The midpoint circle algorithm is similar to Bresenham's circle algorithm and uses the midpoint between pixels to determine whether the pixel is inside or outside a circle. It defines a decision parameter pi based on the midpoint and updates pi by integer amounts at each step to determine the next pixel along the circle. The initial value of pi is set to 5/4 - r when r is an integer to determine the first pixel.

1. Midpoint Circle Algorithm
Mid point algorithm is very similar to
Bresenham’s approach. It is based on the
following function for testing the spatial
relationship between an arbitrary point(x,y)
and a circle of radius centered at the origin:
< 0 (x,y) inside circle
f(x,y) = x2 + y2 – r2 = 0 (x,y) on the circle
> 0 (x,y) outside
circle`

2. Now consider the coordinates of the point
halfway between pixel T & S (xi + 1, yi – ½)
This is called the midpoint and we use it to
define a decision parameter:
pi = f( xi +1, yi – ½)
= (xi + 1)2 + (yi – ½)2 – r2
If pi is –ve , the midpoint is inside the circle,
then we choose pixel T. If pi is +ve, the
midpoint is outside the circle, & we choose S.

3. Similarly, the decision parameter for the next
step is:
pi+1 = (xi+1 + 1)2 + (yi+1 – ½)2 – r2
Since x i+1 = xi + 1
pi+1 - pi = [(xi + 1)+ 1]2 – (xi + 1)2
+ (y i+1 – ½)2 – (yi – ½)2
Hence
pi+1 = pi + 2(xi + 1) + 1 + (y i+12 – yi2)
– (y i+1 – yi)

4. If pixel T is chosen (meaning pi < 0), we
have yi+1 = yi.
If pixel S is chosen (meaning pi > 0), we
have yi+1 = yi – 1. Thus,
pi+1= pi + 2(xi + 1) +1 if pi<0
pi + 2(xi + 1) +1 – 2(yi – 1) if pi>0
In terms of (xi, yi), we have
pi+1= pi + 2xi + 3 if pi<0
pi + 2(xi - yi ) + 5 if pi>0

5. Finally, we compute the initial value for the
decision parameter using the original
definition of pi and (0,r):
pi = (0 + 1)2 + (r – ½)2 – r2 = 5/4 – r
One can see that this is not really integer
computation. However, when r is an integer
we can simply set p1= 1 – r. The error of
being ¼ less than the precise value does not
prevent p1 from getting the appropriate sign.

6. It does not affect the rest of the scan
conversion process, because the decision
variable is only updated with integer increment
in subsequent steps.

7. Algorithm:
The following is a description of this midpoint
circle algorithm that generates the pixel
coordinates in the 90˚ to 45 ˚ octant:
int x = 0, y = r, p = 1-r
while (x <=y)
{ setpixel(x,y);
if (p < 0)
p = p + 2x + 3

8. else
{
p = p + 2(x – y) + 5
y- -
}
x++
}