MidPoint Circle Algorithm

It is based on the following function for testing the spatial relationship between the arbitrary point (x, y) and a circle of radius r centered at the origin:

Now, consider the coordinates of the point halfway between pixel T and pixel S

This is called midpoint (x_i+1,y_i- MidPoint Circle Algorithm ) and we use it to define a decision parameter:

P_i=f (x_i+1,y_i- MidPoint Circle Algorithm ) = (x_i+1)²+(y_i-)²-r² ...............equation 2

If P_i is -ve ⟹midpoint is inside the circle and we choose pixel T

If P_i is+ve ⟹midpoint is outside the circle (or on the circle)and we choose pixel S.

The decision parameter for the next step is:

P_i+1=(x_i+1+1)²+(y_i+1- MidPoint Circle Algorithm )²- r²............equation 3

Since x_i+1=x_i+1, we have

If pixel T is choosen ⟹P_i<0

We have y_i+1=y_i

If pixel S is choosen ⟹P_i≥0

We have y_i+1=y_i-1

We can continue to simplify this in n terms of (x_i,y_i) and get

Now, initial value of P_i (0,r)from equation 2

We can put MidPoint Circle Algorithm ≅1
∴r is an integer
So, P₁=1-r

Algorithm:

Step1: Put x =0, y =r in equation 2
We have p=1-r

Step2: Repeat steps while x ≤ y
Plot (x, y)
If (p<0)
Then set p = p + 2x + 3
Else
p = p + 2(x-y)+5
y =y - 1 (end if)
x =x+1 (end loop)

Step3: End

Program to draw a circle using Midpoint Algorithm:

#include <graphics.h>
#include <stdlib.h>
#include <math.h>
#include <stdio.h>
#include <conio.h>
#include <iostream.h>

class bresen
{
	float x, y,a, b, r, p;
	public:
	void get ();
	void cal ();
};
	void main ()
    {
	bresen b;
	b.get ();
	b.cal ();
	getch ();
   }
	Void bresen :: get ()
   {
	cout<<"ENTER CENTER AND RADIUS";
	 cout<< "ENTER (a, b)";
	cin>>a>>b;
	cout<<"ENTER r";
	cin>>r;
}
void bresen ::cal ()
{
	/* request auto detection */
	int gdriver = DETECT,gmode, errorcode;
	int midx, midy, i;
	/* initialize graphics and local variables */
	initgraph (&gdriver, &gmode, " ");
	/* read result of initialization */
	errorcode = graphresult ();
	if (errorcode ! = grOK)    /*an error occurred */
	{
 		printf("Graphics error: %s \n", grapherrormsg (errorcode);
		printf ("Press any key to halt:");
		getch ();
		exit (1); /* terminate with an error code */
	}
	x=0;
	y=r;
	putpixel (a, b+r, RED);
	putpixel (a, b-r, RED);
	putpixel (a-r, b, RED);
	putpixel (a+r, b, RED);
	p=5/4)-r;
	while (x<=y)
	{
		If (p<0)
		p+= (4*x)+6;
		else
		{
			p+=(2*(x-y))+5;
			y--;
		}
		x++;
		putpixel (a+x, b+y, RED);
		putpixel (a-x, b+y, RED);
		putpixel (a+x, b-y, RED);
		putpixel (a+x, b-y, RED);
		putpixel (a+x, b+y, RED);
		putpixel (a+x, b-y, RED);
		putpixel (a-x, b+y, RED);
		putpixel (a-x, b-y, RED);
	}
}

Output: