![18
• Every end-point is labelled with the
appropriate region code.
wymax
wymin
wxmin wxmax
Window
P3 [0001]
P6 [0000]
P5 [0000]
P7 [0001]
P10 [0100]
P9 [0000]
P4 [1000]
P8 [0010]
P12 [0010]
P11 [1010]
P13 [0101] P14 [0110]](https://image.slidesharecdn.com/windowingclipping-210524103114/85/Windowing-clipping-18-320.jpg)
![19
Lines completely contained within the window
boundaries have region code [0000] for both end-
points so are not clipped. The OR operation can
efficiently check this.
wymax
wymin
wxmin wxmax
Window
P3 [0001]
P6 [0000]
P5 [0000]
P7 [0001]
P10 [0100]
P9 [0000]
P4 [1000]
P8 [0010]
P12 [0010]
P11 [1010]
P13 [0101] P14 [0110]](https://image.slidesharecdn.com/windowingclipping-210524103114/85/Windowing-clipping-19-320.jpg)
![20
Any lines with a common set bit in the region
codes of both end-points can be clipped.
– The AND operation can efficiently check this.
wymax
wymin
wxmin wxmax
Window
P3 [0001]
P6 [0000]
P5 [0000]
P7 [0001]
P10 [0100]
P9 [0000]
P4 [1000]
P8 [0010]
P12 [0010]
P11 [1010]
P13 [0101] P14 [0110]](https://image.slidesharecdn.com/windowingclipping-210524103114/85/Windowing-clipping-20-320.jpg)
Windowing clipping
- 2. Windowing Concepts Introduction to Clipping Point Clipping Line Clipping Area Clipping 2
- 3. 3 Window Image Space Viewport Information outside the viewport is clipped away
- 4. • World Coordinate System (Object Space) - Representation of an object measured in some physical units. • Window - The rectangle defining the part of the world we wish to display. • Image Coordinate System (Image Space) - The space within the image is displayed. • Viewport - The rectangle area in image space where the image from the window will appear. • Viewing Transformation - The process of going from a window in world coordinates to a viewport in image coordinates. 4
- 5. 5 • In 2D, a ‘world’ consists of an infinite plane, defined in ‘world’ coordinates, i.e metres, angstroms etc. • We need to pick an area of the 2D plane to view, referred to as the ‘window’. • On our display device, need to allocate an area for display, referred to as the ‘viewport’ in device specific coordinates. – Clip objects outside of window. – Translate to fit viewport. – Scale to device coordinates.
- 6. 6 Choose Window in World Coordinates Clip to size of Window Translate to origin Scale to size of Viewport Translate to proper position in image
- 7. Any procedure that identifies those portions of a picture that are either inside or outside of a specified region is referred to as a clipping algorithm or clipping. 7 World Coordinates
- 8. 8 • When we display a scene only those objects within a particular window are displayed. wymax wymin wxmin wxmax Window or Clipping Region World Coordinates
- 9. 9 Because drawing things to a display takes time we clip everything outside the window. wymax wymin wxmin wxmax Window or Clipping Region World Coordinates
- 10. For the image below consider which lines and points should be kept and which ones should be clipped. 10 wymax wymin wxmin wxmax Window P1 P2 P3 P6 P5 P7 P10 P9 P4 P8
- 11. Easy - a point (x,y) is not clipped if: wxmin ≤ x ≤ wxmax AND wymin ≤ y ≤ wymax 11 wymax wymin wxmin wxmax Window P1 P2 P5 P7 P10 P9 P4 P8 Clipped Points Within the Window are Not Clipped Clipped Clipped Clipped
- 12. Harder - examine the end-points of each line to see if they are in the window or not. 12 Situation Solution Example Both end-points inside the window Don’t clip One end-point inside the window, one outside Must clip Both end-points outside the window Don’t know!
- 13. 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. 13
- 14. 14 – For lines with both end- points outside the window test the line for intersection with all of the window boundaries, and clip appropriately. However, calculating line intersections is computationally expensive. Because a scene can contain so many lines, the brute force approach to clipping is much too slow.
