1. 2D Viewing and Projection
Dr. Aree Ali Mohammed
Assistant Professor
3rd Stage
University of Sulaimani - School of Science - Computer Dept.

2. Windowing and Clipping
The process of selecting and viewing the picture with different views is called windowing.
The process which divides each element of the picture into its visible and invisible portions, allowing the invisible portion to be discarded is called clipping.
Windowing and Clipping
Window
wymax
wymin
wxmin
wxmax
World Coordinates
World Coordinates

3. Viewing Transformation I
Picture stored in a computer  WCS (World Coordinate System)
Picture displayed on a device  PDCS (Physical Device CS)
Mapping from WCS to PDCS  Viewing Transformation
Normalization Transformation  Device independent
University of Sulaimani - School of Science - Computer Dept.

4. Viewing Transformation II
1. Normalization Transformation  Device independent An interpreter is used to convert the normalized device coordinates to the actual device coordinates.
The transformation which maps the world coordinate to normalized device coordinate is called normalization transformation. It involves scaling of x and y, thus it is also referred to as scaling transformation
University of Sulaimani - School of Science - Computer Dept.

5. Viewing Transformation III
2. Workstation Transformation Maps normalized device coordinates  physical device coordinates The viewing transformation is the combination of normalization transformation and workstation transformation
Normalization Transformation NC
Workstation
Transformation WC DC
2D Viewing V=N.W
University of Sulaimani - School of Science - Computer Dept.

6. Viewing Transformation IV
Workstation Transformation
WCS  infinite  finite  window
Device display  finite  viewport
The window defines what is to be viewed; the viewport defines
where it is to be displays.
University of Sulaimani - School of Science - Computer Dept.

7. Viewing Transformation V
Workstation Transformation Steps

8. Viewing Transformation VI

9. Viewing Transformation VII
Normalization Transformation NC DC WC
Ex: Find the normalization transformation window to viewport, with window, lower left corner at (1,1) and upper right corner at (3,5) onto a viewpoint with lower left corner at (0,0) and upper right corner at (1/2,1/2).
Sol:

10. University of Sulaimani - School of Science - Computer Dept.
2D Clipping
For the image below consider which lines and points should be kept and which ones should be clipped P4
Window P2
wymax P6 P3 P1 P7 P5 P9 P8
wymin
P10
wxmin
wxmax
University of Sulaimani - School of Science - Computer Dept.

11. University of Sulaimani - School of Science - Computer Dept.
Point Clipping
Easy - a point (x,y) is not clipped if:
wxmin ≤ x ≤ wxmax AND wymin ≤ y ≤ wymax
otherwise 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
University of Sulaimani - School of Science - Computer Dept.

12. Line Clipping
Harder - examine the end-points of each line to see if they are in the window or not
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. Line Clipping Algorithms (I)
Sutherland and Cohen Subdivision Algorithm
Reduce the number of intersections  speed up
Use four digit (bit) code to indicate which of nine regions contain the end point of line.
Called region code or outcodes.
1001
1000
1010
0001
0000
Window
0010
0101
0100
0110 4 3 2 1
above
below
right
left
Region Code Legend
University of Sulaimani - School of Science - Computer Dept.

14. Line Clipping Algorithms (II)
Ex: Consider the clipping window and the lines shown in figure. Find the region codes for each end point and identify whether the line is completely visible, partially visible or completely invisible.
University of Sulaimani - School of Science - Computer Dept.

16. University of Sulaimani - School of Science - Computer Dept.

17. Calculating Line Intersections I
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
University of Sulaimani - School of Science - Computer Dept.

18. Calculating Line Intersections II
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)
University of Sulaimani - School of Science - Computer Dept.

19. University of Sulaimani - School of Science - Computer Dept.
Area Clipping
Similarly to lines, areas must be clipped to a window boundary
Consideration must be taken as to which portions of the area must be clipped
University of Sulaimani - School of Science - Computer Dept.

20. Sutherland-Hodgman Area Clipping Algorithm
A technique for clipping areas developed by Sutherland & Hodgman
Put simply the polygon is clipped by comparing it against each boundary in turn
Clip Bottom
Original Area
Clip Left
Clip Right
Clip Top
University of Sulaimani - School of Science - Computer Dept.

21. University of Sulaimani - School of Science - Computer Dept.
2D Polygon Clipping
University of Sulaimani - School of Science - Computer Dept.

22. University of Sulaimani - School of Science - Computer Dept.
Review Questions
Write short note on viewing transformation.
Distinguish between viewport and window.
What do you mean by normalization transformation? Why it is needed?
Derive the transformation matrix for 2D viewing transformation.
What is point and line clipping?
Explain the Sutherland and Cohen Subdivision Algorithm for line clipping.
Use Sutherland and Cohen outcode Algorithm to clip two lines P1 (40,50) – P2 (75,45) and P3(70,20), P4(100,10) against a window A(50,10), B(80,10), C(80,40), D(50,40).
Clip the line P1(-3/2,1/6) and P2(1/2,3/2) against window A(-1,-1), B(-1,1), C(1,1) and D(1,-1) using Sutherland and Cohen Subdivision Algorithm
University of Sulaimani - School of Science - Computer Dept.

23. Group Discussion