1. Clipping Polygon Clipping Polygon : Area primitive
Simple Polygon: Planar set of ordered points
No line crossings
No holes V3 V4 V1 V2
Simple Polygon
Line Crossing
Hole

2. Clipping Polygon Clipping Polygon : Area primitive Non-Convex Polygon

3. Clipping Polygon Clipping Sutherland-Hodgman Window must be convex
Polygon to be clipped can be convex or non-convex
Polygon
Polygon
Window

4. Clipping Polygon Clipping Sutherland-Hodgman Window must be convex
Polygon to be clipped can be convex or non-convex

5. Clipping
Polygon Clipping
Sutherland-Hodgman

6. Clipping Polygon Clipping Sutherland-Hodgman Approach
Polygon to be clipped is given as v1, v2, …, vn
Polygon edge is a pair [vi, vi+1]
Process all polygon edges in succession against
a window edge
polygon (v1, v2, …, vn) polygon (w1, w2, …, wm)
Repeat on resulting polygon with next window edge

7. Clipping Polygon Clipping Sutherland-Hodgman Approach Four Cases
s = vi is the polygon edge starting vertex
p = vi+1 is the polygon edge ending vertex
i is a polygon-edge/window-edge intersection point
wj is the next polygon vertex to be output

8. Clipping Polygon Clipping Sutherland-Hodgman Approach
Case 1: Polygon edge is entirely inside the window edge
Inside
Outside s
p is next vertex of resulting polygon
p wj and j+1 j p
Output

9. Clipping Polygon Clipping Sutherland-Hodgman Approach
Case 2: Polygon edge crosses window edge going out
Inside
Outside
Intersection point i is next vertex
of resulting polygon
i wj and j+1 j p s i
Output

10. Clipping Polygon Clipping Sutherland-Hodgman Approach
Case 3: Polygon edge is entirely outside the window edge
Inside
Outside p
No output s

11. Clipping Polygon Clipping Sutherland-Hodgman Approach
Case 4: Polygon edge crosses window edge going in
Inside
Outside
Output s p i
Intersection point i and p are next two
vertices of resulting polygon
i wj and p wj+1 and j+2 j

12. Clipping Polygon Clipping Sutherland-Hodgman Example s5 s4 s3 s2 s1
Window s1

13. Clipping Polygon Clipping Sutherland-Hodgman Example s5 s4 i2 i1 s3 s2

14. Clipping
Polygon Clipping
Sutherland-Hodgman
Example t4 t3 t2 t1 t5

15. Clipping
Polygon Clipping
Sutherland-Hodgman
Example u3 u2 u1 u5 u4

16. Clipping Polygon Clipping Sutherland-Hodgman Example v2 i3 v1 i6 v5 i5

17. Clipping
Polygon Clipping
Sutherland-Hodgman
Example w1 w6 w5 w4 w3 w2

18. Polygon Scan Conversion
Polygon Filling
Consider first triangle

19. Polygon Scan Conversion
Polygon Filling
Consider first triangle
Color all pixels inside triangle
Inside (containment) test

20. Polygon Scan Conversion
Polygon Filling
Triangle
Use horizontal spans.
Process horizontal spans in
scan-line order.
For the next spans use edge slopes XL XR

21. Polygon Scan Conversion
Polygon Filling
Polygon
How do we decide what parts of the
span should be filled ?

22. Polygon Scan Conversion
Polygon Filling
Polygon
How do we decide what parts of the
span should be filled ?
Parity check
if odd fill
if even don’t fill

23. Polygon Scan Conversion
Polygon Filling
Polygon
What happens here?

24. Polygon Scan Conversion
Polygon Filling
Polygon
What happens here?

25. Polygon Scan Conversion
Polygon Filling
Polygon
Recursive seed filling

27. Geometrical Transformation
Translation P ( x , y ) P ' ( x ' , y ' ) P’ x ' = x + t ; y ' = y + t x y P T P ' = P + T P = [ x , y ]; T = [ t , t ] x y

28. Geometrical Transformation
Scaling P ( x , y ) P ' ( x ' , y ' ) x ' = s x ; y ' = s y x y é s ù x [ x ' , y ' ] = [ x , y ] ê ú s ë û y P ' = P . S

29. Geometrical Transformation
Scaling P ( x , y ) P ' ( x ' , y ' ) x ' = s x ; y ' = s y x y é s ù [ x ' ' x , y ] = [ x , y ] ê ú s ë û y P ' = P . S s = s :
uniform
scaling x y s s :
differenti al
scaling x y

