1. 3/4/04© University of Wisconsin, CS559 Spring 2004 Last Time Clipping Lines –Cohen-Sutherland: The use of outcodes and early reject/accept tests –Liang-Barsky: Parametric clipping, find intersection parameters, label as entering leaving, reason based on ordering of enter/leaving Clipping Polygons –Weiler-Atherton lets you clip general polygons against general clip regions Drawing Lines: –Bresenham’s algorithm draws lines using only addition, no rounding

2. 3/4/04© University of Wisconsin, CS559 Spring 2004 Today Drawing lines –Bresenham example Drawing polygons Basic anti-aliasing

3. 3/4/04© University of Wisconsin, CS559 Spring 2004 Bresenham’s Algorithm For integers, slope between 0 and 1: –x=x 1, y=y 1, d=2dy - dx, draw (x, y) –until x=x 2 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

4. 3/4/04© University of Wisconsin, CS559 Spring 2004 Example: (2,2) to (7,6)  x=5,  y=4 xyd 1 23456781 2 3 4 5 6 7

5. 3/4/04© University of Wisconsin, CS559 Spring 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

6. 3/4/04© University of Wisconsin, CS559 Spring 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 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

7. 3/4/04© University of Wisconsin, CS559 Spring 2004 Polygon ParityNon-zero Winding No. Non-exterior Inside/Outside Rules

8. 3/4/04© University of Wisconsin, CS559 Spring 2004 What is inside - 2? Assume sampling with an array of spikes If spike is inside, pixel is inside

9. 3/4/04© University of Wisconsin, CS559 Spring 2004 What is inside - 2? Assume sampling with an array of spikes If spike is inside, pixel is inside

10. 3/4/04© University of Wisconsin, CS559 Spring 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 Rule: if (x+ , y+  ) is in, (x,y) is in (for arbitrarily small  ) What if a pixel is on a vertex? Does our rule still work?

11. 3/4/04© University of Wisconsin, CS559 Spring 2004 Ambiguous Case 1 Rule: –On edge? If (x+ , y+  ) is in, pixel is in –Which pixels are colored?

12. 3/4/04© University of Wisconsin, CS559 Spring 2004 Ambiguous Case 1 Rule: –Keep left and bottom edges –Assuming y increases in the up direction –If rectangles meet at an edge, how often is the edge pixel drawn?

13. 3/4/04© University of Wisconsin, CS559 Spring 2004 Ambiguous Case 2

14. 3/4/04© University of Wisconsin, CS559 Spring 2004 Ambiguous Case 2 ? ? ? ? or

15. 3/4/04© University of Wisconsin, CS559 Spring 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 2 3 4 5 6

16. 3/4/04© University of Wisconsin, CS559 Spring 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

17. 3/4/04© University of Wisconsin, CS559 Spring 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?

18. 3/4/04© University of Wisconsin, CS559 Spring 2004 Spans Fill rows from bottom to top, one at a time Have x min, x max for each span Define: –floor(x): largest integer < x –ceiling(x): smallest integer >=x Fill from ceiling(x min ) up to floor(x max ) Consistent with convention x min,, ceiling(x min ) x max floor(x max )

19. 3/4/04© University of Wisconsin, CS559 Spring 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? All can be resolved by referring to our convention about what polygon pixel belongs to

20. 3/4/04© University of Wisconsin, CS559 Spring 2004 Updating Edges Each edge is a line of the form: Next row is: So, each current edge can have it’s x position updated by adding a constant stored with the edge Other values may also be updated, such as depth or color information

21. 3/4/04© University of Wisconsin, CS559 Spring 2004 When are Edges Relevant (1) Use figures and convention to determine when edge is irrelevant –For y =y max of edge Similarly, edge is relevant when y>=y min and y<y max of edge What about horizontal edges? –m’ is infinite

22. 3/4/04© University of Wisconsin, CS559 Spring 2004 When are Edges Relevant (2) 1 2 3 4 3,4 1,3 1,2 Convex polygon: Always only two edges active

23. 3/4/04© University of Wisconsin, CS559 Spring 2004 When are Edges Relevant (3) 1,3 Horizontal edges? 1 2 3 4 4? 2?

24. 3/4/04© University of Wisconsin, CS559 Spring 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 In practice, use a version of the midpoint algorithm to update edges –What is the “midpoint” tested? –Fast increment, no rounding, handles floating point

25. 3/4/04© University of Wisconsin, CS559 Spring 2004 yiyi y i +1 x i +1 Midpoint Method For Polygon Edges Consider a left edge with slope [0-1] Always want to draw point on line or below Consider the point (x i +1,y i +1). Is it above or below the line? xixi Choose (x i +1,y i +1) yiyi y i +1 x i +1xixi Choose (x i +1,y i )

26. 3/4/04© University of Wisconsin, CS559 Spring 2004 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

27. 3/4/04© University of Wisconsin, CS559 Spring 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

28. 3/4/04© University of Wisconsin, CS559 Spring 2004 Pre-Filtered Primitives We can simulate filtering by rendering “thick” primitives, with , and compositing Expensive, and requires the ability to do compositing Hardware method: Keep sub-pixel masks tracking coverage Ideal 1/6 2/3 1/6 Filter  =1/6  =2/3  =1/6 over  =1/6  =2/3  =1/6 Pre-Filtered and composited

29. 3/4/04© University of Wisconsin, CS559 Spring 2004 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

30. 3/4/04© University of Wisconsin, CS559 Spring 2004 Where We Stand At this point we know how to: –Convert points from local to window coordinates –Clip polygons and lines to the view volume –Determine which pixels are covered by any given line or polygon –Anti-alias Next thing: –Determine which polygon is in front