1. © University of Wisconsin, CS559 Fall 2004
Last Time
Clipping
Liang-Barksy
Weiler-Atherton
Drawing points
Intro to drawing lines
Midterm Oct 28
10/21/04
© University of Wisconsin, CS559 Fall 2004

2. © University of Wisconsin, CS559 Fall 2004
Today
Drawing lines
Filling polygons
10/21/04
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3. © University of Wisconsin, CS559 Fall 2004
Drawing Points
When points are mapped into window coordinates, they could land anywhere – not just at a pixel center
Solution is the simple, obvious one
Map to window space
Fill the closest pixel
Can also specify a radius – fill a square of that size, or fill a circle
Square is faster
What function are we sampling with?
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4. © University of Wisconsin, CS559 Fall 2004
Drawing Lines
Task: Decide which pixels to fill (samples to use) to represent a line
We know that all of the line lies inside the visible region (clipping gave us this!)
Issues:
If slope between -1 and 1, one pixel per column. Otherwise, one pixel per row
Constant brightness? Lines of the same length should light the same number of pixels (we normally ignore this)
Anti-aliasing? (Getting rid of the “jaggies”)
Sampling theory?
10/21/04
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5. Line Drawing Algorithms
Consider lines of the form y=m x + c, where m=y/x, 0<m<1, integer coordinates
All others follow by symmetry, modify for real numbers
Variety of slow algorithms (Why slow?):
step x, compute new y at each step by equation, rounding:
step x, compute new y at each step by adding m to old y, rounding:
10/21/04
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6. Bresenham’s Algorithm Overview
Aim: For each x, plot the pixel whose y-value is closest to the line
Given (xi,yi), must choose from either (xi+1,yi+1) or (xi+1,yi)
Idea: compute a decision variable
Value that will determine which pixel to draw
Easy to update from one pixel to the next
Bresenham’s algorithm is the midpoint algorithm for lines
Other midpoint algorithms for conic sections (circles, ellipses)
10/21/04
© University of Wisconsin, CS559 Fall 2004

7. Bresenham’s Algorithm Overview
Input: Line with integer endpoints, slope 0<m<1
Aim: For each x, plot the pixel whose y-value is closest to the line
Given (xi,yi), must choose from either (xi+1,yi+1) or (xi+1,yi)
Idea: compute a decision variable
Value that will determine which pixel to draw
Easy to update from one pixel to the next
Bresenham’s algorithm is the midpoint algorithm for lines
Other midpoint algorithms for conic sections (circles, ellipses)
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8. © University of Wisconsin, CS559 Fall 2004
Midpoint Methods
Consider the midpoint between (xi+1,yi+1) and (xi+1,yi)
If it’s above the line, we choose (xi+1,yi), otherwise we choose (xi+1,yi+1)
yi+1
yi+1 yi yi xi
xi+1 xi
xi+1
Choose (xi+1,yi)
Choose (xi+1,yi+1)
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9. Midpoint Decision Variable
Write the line in implicit form:
The value of F(x,y) tells us where points are with respect to the line
F(x,y)=0: the point is on the line
F(x,y)<0: The point is above the line
F(x,y)>0: The point is below the line
The decision variable is the value of di = 2F(xi+1,yi+0.5)
The factor of two makes the math easier
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10. © University of Wisconsin, CS559 Fall 2004
What Can We Decide?
di negative => next point at (xi+1,yi)
di positive => next point at (xi+1,yi+1)
At each point, we compute di and decide which pixel to draw
How do we update it? What is di+1?
10/21/04
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11. Updating The Decision Variable
dk+1 is the old value, dk, plus an increment:
If we chose yi+1=yi+1:
If we chose yi+1=yi:
What is d1 (assuming integer endpoints)?
