![ALGORITHM - STEP 04
Using pk and qk (k1:4), find if the line can be rejected or intersection parameters(t1andt2)must
be adjusted.
•if pk<0, update t1 as max[0,qk/pk] where k1:4
•if pk>0, update t2 as min[1,qk/pk] where k1:4
After the update,
•If t1>t2, reject the line
•If t1>0, calculate new values of x0,y0
•If t2<1, calculate new values of x1,y1](https://image.slidesharecdn.com/liang-barskyalgorithmpolygonclippingpipeline-200610081754/85/Liang-Barsky-Algorithm-Polygon-clipping-pipeline-clipping-of-polygons-12-320.jpg)
Liang- Barsky Algorithm, Polygon clipping & pipeline clipping of polygons
- 1. LIANG- BARSKY ALGORITHM, POLYGON CLIPPING & PIPELINE CLIPPING OF POLYGONS A G A L DANUSHKA
- 2. WHAT IS CLIPPING? The primary use of clipping in computer graphics is to remove objects, lines, or line segments that are outside the viewing pane. The viewing transformation is insensitive to the position of points relative to the viewing volume ( especially those points behind the viewer) and it is necessary to remove these points before generating the view. There different types of clipping Point clipping Line clipping Polygon clipping Curve clipping Text clipping
- 3. LINE CLIPPING In computer graphics, line clipping is the process of removing lines or portions of lines outside an area of interest. Typically, any line or part thereof which is outside of the viewing area is removed. In the line clipping 4 different cases are possible Line intersect clipping window from one point Line intersect clipping window from two points Line completely outside the clipping window Line completely inside the clipping window
- 4. LINE CLIPPING ALGORITHMS There are two common algorithms for line clipping: Cohen–Sutherland clipping algorithm Liang–Barsky clipping algorithm
- 5. WHAT IS LIANG-BARSKY ALGORITHM? The Liang-Barsky algorithm is a line clipping algorithm. This algorithm is more efficient than Cohen–Sutherland line clipping algorithm and can be extended to 3-Dimensional clipping. This algorithm is considered to be the faster parametric line-clipping algorithm. The following concepts are used in this clipping: The parametric equation of the line. The inequalities describing the range of the clipping window which is used to determine the intersections between the line and the clip window.
- 6. PARAMETRIC DEFINITION OF THE LINE Liang – Barsky line clipping algorithm is faster line clipper algorithm based on analysis of the parametric equation of a line segment. Parametric definition of a line; Where ,
- 7. INITIALIZATIONS AND CALCULATIONS Initializations; Set viewport points as xmin, xmax, ymin and ymax Set the line with 2 points A(x0,y0) and B (x1,y1) Set line intersection parameters tentry(t1) = 0.0 and tleaving(t2) =1.0 Calculations: Find dx= x1-x0 and dy= y1-y0 Update t1or t2depending upon dx or dy xmin,ymin xmax,ymax B A
- 8. ALGORITHM 1. Set tmin=0, tmax=1. 2. Calculate the values of t (t(left), t(right), t(top), t(bottom)), (i) If t < tmin ignore that and move to the next edge. (ii) else separate the t values as entering or exiting values using the inner product. (iii) If t is entering value, set tmin = t; if t is existing value, set tmax = t. 3. If tmin < tmax, draw a line from (x1 + tmin(x2-x1), y1 + tmin(y2-y1)) to (x1 + tmax(x2-x1), y1 + tmax(y2-y1)) 4. If the line crosses over the window, (x1 + tmin(x2-x1), y1 + tmin(y2-y1)) and (x1 + tmax(x2-x1), y1 + tmax(y2-y1)) are the intersection point of line and edge.
- 9. ALGORITHM - STEP 01 Accept Line end points A , B and window coordinates Xwmin ,Ywmin, Xwmax ,Ywmax as input. Set line intersection parameters tentry(t1) = 0.0 and tleaving(t2) = 1.0 xmin,ymin xmax,ymax B A
- 10. ALGORITHM - STEP 02 Obtain pk and qk for k 1: 4 Edge & K value p q t Left k=1 p1= -dx q1= x0-xmin t = q1/p1 Right k=2 p2= dx q2= xmax-x0 t = q2/p2 Bottom k=3 p3= -dy q3= y0-ymin t = q3/p3 Top k=4 p4= dy q4= ymax-y0 t = q4/p4
- 11. ALGORITHM - STEP 03 If pk=0,the line is parallel to the corresponding clipping boundary. a)If pk=0 and qk<0, the line is completely outside the boundary. b)If pk=0 and qk>=0, the line is inside the parallel clipping boundary.
