Computer Graphics

MID POINT CIRCLE

ALGORITHM

CIRCLE ALGORITHMS



Use 8-fold symmetry and only compute pixel positions for the 45° sector.

45°

(x, y)

(y, x) (-x, y)

(y, -x)

(x, -y) (-x, -y)

(-y, x)

(-y, -x)

MIDPOINT CIRCLE ALGORITHM

y

_i



y

_i

-1

x

_i

x

_i

+1 x

_i

+2

Midpoin t

x

²

+ y

²

– r

²

= 0

Assuming that we have just plotted the pixels at (x

_i

, y

_i

).

Which is next? (x

_i

+1, y

_i

) OR (x

_i

+1, y

_i

– 1).

- The one that is closer to the circle.

MIDPOINT CIRCLE ALGORITHM

 The decision parameter is the circle at the midpoint between the pixels y

_i

and y

_i

– 1.

 If p

i

< 0, the midpoint is inside the circle and the pixel y

i

is closer to the circle boundary.

 If p

i

≥ 0, the midpoint is outside the circle and the pixel y

i

- 1 is closer to the circle boundary.

1 2

2 1 2 2

2

( 1, )

( 1) ( )

i circle i i

i i

p f x y

x y r

  

    

DECISION PARAMETERS

 Decision Parameters are obtained using incremental calculations

OR

where y

i+1

is either y

i

or y

i

-1 depending on the sign of p

i

Note:

x

_i+1

= x

_i

+1

1 1 1 12

2 1 2 2

1 2

( 1, )

( 2) ( )

i circle i i

i i

p f x y

x y r

  



  

    

2 2 2

1

2( 1) (

1

) (

1

) 1

i i i i i i i

p

_

 p  x   y

_

 y  y

_

 y 

THE ALGORITHM

1. Initial values:- point (0 , r) [

^{x0 = 0 y0 = r}

] 2. Initial decision parameter

3.

At each x

i

position, starting at i = 0

if p

i

< 0, the next point is (x

i

+ 1, y

i

) and

p

i+1

= p

i

+ 2x

i+1

+ 1

if p

i

≥ 0, the next point is (x

i

+1, y

i

-1) and

p

i+1

= p

i

+ 2x

i+1

+ 1 – 2y

i+1

where 2x

i+1

= 2x

i

+ 2 and 2y

i+1

= 2y

i

– 2

4. Determine symmetry points in the other octants 5. Repeat 3 – 4 until x becomes greater or equal to y

2 2 5

1 1

0 _circle

(1,

2

) 1 (

2

)

4

p  f r     r  r   r

10    

9  

8 

7 

6 

5 

4 

3 

2 

1 

0 

0 1 2 3 4 5 6 7 8 9 10

i p

_i

Next 2x

_i+1

2y

_i+1

0 -9 (1, 10) 2 18

1 -6 (2, 10) 4 18

2 -1 (3, 10) 6 18

3 6 (4, 9) 8 18

4 -3 (5, 9) 10 16

5 8 (6, 8) 12 16

6 5 (7, 7)

r = 10

p

₀

= 1 – r = -9 (if r is integer, then consider p

₀

= 1 – r)

Initial point (x

₀

, y

₀

) = (0, 10)

2x

_i+1

= 2x

_i

+ 2 and 2y

_i+1

= 2y

_i

– 2

If (p < 0) => (x

_i

+ 1, y

_i

)

p

_i+1

= p

_i

+ 2x

_i+1

+ 1 Else if (p>=0) =>

(x

_i

+1, y

_i

-1)

p

_i+1

= p

_i

+ 2x

_i+1

+ 1

– 2y

_i+1

MIDPOINT FUNCTION

void plotpoints(int x, int y)

{ setpixel(xcenter+x, ycenter+y);

setpixel(xcenter-x, ycenter+y);

setpixel(xcenter+x, ycenter-y);

setpixel(xcenter-x, ycenter-y);

setpixel(xcenter+y, ycenter+x);

setpixel(xcenter-y, ycenter+x);

setpixel(xcenter+y, ycenter-x);

setpixel(xcenter-y, ycenter-x);

}

void circle(int r) { int x = 0, y = r;

plotpoints(x,y);

int p = 1 – r;

while (x<y) {

x++;if (p<0) p += 2*x + 2;

else {

y--;p += 2*(x-y) + 1;

}plotpoints(x,y);

} }