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Cyrus & Beck Clipping Algorithm
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![Cyrus & Beck Clipping Algorithm
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![Denominator < 0 → point entering clip region, classify as
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Denominator > 0 → point leaving clip region, classify as PL
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![Example Cyrus & Beck Clipping
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nL = i = [+1 0]
nR= -i= [-1 0]
nT = - ĵ = [0 -1]
nB= +ĵ=[0 +1]](https://image.slidesharecdn.com/materi05cg22232-230427215311-32de48ca/75/Materi_05_CG_2223_2_-pdf-14-2048.jpg)
![Example Cyrus & Beck Clipping
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w1 = [-1 1] – [0 0] = [-1 1]
D = [ 9 3] – [-1 1] = [10 2]
w1 ⋅ nL = [1 1]⋅[+1 0] = -1
D ⋅ nL = [10 2]⋅[+1 0] = 10
t = (w1 ⋅ nL / D⋅ nL) = -(-1/10) = 1/10](https://image.slidesharecdn.com/materi05cg22232-230427215311-32de48ca/75/Materi_05_CG_2223_2_-pdf-15-2048.jpg)
Materi_05_CG_2223_2_.pdf
- 1. Clipping 1. Line Clipping(Cyrus & Beck Algorithm) 2. Polygon Clipping 2022/2023 - 2
- 2. Cyrus & BeckClipping Algorithm • Treat line in parametric form. • Note line will cross all clip boundaries somewhere : 1
- 3. Cyrus & BeckClipping Algorithm • Treat line in parametric form. • Note line will cross all clip boundaries somewhere : Find all intersections, test afterwards to determine if they are on the clip rectangle. Define a point on the clipping plane and its normal. Notice the dot product between a vector from this point and the intersection is zero. 2
- 4.
- 5. Cyrus & BeckClipping Algorithm 4
- 6. Cyrus & BeckClipping Algorithm 5
- 7.
- 8. Edge Ej Nj P0 P1 PEJ 0 ] ) ( [ > − •EJ j P t P N 0 ] ) ( [ = − • EJ j P t P N 0 ] ) ( [ < − • EJ j P t P N t P P P t P ) ( ) ( 0 1 0 − + = Cyrus & Beck Clipping Algorithm 7
- 9. Cyrus & BeckClipping Algorithm 8 D N P P N t P P D t P P N P P N P t P P P N P t P N t P P P t P j EJ j j EJ j EJ j EJ j • − − • = − = = − • + − • = − − + • = − • − + = ] [ for t solve ), ( Let 0 ] [ ] [ 0 ] ) ( [ 0 ] ) ( [ ) ( ) ( 0 0 1 0 1 0 0 1 0 0 1 0 Note Sign of denominator is important. Must ≠ 0 ( ignore parallel lines ). Parametric form implies direction. Denominator < 0 → point entering clip region. Denominator > 0 → point leaving clip region.
- 10. Denominator < 0→ point entering clip region, classify as PE Denominator > 0 → point leaving clip region, classify as PL Edge Ej Nj D N P P N t j EJ j • − − • = ] [ 0 θ D Cyrus & Beck Clipping Algorithm 9
- 11. Cyrus & BeckClipping Algorithm 10 Sort PE’s and PL’s according to t. PE PL PL PE t t Draw from a PE → PL pair. PL < PE → no intersection
- 12. Cyrus & BeckClipping Algorithm 11 Note: PEi = f (boundary point) Lat = Latura (at edge)
- 13. Cyrus & BeckClipping Algorithm 12
- 14. Example Cyrus &Beck Clipping 13 nL = i = [+1 0] nR= -i= [-1 0] nT = - ĵ = [0 -1] nB= +ĵ=[0 +1]
- 15. Example Cyrus &Beck Clipping 14 w1 = [-1 1] – [0 0] = [-1 1] D = [ 9 3] – [-1 1] = [10 2] w1 ⋅ nL = [1 1]⋅[+1 0] = -1 D ⋅ nL = [10 2]⋅[+1 0] = 10 t = (w1 ⋅ nL / D⋅ nL) = -(-1/10) = 1/10
- 16. Example Cyrus &Beck Clipping 15
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- 19. Cases 18 Directrix or direction ofline Distance to boundary point from the end points Weighting factor (1) (2) (1) (2) (3) equation (3) & its notes, used to determine whether the line(s) intersects or not with the window.
