1. Graphics Pipeline Rasterization CMSC 435/634

2. Drawing Terms Primitive – Basic shape, drawn directly – Compare to building from simpler shapes Rasterization or Scan Conversion – Find pixels for a primitive – Usually for algorithms that generate all pixels for one primitive at a time – Compare to ray tracing: all primitives for one pixel

3. Line Drawing Given endpoints of line, which pixels to draw?

4. Line Drawing Given endpoints of line, which pixels to draw?

5. Line Drawing Given endpoints of line, which pixels to draw? Assume one pixel per x. Which y? Look at midpoint between candidate pixels ? ? ? ? ? ? ? ?

6. Line Drawing Plug midpoint into implicit line equation Sign decides: called a decision variable Incremental update

7. Line Drawing Implicit line equation Midpoint algorithm y = y 0 d = f(x 0 +1, y 0 +0.5) for x = x 0 to x 1 draw(x,y) if (d < 0) then y = y+1 d = d + (x 1 - x 0 ) + (y 0 - y 1 ) else d = d + (y 0 - y 1 )

8. Polygon Rasterization Problem – How to generate filled polygons (by determining which pixel positions are inside the polygon) – Conversion from continuous to discrete domain Concepts – Spatial coherence – Span coherence – Edge coherence

9. Scanning Rectangles for ( y from y 0 to y 1 ) for ( x from x 0 to x 1 ) Write Pixel (x, y)

10. Scanning Rectangles (2) for ( y from y 0 to y 1 ) for ( x from x 0 to x 1 ) Write Pixel (x, y)

11. Scanning Rectangles (3) for ( y from y 0 to y 1 ) for ( x from x 0 to x 1 ) Write Pixel (x, y)

12. Triangle Rasterization Barycentric coordinates are decision variables

13. Barycentric Triangle Rasterization For all y in y min to y max do For all x in x min to x max do Compute ( , ,  ) for (x,y) If (  ≥ 0 and  ≥ 0 and  ≥ 0) then c =  c 0 +  c 1 +  c 2 Draw pixel(x,y) with color c

14. Incremental Computation , , and  are linear in X and Y What about pixel-to-pixel updates?

15. “Clipless” Homogeneous Rasterization Compute barycentrics using homogeneous coordinates Extra edge equations for clip edges – Compute t for clip plane at each vertex – Only visible (w>near) pixels will be drawn Adds computation – Divide by w per pixel instead of per vertex – But avoids branching and extra triangles – Good for hardware

16. Homogeneous Barycentrics Each barycentric is – Equal to 1 at one vertex – Equal to 0 at the other two

17. Homogeneous Barycentrics Write formula for barycentric coordinate in homogeneous form

18. Homogeneous Barycentrics This defines a system of three equations or

19. Homogeneous Barycentrics Equation (again) Which we can solve:

20. Homogeneous Barycentrics Coefficients for all three:

21. Changes to Rasterization NONE! – Coefficients computed with homogeneous coords – But they’re the same coefficients!

22. Homogenous Clip Plane Clip parameter at each vertex Clipping decision variable coefficients