Bresenham’s circle drawing algorithm

link 1

It is not easy to display a continuous smooth arc on the computer screen as our computer screen is made of pixels organized in matrix form. So, to draw a circle on a computer screen we should always choose the nearest pixels from a printed pixel so as they could form an arc. There are two algorithm to do this:

  1. Mid-Point circle drawing algorithm
  2. Bresenham’s circle drawing algorithm

We have already discussed the Mid-Point circle drawing algorithm in our previous post.In this post we will discuss about the Bresenham’s circle drawing algorithm.
Both of these algorithms uses the key feature of circle that it is highly symmetric. So, for whole 360 degree of circle we will divide it in 8-parts each octant of 45 degree. In order to that we will use Bresenham’s Circle Algorithm for calculation of the locations of the pixels in the first octant of 45 degrees. It assumes that the circle is centered on the origin. So for every pixel (x, y) it calculates, we draw a pixel in each of the 8 octants of the circle as shown below :
circle 1

Now, we will see how to calculate the next pixel location from a previously known pixel location (x, y). In Bresenham’s algorithm at any point (x, y) we have two option either to choose the next pixel in the east i.e. (x+1, y) or in the south east i.e. (x+1, y-1).

circle 2

And this can be decided by using the decision parameter d as:

  • If d > 0, then (x+1, y-1) is to be chosen as the next pixel as it will be closer to the arc.
  • else (x+1, y) is to be chosen as next pixel.

Now to draw the circle for a given radius ‘r’ and centre (xc, yc) We will start from (0, r) and move in first quadrant till x=y (i.e. 45 degree). We should start from listed initial condition:

d = 3 - (2 * r)
x = 0
y = r

Now for each pixel, we will do the following operations:

  1. Set initial values of (xc, yc) and (x, y)
  2. Set decision parameter d to d = 3 – (2 * r).
  3. Repeat steps 4 to 8 until x < = y
  4. call drawCircle(int xc, int yc, int x, int y) function.
  5. Increment value of x.
  6. If d < 0, set d = d + (4*x) + 6
  7. Else, set d = d + 4 * (x – y) + 10 and decrement y by 1.
  8. call drawCircle(int xc, int yc, int x, int y) function

drawCircle() function:

// function to draw all other 7 pixels
// present at symmetric position
drawCircle(int xc, int yc, int x, int y)
{
    putpixel(xc+x, yc+y, RED);
    putpixel(xc-x, yc+y, RED);
    putpixel(xc+x, yc-y, RED);
    putpixel(xc-x, yc-y, RED);
    putpixel(xc+y, yc+x, RED);
    putpixel(xc-y, yc+x, RED);
    putpixel(xc+y, yc-x, RED);
    putpixel(xc-y, yc-x, RED);
}

Below is C implementation of above approach.

// C-program for circle drawing
// using Bresenham’s Algorithm
// in computer-graphics
#include <stdio.h>
#include <dos.h>
#include <graphics.h>
// Function to put pixels
// at subsequence points
void drawCircle(int xc, int yc, int x, int y)
{
    putpixel(xc+x, yc+y, RED);
    putpixel(xc-x, yc+y, RED);
    putpixel(xc+x, yc-y, RED);
    putpixel(xc-x, yc-y, RED);
    putpixel(xc+y, yc+x, RED);
    putpixel(xc-y, yc+x, RED);
    putpixel(xc+y, yc-x, RED);
    putpixel(xc-y, yc-x, RED);
}
// Function for circle-generation
// using Bresenham's algorithm
void circleBres(int xc, int yc, int r)
{
    int x = 0, y = r;
    int d = 3 - 2 * r;
    while (y >= x)
    {
        // for each pixel we will
        // draw all eight pixels
        drawCircle(xc, yc, x, y);
        x++;
        // check for decision parameter
        // and correspondingly
        // update d, x, y
        if (d > 0)
        {
            y--;
            d = d + 4 * (x - y) + 10;
        }
        else
            d = d + 4 * x + 6;
        drawCircle(xc, yc, x, y);
        delay(50);
    }
}
// driver function
int main()
{
    int xc = 50, yc = 50, r2 = 30;
    int gd = DETECT, gm;
    initgraph(&gd, &gm, "");  // initialize graph
    circleBres(xc, yc, r);    // function call
    return 0;
}

Output:
circleout

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rakesh

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