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. Barycentric Coordinates Use non-orthogonal coordinates to describe position relative to vertices – Scaled edge equations 0 on edge, 1 at opposite vertex

13. Barycentric Example

14. Barycentric Coordinates Computing coordinates – Equations for ,  and  in book – Solutions to linear equations of x,y Ratio of areas / ratio of cross products – Area = 0.5*b*h – Length of cross product = 2*area of triangle Matrix form

15. Area Computation

16. Barycentric Matrix Computation Edge equation for each coordinate Plug in known values at the vertices

17. Barycentric Matrix Computation Write linear system in matrix form Solve

18. Barycentric Matrix Computation Solutions = rows of inverse matrix

19. Barycentric Rasterization For all x do For all y do Compute ( , ,  ) for (x,y) If (   [0,1] and   [0,1] and   [0,1] then c =  c 0 +  c 1 +  c 2 Draw pixel (x,y) with color c

20. Barycentric Rasterization x min = floor(min(x 0,x 1,x 2 )) x max = ceiling(max(x 0,x 1,x 2 )) y min = floor(min(y 0,y 1,y 2 )) y max = ceiling(max(y 0,y 1,y 2 )) for y = y min to y max do for x = x min to x max do  =  (x,y)  =  (x,y)  =  (x,y) If (   [0,1] and   [0,1] and   [0,1] then c =  c 0 +  c 1 +  c 2 Draw pixel (x,y) with color c

21. Incremental Computation , , and  are linear in x and y What about  (x+1,y)?