Notice that we don’t need c any more
10/21/04
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12. Bresenham’s Algorithm
For integers, slope between 0 and 1:
x=x1, y=y1, d=2dy - dx, draw (x, y)
until x=x2
x=x+1
If d>0 then { y=y+1, draw (x, y), d=d+2y - 2x }
If d<0 then { y=y, draw (x, y), d=d+2y }
Compute the constants (2y-2x and 2y ) once at the start
Inner loop does only adds and comparisons
For floating point, initialization is harder, x and y will be floating point, but still no rounding required
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13. © University of Wisconsin, CS559 Fall 2004
Example: (2,2) to (7,6)
x=5, y=4
x y d 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8
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14. © University of Wisconsin, CS559 Fall 2004
Filling polygons
Sampling polygons:
When is a pixel inside a polygon?
Given a pixel, which polygon does it lie in? Point location
Polygon representation:
Polygon defined by a list of edges - each is a pair of vertices
All vertices are inside the view volume and map to valid pixels (clipping gave us this).
Assume integers in window coordinates to simplify things for now
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15. © University of Wisconsin, CS559 Fall 2004
What is inside - 1?
Easy for simple polygons - no self intersections or holes
OpenGL requires these. Undefined for other cases
OpenGL also requires convex polygons – easy triangulation
For general polygons, three rules are possible:
Non-exterior rule: A point is inside if every ray to infinity intersects the polygon
Non-zero winding number rule: Draw a ray to infinity that does not hit a vertex, if the number of edges crossing in one direction is not equal to the number crossing the other way, the point is inside
Parity rule: Draw a ray to infinity and count the number or edges that cross it. If even, the point is outside, if odd, it’s inside
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16. © University of Wisconsin, CS559 Fall 2004
Inside/Outside Rules
Polygon
Non-exterior
Non-zero Winding No.
Parity
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17. © University of Wisconsin, CS559 Fall 2004
What is inside - 2?
Assume sampling with an array of spikes
If spike is inside, pixel is inside
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18. © University of Wisconsin, CS559 Fall 2004
Ambiguous Case
Ambiguous cases: What if a pixel lies on an edge?
Problem because if two polygons share a common edge, we don’t want pixels on the edge to belong to both
Ambiguity would lead to different results if the drawing order were different
Aim: A consistent rule that (almost) always results in one polygon “owning” the edge
One Rule: if (x+, y+) is in, (x,y) is in (for arbitrarily small )
Another: edge belongs to polygon on same side as some off-screen point
What if a pixel is on a vertex? Do our rules still work?
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19. © University of Wisconsin, CS559 Fall 2004
Ambiguous Case 1
Rule 1:
On edge? If (x+, y+) is in, pixel is in
Which pixels are colored?
Rule 2:
If polygon is on same side as point (huge,huge)
Which pixels?
Corners?
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20. © University of Wisconsin, CS559 Fall 2004
Ambiguous Case 2
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21. © University of Wisconsin, CS559 Fall 2004
Ambiguous Case 2 ? ? ? or ?
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22. © University of Wisconsin, CS559 Fall 2004
Really Ambiguous
We will accept ambiguity in such cases
The center pixel may end up colored by one of two polygons in this case
Which two? 1 6 2 5 3 4
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23. © University of Wisconsin, CS559 Fall 2004
Exploiting Coherence
When filling a polygon
Several contiguous pixels along a row tend to be in the polygon - a span of pixels
Scanline coherence
Consider whole spans, not individual pixels
The pixels required don’t vary much from one span to the next
Edge coherence
Incrementally update the span endpoints
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24. © University of Wisconsin, CS559 Fall 2004
Sweep Fill Algorithms
Algorithmic issues:
Reduce to filling many spans
Which edges define the span of pixels to fill?
How do you update these edges when moving from span to span?
What happens when you cross a vertex?
10/21/04
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25. © University of Wisconsin, CS559 Fall 2004
Sweep Fill Algorithms
Algorithmic issues:
Reduce to filling many spans
Which edges define the span of pixels to fill?
How do you update these edges when moving from span to span?
What happens when you cross a vertex?
10/21/04
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26. © University of Wisconsin, CS559 Fall 2004
Algorithm
For each row in the polygon:
Throw away irrelevant edges
Obtain newly relevant edges
Fill span
Update current edges
Issues:
How do we update existing edges?
