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. Assume one pixel per column (x index), which row (y index)? Choose based on relation of line to midpoint between candidate pixels ? ? Line Drawing ? ? ? ? ? ?

6. Choose with decision variable Plug midpoint into implicit line equation 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 x in x min to x max do For all y in y min to y 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. “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

15. Homogeneous Barycentrics Each barycentric coordinate is a linear function of X and Y It is 1 at one vertex and 0 at the other two

16. Homogeneous Barycentrics Write formula for barycentric coordinate in homogeneous form This defines a system of three equations

17. Homogeneous Barycentrics Know Simplifies the linear equation to Which we can solve:

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