13.8 Scan Converting Circles

Brute Force:

Second octan, going rightward

x := 0
REPEAT
   y := sqrt( R*R - x*x )
   CirclPixels( x , round(y) )
   x := x+1
UNTIL x>y

|o---o----o----o----o----o----o----o
----------------------------------------
|o---o----o----o----o----o----o----o Y
----------------------------------------
|o---o----o----o----o---|o----o----o
----------------------------------------
|o---o----o----o----o----o---|o||--o
----------------------------------------
o---o----o----o----o----o----o----o||
|

Midpoint Circle Algorithm

90^o 45^o
(x,y) = an intersection point of the circle with vertical mesh line
(x,Y ) = pixel corresponding to (x,y)
L = distance of midpoint $(x, Y - 1) 2$ from center of circle
${ L < R? yes PaintPixel(x,Y) no PaintPixel(x,Y -1)$

L < R
iff L * L < R * R
iff

= L * L - R * R < 0

‘Y = Y - 1’ when

changes sign

x:=0  Y:=R
Delta = (R-0.5) * (R-0.5) - R*R
PaintPixels(x,Y)
REPEAT
   Delta :=  Delta +
             (

_{(x+1,Y -1/2)}-

_{(x,Y -1/2)})
   IF Delta>0 THEN
     Delta := Delta +
              (

_{(x+1,Y -3/2)}-

_{(x+1,Y -1/2)})
     Y := Y-1
   END
   x := x+1
   PaintPixels(x,Y)
UNTIL x>Y

x x+1
Yo•----o----o
D(x,Y -12)-- ---- |D(x+1,Y-12)
Y-1o----o----o
| --------D 3
o----o----o (x+1,Y- 2)
|| ---
| --
| --
| --
|--
||-||

Intersection point (x,y) is ignored for an edge (x_max,y_max) (x_min,y_min), if it satisfies y = y_max, i.e,
- All intersection points in horizontal line
- Highest endpoint in nonhorizontal line
Each intersection point at scanline Y is
- Either a maximal vertex (x_max,y_max) of an edge
- Or a neighbor of a higher intersection point on the same edge. If the higher intersection point has coordinates (x,Y+1) then the current intersection point has coordinates (x+x,Y), where
  $dx = xmax--xmin-- ymax -ymin$
An edge having intersection point at the current scanline Y is called an active edge

1 1 D(x,Y -12) = x *x + (Y - 2) * (Y - 2)- R *R D(x+1,Y -12) = (x + 1) * (x + 1) + (Y - 12) *(Y - 12)- R * R D 1 = (x + 1) * (x + 1) + (Y - 3) *(Y - 3)- R * R (x+1,y- 2-1) 2 2

_{(x+1,Y -1/2)}-_{(x,Y -1/2)}:

2 2 2
(x + 1) + (Y - 0.5) - R
-
-------x2-----+---(Y----0.5)2-------R2-
2x + 1

_{(x+1,Y -3/2)}-_{(x+1,Y -1/2)}:

(x + 1)2 + (Y - 1- 0.5)2 - R2
-
(x + 1)2 + (Y - 0.5)2 - R2
----2(1---Y-)-----------------------------

x:=0
Y:=R
Delta := (R-0.5) * (R-0.5) - R*R
PaintPixels(x,Y)
REPEAT
   Delta :=  Delta + 2 * x + 1
   IF Delta>0 THEN
     Delta := Delta + 2 * (1 - Y)
     Y := Y-1
   END
   x := x+1
   PaintPixels(x,Y)
UNTIL x>Y

Eight-Way Symmetry

(- x,y) (x,y)
-|•|||•||-
(- y,x) -- | -----(y,x)
•- --- •|
•- •-
(-y,- x)-|| |--(y,- x)
|•|||•||
(-x,- y) (x,- y)

PROCEDURE CirclePixels(x,y);
BEGIN
   PaintPixel(x,y)
   PaintPixel(y,x)
   PaintPixel(y,-x)
   PaintPixel(x,-y)
   PaintPixel(-x,-y)
   PaintPixel(-y,-x)
   PaintPixel(-y,x)
   PaintPixel(-x,y)
END