Part VII Polygon Filling

1
Part VII Polygon Filling
2
Scan-line Algorithm for Polygon Filling

• Two tasks in polygon filling
• Which pixels to fill? Only interior pixels! ?
  now!
• What colors for the pixels? ? later!
• Well-known for polygon filling is the scan-line
  algorithm, which computes the x-intersections
  between each scan line and polygons edges, and
  then fill all pixels between the intersections.

inside
outside
scan line
right intersection
left intersection
fill all pixels between them
3
Convex vs. Concave

• A polygon is convex if, given any 2 points in its
  interior, the line segment joining them is also
  in the interior.
• Lets restrict polygon filling to convex
  polygons!!
• In most cases, there are then 2 intersections
  between a scan-line and a convex polygon to be
  filled.
• Why most? Why not all?
• This will be cleared soon.

scan line
concave polygons
convex polygons
4
Scan-line Algorithm Overview

• Suppose that we are given the following polygon.
• Checking the y-coordinates of the vertices, we
  can easily see that the scan-lines 2 through 7
  should be processed.
• For each scan-line, we can compute two
  x-intersections and fill pixels between them.
• For fractional intersections, round up xleft and
  and round down xright.
• For example, scan-line 3 lt2.4,6.5gt ? 3,6.

8 7 6 5 4 3 2 1
y
y7.3
xleft2.4
xright6.5
y1.6
x 1 2 3 4 5 6 7
5
Integer Intersections

• How about integer intersections?
• Problems (on the boundary)
• inconsistency edge ownership may depend on
  drawing order.
• Inefficiency the edges may be overwritten.
• weird rendering possible especially for
  animation.
• For integer intersections, take xleft,
  xright-1.
• In summary, for both fractional and integer
  intersections, ?? xleft ? at the left end, and ?
  ?? xright ? -1 ? at the right end.

?
?
Whose edges?
X16
4 5 6 7 8 9 10 11 12 13 14 15
lt12,16gt is computed as intersections, and pixels
are filled in 12,15.
X3
6
Which Scan-lines to Process?

• So far, x-intersections.
• How about y-intersections? i.e. Which scan-lines
  to process?
• If both ylow and yhigh of the polygon are
  fractional numbers, simply round up ylow and
  round down yhigh.
• What if integer intersections? Take ylow,
  yhigh-1.
• In summary, for both fractional and integer
  intersections, ? ylow ?? at the bottom, and ?
  yhigh ?-1 ? at the top.

yhigh
ylow
7
Computation of X-intersections

• Suppose that we have computed x-intersection for
  scan-line i.
• Then, for scan-line i1, the x-intersection can
  be incrementally computed using the inverse of
  the slope.
• Once we get x-intersection at scan-line 2, we can
  compute that for scan-line 3, and then scan-line
  4, etc. Then, how about scan-line 2?

i1
1
m
i
ymxB
1
1/m
new-x old-x 1/m
old-x
8 7 6 5 4 3 2 1
y
(5.4,7.3)
(1.5?,2)
2 1
0.4
(1.5,1.6)
1
?
(1.5,1.6)
1/m
x 1 2 3 4 5 6 7
8
Edge Orientations

• Assume that, for every polygon, vertices are
  given in CCW order.
• An edge is at left side if it is downward. If
  upward, right side.
• Horizontal edges (whose end points y-coordinates
  are identical) are simply ignored.

3
1
4
2
3
2
5
1
v1
v1
v1
y1
y1
c
a
y3
y3
v3
v3
v3
b
y2
y2
v2
v2
v2
y1 gt y2 left-side edge
y2 lt y3 right-side edge
y3 lt y1 right-side edge
9
Interpolation along Y

• Reserve two arrays for the whole scene left- and
  right-bounds.
• For left-side edges, process the left-side bound.
  For right-side edges, process the right-side
  bound.
• For each edge E, determine the y-range ymin,
  ymax to be processed ymin ?? ylow? ? and ymax
  ?? yhigh? -1 ?.
• Compute the x-intersection either at ymin or at
  ymax.
• Incrementally compute the x-intersections of all
  other scan-lines in E.

x4
x3
amax cmax


c
a
cmin
x6
b
x5
bmax
x1
x2


amin bmin
x1
x2 x3 x4 x5
x6
left-side bound
right-side bound
10
Pixel Filling along X

• Process the set S of scan-lines ymin, ymax of
  the triangle one by one.
• For each scan-line, retrieve xleft from the
  left-side bound and xright from the right-side
  bound.
• Then, fill pixels xmin, xmax ? ?? xleft? ? ,
  ??? xright? -1? .

ymax


xright
xleft


ymin
right-side bound
left-side bound
11
Convex vs. Concave (revisited)

• The scan-line algorithm weve discussed follows
  OpenGLs convention and assumes convex polygons.
  
• OpenGL does not restrict the number of edges of a
  polygon, but the polygon must be convex. Concave
  polygons are often be drawn awkwardly.
• How to draw/fill concave polygons?
• The simple solution is to subdivide it into a set
  of convex polygons and to draw each convex
  polygon.

scan line
concave polygons