30. Geometrical Transformation
Rotation P ( x , y ) P ' ( x ' , y ' ) x = r
cos( φ ); y = r
sin( φ ) x ' = r
cos( φ + θ ) P’ y ' = r
sin( φ + θ ) q P ' = - f x x
cos( θ ) y
sin( θ ) y ' = x
sin( θ ) + y
cos( θ ) é
cos( θ )
sin( θ ) ù [ x ' , y ' ] = [ x , y ] ê ú - ë
sin( θ )
cos( θ ) û P ' = P . R θ

31. Geometrical Transformation
General 2x2 Matrix X X ' X ' = X . T é a b ù [ x ' , y ' ] = [ x , y ] = [( ax + cy ), ( bx + dy )] ê ú ë c d û
If a = d = 1 and b = c = 0
T = Identity Matrix
X’ = X

32. Geometrical Transformation
General 2x2 Matrix y
If b = c = 0
X’ = [ax, dy] : Scaling
If b = c = 0 and a = -1, d = 1
X’ = [-x, y] : Reflection P’ P x

33. Geometrical Transformation
General 2x2 Matrix é 1 ù
Shearing in X [ x ' , y ' ] = [ x , y ] = [ x + cy , y ] ê ú ë c 1 û

34. Geometrical Transformation
General 2x2 Matrix é 1 b ù
Shearing in Y [ x ' , y ' ] = [ x , y ] = [ x , bx + y ] ê ú ë 1 û

35. Geometrical Transformation
Homogenous Coordinates
Scale/Rotate/Reflect/Shear: X’ = XT
Translate: X’ = X + T P = [ x , y ] P = [ x , y , w ] h
Multiple values for the same point
e.g., (2, 3, 6) and (4, 6, 12) are same points

36. Geometrical Transformation
Homogenous Coordinates w
(x, y,w)
(x/w, y/w,1)
w=1 x y

37. Geometrical Transformation
Homogenous Coordinates
Unifying representation for transformation
Transformation matrix from 2x2 to 3x3 X X ' X ' = X . T é a b ù ê ú [ x ' , y ' , w ] = [ x , y , 1 ] c d ê ú ë ê l m 1 ú û = [( ax + cy + l ), ( bx + dy + m ), 1 ]

38. Geometrical Transformation
Homogenous Coordinates
Translation é 1 ù ê ú T = 1 ê ú ê l m 1 ú ë û X ' = XT [ x ' , y ' ] = [ x + l , y + m ]

39. Geometrical Transformation
Homogenous Coordinates
Scaling/Rotation/Shear é a b ù ê ú T = c d ê ú ë ê 1 ú û

40. Geometrical Transformation
Homogenous Coordinates
Successive translations: T1 = (l1, m1), T2 = (l2, m2)
After T1
After T1 and T2
Successive translations are additive

41. Geometrical Transformation
Homogenous Coordinates
Successive scaling: S1 = (sx1, sy1), S2 = (sx2, sy2)
After S1
After S1 and S2
Successive scaling is multiplicative

42. Geometrical Transformation
Homogenous Coordinates
Successive rotations: R(q), R(f)
After R(q)
After R(q) and R(f)
Successive rotations are additive

43. Geometrical Transformation
Composition of transformation
Rotation about arbitrary point y
Rotation about origin is known
Translate such that P becomes O
Rotate (about O)
Translate back to P P O x

44. Geometrical Transformation
Composition of transformation
Rotation about arbitrary point y
Rotation about origin is known
Translate such that P becomes O P O x

45. Geometrical Transformation
Composition of transformation
Rotation about arbitrary point y
Rotation about origin is known
Rotate about O O x

46. Geometrical Transformation
Composition of transformation
Rotation about arbitrary point y
Rotation about origin is known
Translate back to P P O x

47. Geometrical Transformation
Composition of transformation
Composite transformation

48. Geometrical Transformation
Composition of transformation
Reflection about an arbitrary line B y C A O x

49. Geometrical Transformation
Composition of transformation
Reflection about an arbitrary line y B
Translation C A O x

50. Geometrical Transformation
Composition of transformation
Reflection about an arbitrary line y B
Rotation A C O x

51. Geometrical Transformation
Composition of transformation
Reflection about an arbitrary line y
Reflection O x A C B

52. Geometrical Transformation
Composition of transformation
Reflection about an arbitrary line y
Rotation C O A x B

53. Geometrical Transformation
Composition of transformation
Reflection about an arbitrary line y
Translation C A O x B

54. Geometrical Transformation
Composition of transformation
Given T1 and T2
In general,

55. Geometrical Transformation
Rigid Transformations
Square remains square
Preserves length and angles
Sequence of rotations and translations é r r ù 11 12 ê ú T = r r ê 21 22 ú ê ë l m 1 ú û

56. Geometrical Transformation
Affine Transformations
Preserves parallelism
Sequence of rotations, translations, scaling and shear
Linear transformation is when no translation

57. Geometrical Transformation
General Transformation
3x3 matrix