When is an edge relevant/irrelevant?
What are the endpoints of a span
All can be resolved by referring to our convention about what polygon a pixel belongs to
10/21/04
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27. When are Edges Relevant (1)
Use figures and convention to determine which edges are needed for which scanlines (rows of pixels)
Edges from ymin to ymax are relevant for what y values?
What about horizontal edges?
ymax
ymin
10/21/04
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28. When are Edges Relevant (2)
Convex polygon:
Always only two edges active 2
1,2 1
1,3 3
3,4 4
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29. © University of Wisconsin, CS559 Fall 2004
Spans
Fill rows from bottom to top, one at a time
Have pixels xmin, xmax for each span
Fill [xmin, xmax)
Include xmin, exclude xmax
xmax
xmin,
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30. © University of Wisconsin, CS559 Fall 2004
Tracking xmin and xmax
Use a variant of the midpoint method, but not quite the same
Which pixel, with respect to the edge, do we wish to choose for xmin and xmax?
What is the decision variable?
10/21/04
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31. Midpoint Method For Polygon Edges
Consider a left edge with slope [0-1]
Always want to choose point on line or below
Consider the point (xi+1,yi+1). Is it above or below the line?
yi+1
yi+1 yi yi xi
xi+1 xi
xi+1
Choose (xi+1,yi+1)
Choose (xi+1,yi)
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32. © University of Wisconsin, CS559 Fall 2004
Sweep Fill Details
For convex polygons there are always 2 edges, a left and a right
Fixes the amount of memory required, good for hardware
Can generate memory addresses (for pixel writes) efficiently
You know address of leftmost pixel, and address of rightmost, can fill all in between quickly
Other values may also be updated, such as z, w or color information
Use a midpoint-like rule for these too
Modern hardware, with floating point frame buffers, just interpolates
10/21/04
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33. Extending to Arbitrary Polygons
Can be more than one span in any row
Keep sorted list of edges across row, fill between pairs of edges
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34. © University of Wisconsin, CS559 Fall 2004
Anti-Aliasing
Recall: We can’t sample and then accurately reconstruct an image that is not band-limited
Infinite Nyquist frequency
Attempting to sample sharp edges gives “jaggies”, or stair-step lines
Solution: Band-limit by filtering (pre-filtering)
What sort of filter will give a band-limited result?
But when doing computer rendering, we don’t have the original continuous function
10/21/04
© University of Wisconsin, CS559 Fall 2004

35. Alpha-based Anti-Aliasing
1/8
Set the  of a pixel to simulate a thick line
The pixel gets the line color, but with <=1
This supports the correct drawing of primitives one on top of the other
Draw back to front, and composite each primitive over the existing image
Only some hidden surface removal algorithms support it
1/4
.914
1/8
1/4
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1/4
1/8
.914
1/4
1/8
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36. © University of Wisconsin, CS559 Fall 2004
Calculating 
Consider a line as having thickness (all good drawing programs do this)
Consider pixels as little squares
Set  according to the proportion of the square covered by the line
The sub-pixel coverage interpretation of 
1/8
1/4
.914
1/8
1/4
.914
1/4
1/8
.914
1/4
1/8
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37. © University of Wisconsin, CS559 Fall 2004
Weighted Sampling
Instead of using the proportion of the area covered by the line, use convolution to do the sampling
Equivalent to filtering the line then point sampling the result
Place the “filter” at each pixel, and integrate product of pixel and line
Common filters are cones (like Bartlett) or Gaussians
10/21/04
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38. Post-Filtering (Supersampling)
Sample at a higher resolution than required for display, and filter image down
Easy to implement in hardware
Typical is 2x2 sampling per pixel, with simple averaging to get final
What kind of filter?
More advanced methods generate different samples (eg. not on regular grid) and filter properly
Issues of which samples to take, and how to filter them
10/21/04
© University of Wisconsin, CS559 Fall 2004