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Different types of Polygons
Simple Convex
Simple Concave
Non-simple : self-intersecting
With holes
Convex
Concave
Self-intersecting
Polygon Scan Conversion
• Scan Conversion = Fill
• How to tell inside from outside
– Convex easy
– Nonsimple difficult
– Odd even test
• Count edge crossings
odd-even fill
Winding Number
• Count clockwise crossings as positive and
counterclockwise crossings as negative
winding number = 1
winding number = 2
• Alternate definition of inside: inside if winding number  0
Filling in the Frame Buffer
• Fill at end of pipeline
– Convex Polygons only
– Nonconvex polygons assumed to have been
tessellated
– Shades (colors) have been computed for vertices
(Gouraud shading)
– March across scan lines interpolating shades
• Incremental work small
Using Interpolation
C1 C2 C3 specified by glColor or by vertex shading
C4 determined by interpolating between C1 and C2
C5 determined by interpolating between C2 and C3
interpolate between C4 and C5 along span
C1
C4
scan line
C2
C5
span
C3
Scan Line Polygon Fill
• Determine overlap Intervals for scan lines that cross that area.
– Requires determining intersection positions of the edges of the
polygon with the scan lines
– Fill colors are applied to each section of the scanline that lies
within the interior of the region.
– Interior regions determined as in odd-even test
10 13 16 19
Interior pixels along a scan line passing through a
polygon area
10 13 16
19
• In the given example, four pixel intersections are at
x=10, x=13, x=16 and x=19
• These intersection points are then sorted from left to
right , and the corresponding frame buffer positions
between each intersection pair are set to specified color
– 10 to 13, and 16 to 19
Scan Line Polygon Fill
• Some scan line intersections at polygon
vertices require special handling:
– A scan line passing through a vertex
intersects two polygon edges at that position
Scan Line y’
1
2
1
Scan Line y
1
2
1
1
Intersection points along the scan lines that intersect polygon
vertices.
Scan line y’ generates an even number of intersections that
can be paired to identify correctly the interior pixel spans.
To identify the interior pixels for scan line y, we must count
the vertex intersection as only one point.
Scan Line Polygon Fill
• The topological difference between scan Line y and scan
Line y’
• For scan line y, the two edges sharing an intersection
vertex are on opposite sides of the scan line.
– Count this vertex as one intersection point
• For scan line y’, the two intersecting edges are both
above the scan line
Scan Line Polygon Fill
• We can distinguish these cases by tracing around the
polygon boundary either in clockwise or
counterclockwise order and observing the relative
changes in vertex y coordinates as we move from one
edge to the next.
• Let (y1, y2) and (y2, y3) be the endpoint y values of two
consecutive edges. If y1, y2, y3 monotonically increase
or decrease, we need to count the middle vertex as a
single intersection point for any scan line passing
through that vertex.
Scan Line Polygon Fill
•
One method for implementing the adjustment
to the vertex intersection count is to shorten
some polygon edges to split those vertices that
should be counted as one intersection
– When the end point y coordinates of the two edges
are increasing , the y value of the upper endpoint for
the current edge is decreased by 1
– When the endpoint y values are monotonically
decreasing, we decrease the y coordinate of the
upper endpoint of the edge following the current
edge
Scan Line Polygon Fill Algorithm
(a)
(b)
Adjusting endpoint values for a polygon, as we process edges in order
around the polygon perimeter. The edge currently being processed is
indicated as a solid like. In (a), the y coordinate of the upper endpoint of
the current edge id decreased by 1. In (b), the y coordinate of the upper
end point of the next edge is decreased by 1
The scan conversion algorithm works as follows
i. Intersect each scanline with all edges
ii. Sort intersections in x
iii. Calculate parity of intersections to determine in/out
iv. Fill the “in” pixels
Special cases to be handled:
i. Horizontal edges should be excluded
ii. Vertices lying on scanlines handled by shortening of edges,
• Coherence between scanlines tells us that
Edges that intersect scanline y are likely to intersect y + 1
X changes predictably from scanline y to y + 1 (Incremental Calculation
Possible)
(Xk + 1, Yk + 1)
(Xk , Yk )
Scan Line yk + 1
Scan Line yk
•
The slope of the edge is constant from one scan line to the next:
– let m denote the slope of the edge.
yk 1  yk  1
1
xk 1  xk 
m
•
Each successive x is computed by adding the inverse of the slope and
rounding to the nearest integer
Integer operations
•
