Mid-Point Circle Drawing Algorithm

Mid-Point Circle Drawing Algorithm

by Foysal Ahmed -
Number of replies: 0

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 it’s centre.

Circle octants

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

example

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 + (yk + 1) + 1



= Pk + (xk+1 – 0.5)2 – (xk – 0.5)2 + 2(yk + 1) + 1
= Pk + (x2k+1 – x2k)2 + (xk+1 – xk)2 + 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)
  • C
  • CPP
  • Java
  • Python3
  • C#
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// 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 centred at (0, 0)
    midPointCircleDraw(0, 0, 3);
    return 0;
}

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)

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 for improving this article.
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