- 12. ALGORITHM - STEP 04 Using pk and qk (k1:4), find if the line can be rejected or intersection parameters(t1andt2)must be adjusted. •if pk<0, update t1 as max[0,qk/pk] where k1:4 •if pk>0, update t2 as min[1,qk/pk] where k1:4 After the update, •If t1>t2, reject the line •If t1>0, calculate new values of x0,y0 •If t2<1, calculate new values of x1,y1
- 13. EXAMPLE dx= P1x -P0x // Horizontal diff between P0and P1 dx= x1-x0= 280-30 = 250 dy= P1y -P0y // Vertical diff between P0 and P1 dy= y1-y0= 160-20 = 140 dx= 250 , dy= 140 Initially tentry(t1) = 0.0 tleaving(t2) = 1.0
- 14. EXAMPLE CONT.. Left edge check: p = -dx= -250 q = x0-xmin= 30-70 = -40 t = q/p = 0.16 If p< 0 // update tentry(t1) if t > t1 set t1 = t -250 < 0 TRUE check if t > t1 0.16 > 0.0 TRUE t1 = t t1 = 0.16 t2 = 1.0
- 15. EXAMPLE CONT.. Right edge check: p = dx= 250 q = xmax–x0= 230-30 = 200 t = q/p = 0.8 If p< 0 FALSE If p > 0 // update tleaving(t2) if t < t2 set t2 = t 250 > 0 TRUE check if t < t2 ---> 0.8 < 1.0 TRUE t2 = t = 0.8 t1 = 0.16 t2 = 0.8
- 16. EXAMPLE CONT.. Bottom edge check: p = -dy= -140 q = y0-ymin= 20-60 = -40 t = q/p = 0.2857 If p< 0 // update tentry(t1) if t > t1 set t1 = t -140 < 0 TRUE check if t > t10.2857 > 0.16 TRUE t1 = tt1 = 0.2857 and t2 = 0.8
- 17. EXAMPLE CONT.. Top edge check: p = dy= 140 q = ymax–y0= 150 -20 = 130 t = q/p = 0.928 If p< 0 FALSE If p > 0 // update tleaving(t2)if t < t2 set t2 = t 140 > 0 TRUE check if t < t2 0.928 < 0.8 FALSE t1 = 0.2857 t2 = 0.8
- 18. EXAMPLE CONT.. t1 = 0.2857 t2 = 0.8 dx= 250dy= 140 if( tentry> 0.0 ) calculate new x0, y0 0.2857 > 0.0 TRUE x0new= x0 + tentry *dx x0new= 30 + 0.2857 * 250 = 101.425 y0new= y0 + tentry *dy y0new= 20 + 0.2857 * 140 = 60.0
- 19. EXAMPLE CONT.. t1 = 0.2857 t2 = 0.8 dx= 250dy= 140 if( tleaving< 1.0 ) calculate new x1, y1 0.8 > 0.0 TRUE x1new= x0 + tleaving*dx x1new= 30 + 0.8 * 250 = 230 y1new= y0 + tleaving*dy y1new= 20 + 0.8 * 140 = 132 Draw clipped line with the points A(x0new , y0new) and B(x1new , y1new)
- 20. WHAT IS POLYGON CLIPPING? A set of connected lines are considered as polygon. To clip a polygon, we cannot directly apply a line-clipping method to the individual polygon edges because this approach would produce a series of unconnected line segments. polygons are clipped based on the window and the portion which is inside the window is kept as it is and the outside portions are clipped. Polygon clipping is difficult compared to line clipping
- 21. DIFFERENT CASE CATEGORIES The polygon clipping is required to deal different cases. Usually it clips the four edges in the boundary of the clip rectangle. The clip boundary determines the visible and invisible regions of polygon clipping and it is categorized as four. Visible region is wholly inside the clip window – saves endpoint Visible exits the clip window – Save the interaction Visible region is wholly outside the clip window – nothing to save Visible enters the clip window – save endpoint and intersection
- 22. POLYGON CLIPPING ALGORITHMS There several algorithms to polygon clipping; Greiner–Hormann clipping algorithm Sutherland–Hodgman algorithm Vatti clipping algorithm Weiler–Atherton clipping algorithm
- 23. SUTHERLAND-HODGMAN ALGORITHM For polygon clipping, we require an algorithm that will generate one or more closed areas that are then scan converted for the area fill. The output of a polygon clipper should be a sequence of vertices that defines the clipped polygon boundaries. Clip a polygon by processing the polygon boundary against each window edge. (Left, Top, Bottom, Right). Apply four cases to process all polygon edges to produce sequence of vertices . This sequence of vertices will make a bounded area. Beginning with the initial set of polygon vertices, we could first clip the polygon against the left rectangle boundary to produce a new sequence of vertices. The new set of vertices could be successively passed to a right boundary clipper, a bottom boundary clipper, and a top boundary clipper.
- 24. CLIPPING PROCESS At each step, a new sequence of output vertices is generated and passed to the next window boundary clipper. There are four possible cases when processing vertices in sequence around the perimeter of a polygon. As each pair of adjacent polygon vertices is passed to a next window boundary clipper, we make some tests:
- 25. CASE 1 If the first vertex is outside the window boundary and the second vertex is inside Then , both the intersection point of the polygon edge with the window boundary and the second vertex are added to the output vertex list.
- 26. CASE 2 If both input vertices are inside the window boundary. Then, only the second vertex is added to the output vertex list.
- 27. CASE 3 If the first vertex is inside the window boundary and the second vertex is outside. Then, only the edge intersection with the window boundary is added to the output vertex list.
- 28. CASE 4 If both input vertices are outside the window boundary. Then, nothing is added to the output vertex list.
- 29. EXAMPLE We illustrate this algorithm by processing the area in figure against the left window boundary. Vertices 1 and 2 are outside of the boundary. Vertex 3, which is inside, 1' and vertex 3 are saved. Vertex 4 and 5 are inside, and they also saved. Vertex 6 is outside, 5' is saved. Using the five saved points, we would repeat the process for the next window boundary. The Sutherland-Hodgman algorithm correctly clips convex polygons, but concave polygons may be displayed with extraneous lines as demonstrated in figure. Since there is only one output vertex list, the last vertex in the list is always joined to the first vertex.
- 30. IMPLEMENTATION There are two sub-problems that need to be discussed before implementing the algorithm To decide if a point is inside or outside the clipper polygon If the vertices of the clipper polygon are given in clockwise order then all the points lying on the right side of the clipper edges are inside that polygon. This can be calculated using : Formula-for-position-of-point
- 31. IMPLEMENTATION CONT To find the point of intersection of an edge with the clip boundary If two points of each line(1,2 & 3,4) are known, then their point of intersection can be calculated using the formula
- 32. COMPLEXITY Complexity Of this algorithm will increase if number of edges of polygon increase. Algorithm has to calculate more number of intersection points over window boundary.