Midpoint Circle Algorithm

Midpoint Circle Algorithm:
A circle is defined as a set of points that are all at a given distance r from a center positioned at. (xc,yc). 
This is represented mathematically by the equation
 ------------- (1)
Using equation (1) we can calculate the value of y for each given value of x as


---------------- (2)

Thus one could calculate different pairs by giving step increments to x and calculating the corresponding value of y. But this approach involves considerable computation at each step and also the resulting circle has its pixels sparsely plotted for areas with higher values of the slope of the curve.
Midpoint Circle Algorithm uses an alternative approach, wherein the pixel positions along the circle are determined on the basis of incremental calculations of a decision parameter.

Let 
 -------------- (3)
Thus f(x,y)=0 represents the equation of a circle.
Further, we know from coordinate geometry, that for any point , the following holds:



In Midpoint Circle Algorithm, the decision parameter at the kth step is the circle function evaluated using the coordinates of the midpoint of the two pixel centres which are the next possible pixel position to be plotted.
Let us assume that we are giving unit increments to x in the plotting process and determining the y position using this algorithm. Assuming we have just plotted the kth
pixel at (
Xk,Yk), we next need to determine whether the pixel at the position ( Xk+1,Yk ) or the one at ( Xk+1 , Yk-1) is closer to the circle.



Our decision parameter pk at the kth step is the circle function evaluated at the midpoint of these two pixels.
The coordinates of the midpoint of these two pixels are ( Xk+1, Yk-1/2).
Thus pk

--------------- (4)
x
Successive decision parameters are obtained using incremental calculations, thus avoiding a lot of computation at each step. We obtain a recursive expression for the next decision parameter i.e. at the k+1th step, in the following manner.
Using Equ. (4), at the k+1th step, we have:


Now if Pk <=0, then the midpoint of the two possible pixels lies within the circle, thus north pixel is nearer to the theoretical circle. Hence, Yk+1= Yk . Substituting this value of in Equ. (6), we have 


If pk > 0 then the midpoint of the two possible pixels lies outside the circle, thus south pixel is nearer to the theoretical circle. Hence, Yk+1= Yk-1. Substituting this value of in Equ. (6), we have  

 


For the boundary condition, we have x=0, y=r. Substituting these values in (4), we have 


For integer values of pixel coordinates, we can approximate P0=1-r,

Thus we have:




Drawing the circle:
We can reduce our calculation drastically (8th fraction ) by making use of the fact that a circle has 8 way symmetry. Thus after calculating a pixel position (x,y) to be plotted, we get 7 other points on the circle corresponding to it. These are:
(x,y); (x,-y); (-x,y); (-x,-y); (y,x); (y,-x); (-y,x); (-y,-x) 





The following code implements Mid Point Circle Algorithm in C++
# include
<iostream.h>

# include
<conio.h>

# include
<graphics.h>

# include
<math.h>



void MidPointCircleAlgo(int Radius,int xC,int yC); // Prototype



void main()

{

    int gDriver=DETECT, gMode;

    initgraph(&gDriver,&gMode,"c:\\tc\\bgi");



    int Radius, xC, yC;

    cout<< endl << "Enter Center point coordinates...";

    cout<<endl<<" Xc : ";

    cin>>xC;

    cout<<endl<<" Yc : ";

    cin>>yC;

    cout<<endl<<"Radius : ";

    cin>>Radius;



cleardevice();

    MidPointCircleAlgo(Radius,xC,yC);

    getch();

}



void MidPointCircleAlgo(int Radius,int xC,int yC)

{

    int P;

    int x,y;



void Draw(int x,int y,int xC,int yC); // Function to plots the points


P = 3 -2* Radius;

    x = 0;

    y = Radius;

    Draw(x,y,xC,yC);




while (x<=y)

    {

        x++;

        if (P<0)

        {

            P += 4 * x + 6;

        }

        else

        {

            P += 4 * (x - y) + 10;

            y--;

        }

        Draw(x,y,xC,yC);

    }



}









void Draw(int x,int y,int xC,int yC)

{

xC=320+xC;

yC=240-yC;

    putpixel(xC + x,yC + y,1);

    putpixel(xC + x,yC - y,1);

    putpixel(xC - x,yC + y,1);

    putpixel(xC - x,yC - y,1);

    putpixel(xC + y,yC + x,1);

    putpixel(xC - y,yC + x,1);

    putpixel(xC + y,yC - x,1);

    putpixel(xC - y,yC - x,1);

}


 

15 comments:

  1. thanks for the explanation n algorithm
    i get an error when i run the algorithm in c++with dos box.the error is....

    BGI ERROR:graphics not initialized
    (use 'initgraph')

    ReplyDelete
  2. Can you give and explain algorithm for anti aliased circle algorithm

    ReplyDelete