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Computer Graphics Midpoint Circle Algorithm

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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:

MidPoint Circle Algorithm
MidPoint Circle Algorithm

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

This is called midpoint (xi+1,yiMidPoint Circle Algorithm) and we use it to define a decision parameter:

            Pi=f (xi+1,yiMidPoint Circle Algorithm) = (xi+1)2+(yiMidPoint Circle Algorithm)2-r2 ……………equation 2

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

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

The decision parameter for the next step is:

Pi+1=(xi+1+1)2+(yi+1MidPoint Circle Algorithm)2– r2…………equation 3

Since xi+1=xi+1, we have

MidPoint Circle Algorithm

If pixel T is choosen ⟹Pi<0

We haveyi+1=yi

If pixel S is choosen ⟹Pi≥0

We haveyi+1=yi-1

MidPoint Circle Algorithm

We can continue to simplify this in n terms of (xi,yi) and get

MidPoint Circle Algorithm

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

MidPoint Circle Algorithm

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


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
            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:


MidPoint Circle Algorithm

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