Chapter 5
Scan Converting Circles

5.1 Brute Force

Second octan, going rightward

DEFICIENCIES

• ‘*’, ‘sqrt’, ‘round’
• arithmetics with reals

5.2 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 + 1,Y - 1 2) from center of circle

• L < R

• L < R iff L * L < R * R iff = L * L - R * R < 0

‘Y := Y - 1’ when changes sign

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

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

WHY ‘d2 + 4’ and not ‘d2 + 6’

5.3 Eight-Way Symmetry

5.4 Scanline Fill (Polygons)

For each scanline Y

• Find intersection points with polygon’s edges

• Sort intersection points by x

• Fill spans: (x₀ , Y ) (x₁,Y ), (x₂,Y ) (x₃,Y ), ...

• 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

    x = x_max-x_min y_max-y_min

• An edge having intersection point at the current scanline Y is called an active edge

COHERENCE -- Local similarity (used to speed up algorithms)

Within the scanline fill algorithm

• Span coherence -- Properties of pixels change slowly along spans
• Scanline coherence -- Properties of spans change slowly between scanlines
• Edge coherence -- Intersection points of scanlines move slowly across edges

Edge Representation

• (x_max , y_max )--Highest coordinate

• N--Number y_max - y_min + 1 of x-intersects

• x--Horizontal distance x_max-x_min y_max-y_min between x-intersects

AET[Y] + ET[Y] AET[Y-1]

AET -- ACTIVE EDGE TABLE

• Possible Polygons.

  

• Other shapes with easily computable intersection points (e.g., circles)

• Sorting is needed even if no new points are added to AET, because edges can intersect.

• Sorting is fast because the lists are short and almost ordered (e.g., insertion sort)

• Fractions can be avoided with representations ‘(integer part, numerator, denominator)’, e.g., (1,2,5) for 7 5.

• Sliver problem: regions with small bottle necks

5.5 Seed Fill (Flood Fill)

• simple algorithm
• Does not require mathematical description for the region (e.g., free hand drawing)

• Heavy on memory and time (4 inquiries; 1 push/pop)

5.6 Scanline Seed Fill

• Variant of seed fill with smaller stack
   
   S := seed span ; paint_push S
   REPEAT
     S := pop
     FOR all nonpainted spans S'
         that are adjacent to S  DO
       paint_push  S'
     END
   UNTIL stack is empty

• span (x₁ , x₂ , y)

  avoids cycles

• Span (x, y). Ignores cycles (push_paint push; pop pop_paint)
   
    Find adjucent spans (x,y+1)
   flag := true
   FOR x := x2 TO x1 DO
     IF (x,y+1) is boundary  THEN
        flag := true
     ELSE IF upflag THEN
        found span (x,y+1)
        flag := false
     END
   END

  (2-3 inquiries; 1 push/pop)

CASES TO CONSIDER

• 4-connected -- Pixels up, down, right, and left are considered to be adjacent to current pixel.

• 8-connected --

• Interior defined -- largest connected region of pixels of same color

• Boundary defined -- largest connected region of pixels that differ from the boundary color