- 15. 15 An efficient line clipping algorithm. The key advantage of the algorithm is that it vastly reduces the number of line intersections that must be calculated. Dr. Ivan E. Sutherland c o - d e v e l o p e d t h e C o h e n - S u t h e r l a n d clipping algorithm. Sutherland is a graphics giant and includes a m o n g s t h i s a c h i e v e m e n t s t h e invention of the head Cohen is something of a mystery – can anybody find out who he was?
- 16. Algorithm Step 1 − Assign a region code for each endpoints. Step 2 − If both endpoints have a region code 0000 then accept this line. Step 3 − Else, perform the logical AND operation for both region codes. Step 3.1 − If the result is not 0000, then reject the line. Step 3.2 − Else you need clipping. Step 3.2.1 − Choose an endpoint of the line that is outside the window. Step 3.2.2 − Find the intersection point at the window boundary (base on region code). Step 3.2.3 − Replace endpoint with the intersection point and update the region code. Step 3.2.4 − Repeat step 2 until we find a clipped line either trivially accepted or trivially rejected. Step 4 − Repeat step 1 for other lines. 16
- 17. World space is divided into regions based on the window boundaries. Each region has a unique four bit region code. Region codes indicate the position of the regions with respect to the window. 17 above below right left 3 2 1 0 Region Code Legend 1001 1000 1010 0001 0000 Window 0010 0101 0100 0110
- 18. 18 • Every end-point is labelled with the appropriate region code. wymax wymin wxmin wxmax Window P3 [0001] P6 [0000] P5 [0000] P7 [0001] P10 [0100] P9 [0000] P4 [1000] P8 [0010] P12 [0010] P11 [1010] P13 [0101] P14 [0110]
- 19. 19 Lines completely contained within the window boundaries have region code [0000] for both end- points so are not clipped. The OR operation can efficiently check this. wymax wymin wxmin wxmax Window P3 [0001] P6 [0000] P5 [0000] P7 [0001] P10 [0100] P9 [0000] P4 [1000] P8 [0010] P12 [0010] P11 [1010] P13 [0101] P14 [0110]
- 20. 20 Any lines with a common set bit in the region codes of both end-points can be clipped. – The AND operation can efficiently check this. wymax wymin wxmin wxmax Window P3 [0001] P6 [0000] P5 [0000] P7 [0001] P10 [0100] P9 [0000] P4 [1000] P8 [0010] P12 [0010] P11 [1010] P13 [0101] P14 [0110]
- 21. Lines that cannot be identified as completely inside or outside the window may or may not cross the window interior. These 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. 21
- 22. 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 found. We can use the region codes to determine which window boundaries should be considered for intersection. 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. 22
- 23. Intersection points with the window boundaries are calculated 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 23
- 24. 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) 24
- 25. Polygons can be clipped against each edge of the window one edge at a time. Window/edge intersections, if any, are easy to find since the X or Y coordinates are already known. Vertices which are kept after clipping against one window edge are saved for clipping against the remaining edges. Note that the number of vertices usually changes and will often increase. 25
- 26. A technique for clipping areas developed by Sutherland & Hodgman. Put simply the polygon is clipped by comparing it against each boundary in turn. 26 Sutherland turns up again. This time with Gary Hodgman with whom he worked at the first ever graphics company Evans & Sutherland Original Area Clip Left Clip Right Clip Top Clip Bottom
- 27. 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. 27
- 28. Each example shows the point being processed (P) and the previous point (S). Saved points define area clipped to the boundary in question. 28 S P Save Point P S P Save Point I I P S No Points Saved S P Save Points I & P I
- 29. Instead of always proceeding around the polygon edges as vertices are processed, we sometime want to follow window boundary. This algorithm is used for clipping concave polygons. V1, V2, V3, V4, V5 are the vertices of the polygon. C1, C2, C3, C4 are the vertices of the clip polygon I1, I2, I3, I4 are the intersection points of 29
- 30. Thank You !!! 30