Clipping Algorithm In Computer Graphics
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  • 1. Windowing ConceptsClipping ◦ Introduction ◦ Brute Force ◦ Cohen-Sutherland Clipping AlgorithmArea Clipping ◦ Sutherland-Hodgman Area Clipping Algorithm
  • 2. A scene is made up of a collection of objectsspecified in world coordinates World Coordinates
  • 3. When we display a scene only those objects within aparticular window are displayed Window wymax wymin wxmin wxmax World Coordinates
  • 4. Because drawing things to a display takes time weclip everything outside the window Window wymax wymin wxmin wxmax World Coordinates
  • 5. For the image below consider which lines and pointsshould be kept and which ones should be clipped P4 Window P2 wymax P6 P3 P1 P7 P5 P9 P8 wymin P10 wxmin wxmax
  • 6. Easy - a point (x,y) is not clipped if: wxmin ≤ x ≤ wxmax AND wymin ≤ y ≤ wymaxotherwise it is clipped P4 Clipped Clipped Window P2 wymax Clipped P5 P1 P7 Points Within the Window are Not Clipped P9 P8 wymin Clipped P10 wxmin wxmax
  • 7. Harder - examine the end-points of each line to see ifthey are in the window or not Situation Solution Example Both end-points inside Don’t clip the window One end-point inside the window, one Must clip outside Both end-points Don’t know! outside the window
  • 8. Brute force line clipping can be performed as follows: ◦ Don’t clip lines with both end-points within the window ◦ For lines with one end- point inside the window and one end-point outside, calculate the intersection point (using the equation of the line) and clip from this point out
  • 9. ◦ For lines with both end-points outside the window test the line for intersection with all of the window boundaries, and clip appropriatelyHowever, calculating line intersections iscomputationally expensiveBecause a scene can contain so many lines,the brute force approach to clipping is muchtoo slow
  • 10. An efficient line clipping algorithmThe key advantage of the algorithm is that it vastlyreduces the number of line intersections that mustbe calculated
  • 11. World space is divided into regions based on thewindow boundaries ◦ Each region has a unique four bit region code ◦ Region codes indicate the position of the regions with respect to the window 1001 1000 1010 3 2 1 0 0000 above below right left 0001 0010 Window Region Code Legend 0101 0100 0110
  • 12. Every end-point is labelled with the appropriateregion code P11 [1010] P4 [1000] Window wymax P6 [0000] P3 [0001] P5 [0000] P12 [0010] P7 [0001] P9 [0000] P8 [0010] wymin P10 [0100] P13 [0101] P14 [0110] wxmin wxmax
  • 13. Lines completely contained within thewindow boundaries have region code [0000]for both end-points so are not clipped P11 [1010] P4 [1000] Window wymax P6 [0000] P3 [0001] P5 [0000] P12 [0010] P7 [0001] P9 [0000] P8 [0010] wymin P10 [0100] P13 [0101] P14 [0110] wxmin wxmax
  • 14. Any lines with a common set bit in theregion codes of both end-points can beclipped – The AND operation can efficiently check this P [1010] 11 P4 [1000] Window wymax P6 [0000] P3 [0001] P5 [0000] P12 [0010] P7 [0001] P9 [0000] P8 [0010] wymin P10 [0100] P13 [0101] P14 [0110] wxmin wxmax
  • 15. Lines that cannot be identified as completely inside oroutside the window may or may not cross the windowinteriorThese lines are processed as follows: ◦ Compare an end-point outside the window to a boundary (choose any order in which to consider boundaries e.g. left, right, bottom, top) and determine how much can be discarded ◦ If the remainder of the line is entirely inside or outside the window, retain it or clip it respectively
  • 16. ◦ Otherwise, compare the remainder of the line against the other window boundaries ◦ Continue until the line is either discarded or a segment inside the window is foundWe can use the region codes to determine whichwindow boundaries should be considered forintersection ◦ To check if a line crosses a particular boundary we compare the appropriate bits in the region codes of its end-points ◦ If one of these is a 1 and the other is a 0 then the line crosses the boundary
  • 17. Consider the line P9 to P10 below ◦ Start at P10 Window ◦ From the region codes wymax of the two end-points we know the line doesn’t P [0000] cross the left or right 9 wymin P ’ [0000] boundary 10 P [0100] ◦ Calculate the 10 wxmin wxmax intersection of the line with the bottom boundary to generate point P10’ ◦ The line P9 to P10’ is completely inside the window so is retained
  • 18. Consider the line P7 to P8 below ◦ Start at P7 ◦ From the two region wymax Window codes of the two end-points we know P7’ [0000] the line crosses the P7 [0001] P8 [0010] left boundary so wymin P8’ [0000] calculate the intersection point to generate P7’ wxmin wxmax
  • 19. Consider the line P7’ to P8 ◦ Start at P8 ◦ Calculate the wymax Window intersection with the right boundary to P7’ [0000] generate P8’ P7 [0001] P8 [0010] P8’ [0000] ◦ P7’ to P8’ is inside wymin the window so is retained wxmin wxmax
  • 20. Windowwymaxwymin wxmin wxmax
  • 21. Intersection points with the window boundaries arecalculated using the line-equation parameters ◦ Consider a line with the end-points (x1, y1) and (x2, y2) ◦ The y-coordinate of an intersection with a vertical window boundary can be calculated using: y = y1 + m (xboundary - x1) where xboundary can be set to either wxmin or wxmax
  • 22. ◦ The x-coordinate of an intersection with a horizontal window boundary can be calculated using: x = x1 + (yboundary - y1) / m where yboundary can be set to either wymin or wymax◦ m is the slope of the line in question and can be calculated as m = (y2 - y1) / (x2 - x1)
  • 23. Similarly to lines, areas mustbe clipped to a windowboundaryConsideration must be takenas to which portions of thearea must be clipped
  • 24. Sutherland A technique for clipping areas turns up developed by Sutherland & again. This time with Hodgman Gary Hodgman with Put simply the polygon is clipped whom he worked at by comparing it against each the first ever graphics company boundary in turn Evans & SutherlandOriginal Area Clip Left Clip Right Clip Top Clip Bottom
  • 25. To clip an area against an individual boundary: ◦ Consider each vertex in turn against the boundary ◦ Vertices inside the boundary are saved for clipping against the next boundary ◦ Vertices outside the boundary are clipped ◦ If we proceed from a point inside the boundary to one outside, the intersection of the line with the boundary is saved ◦ If we cross from the outside to the inside intersection point and the vertex are saved
  • 26. Each example shows Sthe point beingprocessed (P) and the P S Iprevious point (S) PSaved points define Save Point P Save Point Iarea clipped to the P Sboundary in question I P S No Points Saved Save Points I & P
  • 27. Clipping concave areas can be a little more tricky asoften superfluous lines must be removedClipping curves requires more work ◦ For circles we must find theWindow Window Window two intersection points on Window the window boundary
  • 28. Objects within a scene must be clipped to displaythe scene in a windowBecause there are can be so many objects clippingmust be extremely efficientThe Cohen-Sutherland algorithm can be used forline clippingThe Sutherland-Hodgman algorithm can be used forarea clipping
  • 29. In general, methods depend on how characters are representedHowever, three strategies can be followed: all-or-none string clipping ◦ use a bounding rectangle for the string all-or-none character clipping ◦ use a bounding rectangle for the character individual character clipping or component clipping ◦ like line/curve clipping (outlined char’s) ◦ compare individual pixels (bit-mapped)
  • 30. So far, the clipping discussion has been about what is specified as internal clipping (throw away outside parts).But also external clipping (throw away inside parts)