
adnanirshad158
tevazanker
computer-graphics
Algorithms
Algorithms
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.