1. NOTE

2. DDA Line Drawing Algorithmy y x x

3. Bresenham’s Line Drawing Algorithm

4. Bresenham’s Line Drawing Algorithm

5. Scan Line Polygon FillingB A

6. 2
If the scan line is crossing a vertex, treat it as two intersection points 2 2 2 2

7. 2
If the scan line is crossing a vertex, treat it as two intersection points
Except the slops of neighboring edges are increasing/decreasing monotonically. 1 2 2 2

8. If the scan line is crossing a vertex, treat it as two intersection points
Except the slops of neighboring edges are increasing/decreasing monotonically.
Separate these edges
 Is it really necessary?

9. Edge Table
Active Edge List

10. Edge Table
Add & Sort !!
Active Edge List

11. Edge Table
Sort !!
Active Edge List

12. Edge Table
Add & Sort !!
Active Edge List

13. Edge Table
Active Edge List

14. Edge Table
Active Edge List

15. Edge Table
Active Edge List

16. Edge Table
Active Edge List

17. Edge Table
Active Edge List

18. Edge Table
Active Edge List

19. Edge Table
Active Edge List

20. Edge Table
Active Edge List

21. Edge Table
Sort !!
Active Edge List

22. Edge Table
Active Edge List

23. Edge Table
Active Edge List

24. Edge Table
Empty
Active Edge List

25. Scan Line Polygon Filling
Construct the Edge Table (ET) excluding horizontal edge;
Active Edge List (AEL) = null;
for y = Ymin to Ymax
Merge-sort ET[y] into AEL by x value
Fill between pairs of x in AEL
for each edge in AEL
if edge.ymax = y
remove edge from AEL
else
edge.x = edge.x + dx/dy
sort AEL by x value
end scan_fill

26. Boundary Polygon Filling
Proceed to Neighboring Pixels
4-Connected
8-Connected

27. Inside/Outside Test Self-Intersections Odd-Even rule
Nonzero winding number rule

28. Anti-aliasing - Supersampling
Subpixels increase resolution
Maximum intensity level: 9 or more 22 1 2 4 21 20 10 11 12

29. Anti-aliasing - Filtering
Box Filter
Cone Filter
Gaussian Filter

30. Anti-aliasing – Area sampling22 21 20 10 11 12

31. Transformation Matrices using Homogeneous Coordinates
2D Affine Transformation
Translation
Rotation
Scaling
Shear
Mirror

32. Transformation Matrices using Homogeneous Coordinates
Basic 2D transformations as 3x3 matrices
Translation
Scaling
Rotation
Shear

33. Transformation Matrices using Homogeneous Coordinates
Reflection with respect to an axis
X Axis
Y Axis
Origin 1 3 2 1’ 3’ 2’ x y 3 1’ 3’ 2’ 1 2 x y x y 1 3 2 1’ 3’ 2’

34. Transformation Matrices using Homogeneous Coordinates
Combined transformations
By matrix multiplication
Efficiency with pre-multiplication
Matrix multiplication is associative

35. Pivot Point Rotation
Fixed Point Scaling

36. Pivot Point Rotation (0,0) (1,0) (1,1) (0,1) hx= 0.5 yref= -1 yref= –1
(0.5,0)
(1.5,0)
(2,1)
(0,0)
(1,0)
(1,1)
(0,1)
hy= 0.5
xref= -1 x y
(0,0.5)
(1,2)
(0,1.5)
xref= –1

37. Is the point (x, y) inside the clipping window?
Point Clipping
Is the point (x, y) inside the clipping window?
wy2
Inside =
(x>=wx1) &&
(x<=wx2) &&
(y>=wy1) &&
(y<=wy2);
(x, y)
wy1
wx1
wx2

38. Cohen-Sutherland Line Clipping
Classify some lines quickly
AND of bit codes of two endpoints (0 if not fully outside) P7
1001
0001
0101
Bit 4 P1 P4 P8
1000 P3
0000
0100 P2 P6
P10
Bit 3
1010
0010
0110 P5 P9
Bit 1
Bit 2

39. Sutherland-Hodgeman Polygon Clipping
Clip to each window boundary at a time

40. Sutherland-Hodgeman Polygon Clipping
Clip to each window boundary at a time

41. Sutherland-Hodgeman Polygon Clipping
Clip to each window boundary at a time

42. Sutherland-Hodgeman Polygon Clipping
Clip to each window boundary at a time

43. Sutherland-Hodgeman Polygon Clipping
Clip to each window boundary at a time

44. Sutherland-Hodgeman Polygon Clipping
Clip to each window boundary at a time P2 P1
Window
Boundary
Inside
Outside P5 P3 P4

45. Sutherland-Hodgeman Polygon Clipping
Clip to each window boundary at a time P2 P1
Window
Boundary
Inside P’ P”
Outside P5 P3 P4

46. Sutherland-Hodgeman Polygon Clipping
Clip to each window boundary at a time P2 P1
Window
Boundary
Inside P’ P”
Outside

47. Viewport Transformation
Transform 2D geometric primitives
From screen coordinate (projection coordinates)
To image coordinate (device coordinates)
Screen
Image
Viewport

48. Viewport Transformation
Window-to-viewport mapping
Window
Viewport
wy2
vy2
(wx, wy)
(vx, vy)
wy1
vy1
wx1
wx2
vx1
vx2
Screen Coordinates
Image Coordinates
vx = vx1 + (wx – wx1) * (vx2 – vx1) / (wx2 – wx1);
vy = vy1 + (wy – wy1) * (vy2 – vy1) / (wy2 – wy1);