- 20. Polygon Clipping • Clippolygons by clipping successively against 4 sides. 19
- 21. Polygon Clipping • Clippolygons by clipping successively against all 4 sides • Can implement as pipelined algorithm (ie special hardware can do the work) • Recursively test each edge. Form new edge with next vertex Call with new edge 20
- 22. Area Clipping • Similarwith lines, areas must be clipped to a window boundary • Consideration must be taken as to which portions of the area must be clipped 21
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- 26. Sutherland-Hodgeman Polygon Clipping • Loopround vertices, test each against all 4 clipping planes in sequence • Call algorithm again with new edge formed by next vertex -re-entrant • No storage requirement between stages Easy to implement in hardware 25
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- 28. Output Vertex First Output Four cases ofpolygon clipping: Inside Outside Inside Outside Inside Outside Inside Outside Case 3 No output. Case 1 Case 2. Output Intersection Case 4 Second Output Sutherland-Hodgeman Polygon Clipping 27
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- 31. Sutherland-Hodgeman Polygon Clipping • Example 30 Startat the left boundary 1,2 (out,out) → clip 2,3 (in,out) → save 1’, 3 3,4 (in,in) → save 4 4,5 (in,in) → save 5 5,6 (in,out) → save 5’ Saved points → 1’,3,4,5,5’ 6,1 (out,out) → clip Using these points we repeat the process for the next boundary.
- 32. Weiler-Atherton Polygon Clipping •Convex polygons are correctly clipped by the Sutherland- Hodgeman algorithm, but concave polygons may be displayed with extra areas (area inside the red circle), as demonstrated in the following figure. 31
- 33. Weiler-Atherton Polygon Clipping •This occurs when the clipped polygon should have two or more separate sections. But since there is only one output vertex list, the last vertex in the list is always joined to the first vertex. • There are several things we could do to correctly display concave polygons. • For one, we could split the concave polygon into two or more convex polygons and process each convex polygon separately • Another possibility is to modify the Sutherland-Hodgeman approach to check the final vertex list for multiple vertex points along any Clip window boundary and correctly join pairs of vertices. • Finally, we could use a more general polygon clipper, such as either the Weiler-Atherton algorithm or the Weiler algorithm. 32
- 34. Weiler-Atherton Polygon Clipping •InWeiler-Atherton Polygon Clipping, the vertex- processing procedures for window boundaries are modified so that concave polygons are displayed correctly. •This clipping procedure was developed as a method for identifying visible surfaces, and so it can be applied with arbitrary polygon-clipping regions. 33
- 35. Weiler-Atherton Polygon Clipping •The basic idea in this algorithm is that instead of always proceeding around the polygon edges as vertices are processed, we sometimes want to follow the window boundaries. • Which path we follow depends on the polygon-processing direction (clockwise or counterclockwise) and whether the pair of polygon vertices currently being processed represents an outside-to-inside pair or an inside-to- outside pair. 34
- 36. Weiler-Atherton Polygon Clipping Forclockwise processing of polygon vertices, we use the following rules: 1. For an outside-to-inside pair of vertices, follow the polygon boundary 2. For an inside-to-outside pair of vertices, follow the window boundary in a clockwise direction. 35
- 37. Weiler-Atherton Polygon Clipping •Example In the following figure, the processing direction in the Weiler-Atherton algorithm and the resulting clipped polygon is shown for a rectangular clipping window. 36
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- 41. Viewing & Clipping- Summary • Window and View Port • Clipping cuts out what we do not “see” • Also reduces unnecessary computation • Can be done at various levels 40