Recall that slope is the ratio of two integers:
y
m
x
•
So, incremental calculation of x can be expressed as
y
xk 1  xk 
x
Integer operations
• How to compute x intercepts incrementally using
integer operations:
– Initialize a counter to 0
– Increment counter by x each time we move up to a
new scan line.
– If counter becomes greater or equal to y, increment
the current x intersection value by 1 and decrease the
counter by y
• Example: m=7/3
• Initial scanline counter = 0
• Next scanlines
– Counter =3
– Counter =6
– Counter =9
• increment x intercept by
• Reset counter to 9-7 = 2
Line with slope 7/3
yk
xk
decrement
decrement
0
4
1
5
2
6
3
0 y0
counter
decrement
x0
• Above scheme truncates integers.
• How do we round to nearest integer?
– We need to compare the counter to y/2.
– Can be done by integer arithmetic by incrementing counter by
2x at each step and comparing with y
– When the counetr is greater than or equal to y, increase x value
by 1 and decrement the counter by 2y
• Example: m=7/3
• Initial scanline counter = 0
• Next scanlines
– Counter =6
– Counter =12
• increment x intercept by
• Reset counter to 12-(2*7) = -2
decrement
decrement
0
-6
2
-4
4
decrement -2
6
0 y0
counter
x0
Sorted Edge Table (SET)
In SET, there is an entry for each scanline.
Traverse edges of the polygon to construct a Sorted Edge Table (SET)
1. Eliminate horizontal edges
2. Add edge to linked-list for the scan line corresponding to the y_lower
vertex. Shorten edges if necessary to resolve the vertex-intersection
problem.
3. For each edge entry, store the following:
-
y_upper: the largest y value on that edge (last scanline to consider)
x_lower: the x intercept at that scanline (initial x value)
1/m: for incrementing x
4. For each scan line the edges are sorted from left to right (based on x)
B
C
E
C’
yC
yB xC 1/ mBC
yA
yE xA 1 / mAE
D
A
1
0
yB xA 1 / mAB
Active Edge List (AEL)
• Construct Active Edge List during scan conversion. AEL
is a linked list of active edges on the current scanline, y.
The active edges are kept sorted by x
– The active edge list contains all the edges crossed by that scan
line.
– As we move up, update the active edge list using the sorted
edge table if necessary.
Algorithm
1.
2.
3.
Set y to the smallest y coordinate that has an entry in the SET; i.e, y for the
first nonempty bucket.
Initialize the AEL to be empty.
For each scanline y repeat:
3.1 Copy from SET bucket y to the AEL those edges whose y_min = y
(entering edges).
3.2 The sort the AEL on x is easier because SET is presorted.
3.3 Fill in desired pixel values on scanline y by using pairs of x coordinates
from AEL.
3.5 Remove from the AEL those entries for which y = y_max (edges not
involved in the next scanline.)
3.4 Increment y by 1 (to the coordinate of the next scanline).
3.5 For each nonvertical edge remaining in the AEL, update x for the new y.
Flood Fill Algorithm
• These algorithms assume that at least one pixel interior
to a polygon or region is known
-assume that the boundary is defined by Bresenham’s
Flood Fill
• Fill can be done recursively if we know a seed
point (x,y) located inside (WHITE)
• Scan convert edges into buffer in edge/inside
color (BLACK)
flood_fill(int x, int y) {
if(read_pixel(x,y)= = WHITE) {
write_pixel(x,y,BLACK);
flood_fill(x-1, y);
flood_fill(x+1, y);
flood_fill(x, y+1);
flood_fill(x, y-1);
}
}
Depth or Z-Buffer
• For each pixel, we store COLOR (color buffer)
and DEPTH (depth buffer).
• Algorithm:
Initialize all elements of buffer COLOR(row, col) to
background color, and DEPTH(row, col) to
maximum-depth;
FOR EACH polygon:
Rasterize polygon to frame;
FOR EACH pixel center (x, y) that is covered:
IF polygon depth at (x, y) < DEPTH(x, y)
THEN COLOR(x, y) = polygon color at (x, y)
AND DEPTH(x, y) = polygon depth at (x, y)
Depth Buffer Operation
Frame
Depth


















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Initialize (“New Frame”)
Depth Buffer Operation – First
Polygon
Frame
Depth






24




26
27



28
29
30

Pink Triangle -- depths computed at pixel centers
Depth Buffer Operation -- First
Polygon
Frame
Depth






24




26
27



28
29
30

Pink Triangle -- pixel values assigned
Depth Buffer Operation – Second
Polygon
Frame
Depth


10
9


24
9
8


26
8
7


28
7
6

Green Rectangle -- depths computed at pixel centers
Depth Buffer Operation – Second
Polygon
Frame
Depth


10
9


24
9
8


26
8
7


28
7
6

Green Rectangle -- pixel values assigned: NOTE REPLACEMENTS!
Depth Buffer Operation – Third
Polygon
Frame
68 not stored
Depth


10
9
72

24
9
8
75

26
8
7


28
7
6

Blue Pentagon -- depths computed at pixel centers
71 not
stored
Depth Buffer Operation – Third
Polygon
Frame
Depth


10
9
72

24
9
8
75

26
8
7


28
7
6

Blue Pentagon -- pixel values assigned: NOTE ‘GOES BEHIND’!
screen
scan-line