1. o عَلَّمَهُ الْبَيَانَ
Allah says
o عَلَّمَهُ الْبَيَانَ
He has taught him speech (and intelligence).
Al-Qur'an, (Ar-Rahman)

2. Bouchahma Maged boumaged@unizwa.edu.om
Lecture-6 Viewing in 2D
Bouchahma Maged

3. Contents In this lecture we’ll cover the following:
Concepts from Linear Algebra
2D Viewing
Viewing Pipeline
Window to Viewport Transform
Clipping
Line Clipping
Polygon Clipping
Curve Clipping
Viewing in OpenGL

4. 2D Viewing pipeline MC WC PVC CVC DC VC NVC
Construct World-Coordinate Scene using modeling coordinates
Convert World Coordinates to Viewing Coordinates
Map normalized viewing coordinates to Device coordinates MC WC
PVC
CVC DC
Projection Transformation
Clipping
Window to Viewport Mapping
(Conversion of VC to Normalized VC) VC
NVC
Modeling Transformation: gltranslatef, …
Viewing Transformation: glulookat
Projection Transformation: gluortho2d, glperspective…
Clipping: automatic (uses viewing coordinates specified in gluortho2d for 2d points)
Window to viewport mapping: automatic, rotated viewports not allowed

5. 2D Viewing pipeline… World Coordinate Scene Description
Objects are first modeled
For example: A star can be modeled by a set of connected points in an axes specific to the star shape
Different objects can have different object specific axes
A scene is typically composed of different modeled objects in a world coordinate system
Thus transformations of each object model to the world coordinate system are used to draw the object in the world as required
ymodel
ymodel
xmodel
xmodel
xworld
yworld

6. Modeling Transformation in OpenGL
MODE = GL_MODELVIEW
glTranslate
glRotate
glScale

7. 2D Viewing pipeline…
Conversion of world coordinates to viewing coordinates
The viewing coordinate system describes how the world is viewed
We may want to choose the origin of the viewing coordinates to be different from that of the world coordinates
We choose the origin of the viewing coordinate system at a point P(x0,y0) in the world coordinates
We may want to have the viewing coordinates oriented at an angle to the world coordinates
We also choose the orientation of the view-up vector V in the world coordinates

8. 2D Viewing pipeline: Translation
yworld
yview
yview
xview
xworld
xview

9. 2D Viewing pipeline: Viewing Orientation
View-up vector

10. Converting world coordinates to viewing coordinates
Thus a transformation between axis will be involved
View up vector is used to calculate the orientation matrix
xview
yview
(0view,0view)
xworld
yworld
(0world,0world) θ
(x0,y0)

11. Viewing Transformation in OpenGL
MODE = GL_MODELVIEW
Glulookat
If gluLookAt is not used then the world coordinates are used hence forth

12. Projection Transformation
The display of a scene requires projection onto the 2D screen
Types of Projection Transformations
Perspective Projections
Parallel Projections
Projection matrix is used to convert the viewing coordinates into projected Viewing Coordinates 12

13. Parallel Orthographic Projection
In 2D viewing only orthographic projections are used

14. Perspective Projection

15. Projection Transformation in OpenGL
MODE = GL_PROJECTION
gluOrtho2D
Produces a one to one mapping from the world coordinates to the projected viewing coordinates
Projected viewing coordinates = world coordinates if gluLookAt is not used

16. Projected Viewing Coordinates
Clipping
Because drawing things to a display takes time we clip everything outside the window
The extremities of the window are specified in viewing coordinates
yview
Window
wymax
wymin
wxmin
wxmax
xview
Projected Viewing Coordinates

17. Clipping Area Definition in OpenGL
gluOrtho2D
specifies the extremities of the window clipping area in projected viewing coordinates which are the same as world coordinates if glulookat is not used

18. Window to View-Port Transformation
Once the viewing coordinates have been obtained, these are mapped onto normalized device coordinates
Done by specifying a view-port in normalized device coordinates and then mapping the window contents onto the view-port
viewport
window
yview 1
yw_max
(xw,yw)
Window is specified in viewing coordinates
(0,0) 1
yw_min
xw_min
xw_max
xview

19. Window to view-port transformation
yDevice
yv_min
(xv,yv)
yv_min
xDevice
xv_min
xv_min
This may require scaling and translation

20. Defining the View Port in OpenGL
glViewPort
Defines the extremities of the view port but in device coordinates

21. Workstation Transformation
Mapping from the normalized device coordinates to device coordinates
yDevice
yv_min
yv_min
xDevice
xv_min
xv_min

22. 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

23. Points Within the Window are Not Clipped
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

24. 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!

25. Brute Force Line Clipping
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

26. Brute Force Line Clipping (cont…)
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

27. Cohen-Sutherland Clipping Algorithm
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

28. Cohen-Sutherland: World Division
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
1001
1000
1010
0001
0000
Window
0010
0101
0100
0110
above
below
right
left 3 2 1
Region Code Legend

29. Cohen-Sutherland: Labelling
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]

30. Cohen-Sutherland: Lines In The Window
Lines completely contained within the window boundaries have region code [0000] for both end-points so are not clipped
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]

31. Cohen-Sutherland: Lines Outside The Window
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]

32. Cohen-Sutherland Examples
Consider the line P9 to P10 below
Start at a point not lying within the window e.g. P10
From the region codes of the two end-points we know the line doesn’t cross the left or right boundary
Calculate the 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
wymax
wymin
wxmin
wxmax
Window
P10 [0100]
P9 [0000]
P10’ [0000]
We have started at a point not lying within the window because the code for that point would automatically indicate which vertices to consider for intersection whereas the code for a point inisde the window (0000) provides no such information

33. Cohen-Sutherland Examples (cont…)
Consider the line P3 to P4 below
Start at (point outside the boundary) P4
From the region code of P4 we know the line crosses the top boundary
Calculate the intersection point to generate P4’ with code (0001)
Discard the line from P4 to P4’ as it lies completely outside (codes!)
Now the line from P4’ to P3 can be completely clipped
wymax
wymin
wxmin
wxmax
Window
P4’ [0001]
P3 [0001]
P4 [1000]

34. Cohen-Sutherland Examples (cont…)
Consider the line P7 to P8 below
Start at P7
P7 is to the left so we find its intersection with the left boundary giving P7’ with code (0000)
Now the outside point is P8 with code (0010) so we find the intersection with the right boundary P8’ with code (0000)
Now as both points P7’ & P8’ are inside the window, a line is drawn between them
wymax
wymin
wxmin
wxmax
Window
P7’ [0000]
P7 [0001]
P8 [0010]
P8’ [0000]

35. Calculating Line Intersections
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

36. Calculating Line Intersections (cont…)
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)

37. Cohen-Sutherland Clipping
Input:
End points of the line P1 and P2
Window boundaries
Output
Draw Flag:
1 indciates the line (or a part of it) is to be drawn
In this case the algorithm also changes P1 and P2 to be the end points of the part of the line to be drawn
0 indicates that the line need not be drawn
The Clipping algorithm is applied on all lines to be drawn one by one

38. Cohen-Sutherland Clipping Pseudo Code
Procedure
Set done=0, draw=0
While (NOT(done))
Encode the points P1 and P2 as C1 and C2
If the codes of P1 and P2 are both 0000, then (Trivial Accept)
done=1, draw=1 and output P1 and P2 as the end points
Else If AND(C1,C2) is non-zero then (Trivial Reject)
done=1, draw=0
Else
If P1 is inside the window then swap the points and their codes (so that P1 is the point which is always outside the window)
Find an intersection of the line from P1 to P2 with the window boundary on the basis of the code of C1
Set P1= intersection point
End While

39. 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

40. 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
Original Area
Clip Left
Clip Right
Clip Top
Clip Bottom

41. Sutherland-Hodgman Area Clipping Algorithm (cont…)
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

42. Sutherland-Hodgman Example
Each example shows the point being processed (P) and the previous point (S)
Saved points define area clipped to the boundary in question S P
Save Point P S P
Save Point I I P S
No Points Saved S P
Save Points I & P I

43. Other Area Clipping Concerns
Clipping concave areas can be a little more tricky as often superfluous lines must be removed
Clipping curves requires more work
For circles we must find the two intersection points on the window boundary
Window
Window

44. Summary
Objects within a scene must be clipped to display the scene in a window
Because there are can be so many objects clipping must be extremely efficient
The Cohen-Sutherland algorithm can be used for line clipping
The Sutherland-Hodgman algorithm can be used for area clipping

45. Many of the brightly colored tile-covered walls and floors of the Alhambra in Spain show us that the Moors were masters in the art of filling a plane with similar inter-locking figures, bordering each other without gaps. What a pity that their religion forbade them to make images!
M.C. Esclier
End of Lecture-6