1. EEL 5771-001 Introduction to Computer Graphics
PPT5: Two-dimensional viewing
Presented by: Yerrapati Raviteja (U )

2. OUTLINE 2D VIEWING PIPELINE CLIPPING TYPES OF CLIPPING Line clipping
Cohen-Sutherland Clipping
Liang-Barsky Clipping
NLN Clipping Algorithm
Point clipping
Area clipping
Sutherland-Hodgman Area Clipping Algorithm
Curve clipping
Text clipping
All-or-none string clipping
All-or-none character clipping
Individual character clipping or component clipping
Exterior clipping
Polygon Clipping
Weiler-Atherton Polygon Clipping

3. CLIPPING
Clipping is a process of extracting a portion of a data base or identifying elements of a scene or picture inside or outside a specified region, called the clipping region.
In the below figure, which lines and points should be kept and which ones should be clipped??

4. Windowing I
A 2D viewing pipeline is made up of a bunch of objects specified in space of coordinates
World Coordinates

5. Windowing II
While we display a scene, only those objects within a particular clipping window are seen
Window
wymax
wymin
wxmin
wxmax
World Coordinates

6. Windowing III
As it takes lots of time to draw things, we simply clip everything which is outside the window.
Window
wymax
wymin
wxmin
wxmax
World Coordinates

7. TYPES OF CLIPPINGS
Based on different output primitives. Further presentation deals with these clipping types with examples:
Line clipping
Point clipping
Area clipping
Curve clipping
Text clipping
Exterior clipping

8. LINE CLIPPING
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
Clipping is not needed
One end-point inside the window, one outside
Clipping is needed
Both end-points outside the window
Clipping decision should be made accordingly

9. How line clipping works:
LINE CLIPPING (cont..)
How line clipping works:
Before
After
Let us discuss about the following Line clipping algorithms in the coming slides:
1. Cohen-Sutherland Clipping
2. Liang-Barsky Clipping
3. NLN Clipping Algorithm

10. Cohen-Sutherland Clipping Algorithm
Oldest, most popular & efficient line clipping algorithm
It reduces the number of line intersections that must be calculated by performing initial tests & speeds up processing
Every line endpoint in a picture is assigned a 4-digit binary code called a region code
Region code identifies the location of a point relative to the boundaries of the clipping rectangle

11. 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

12. 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]

13. 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]

14. 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 (result – not 0000) 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]

15. Cohen-Sutherland: Other Lines
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

16. Cohen-Sutherland: Other Lines (cont…)
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

17. Cohen-Sutherland Examples
Consider the line P9 to P10 below
Start at 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]

18. Cohen-Sutherland Examples (cont…)
Consider the line P3 to P4 below
Start at P4
From the region codes of the two end-points we know the line crosses the left boundary so calculate the intersection point to generate P4’
The line P3 to P4’ is completely outside the window so is clipped
wymax
wymin
wxmin
wxmax
Window
P4’ [1001]
P3 [0001]
P4 [1000]

19. Cohen-Sutherland Examples (cont…)
Consider the line P7 to P8 below
Start at P7
From the two region codes of the two end-points we know the line crosses the left boundary so calculate the intersection point to generate P7’
wymax
wymin
wxmin
wxmax
Window
P7’ [0000]
P7 [0001]
P8 [0010]
P8’ [0000]

20. Cohen-Sutherland Examples (cont…)
Consider the line P7’ to P8
Start at P8
Calculate the intersection with the right boundary to generate P8’
P7’ to P8’ is inside the window so is retained
wymax
wymin
wxmin
wxmax
Window
P7’ [0000]
P7 [0001]
P8 [0010]
P8’ [0000]

21. Liang-Barsky Line Clipping Algorithm:-
The Liang–Barsky algorithm uses the parametric equation of a line and inequalities describing the range of the clipping window to determine the intersections between the line and the clipping window. With these intersections it knows which portion of the line should be drawn.
Consider first the usual parametric form of a straight line:
X= x0+u(x1-x0)= x0+uΔx,
Y= y0+u(y1-y0)= y0+uΔy,
where 0<=u<=1.

22. Liang-Barsky line clipping (cont..)
Faster line clipper algorithm
Based on the analysis of parametric equation of a line segment of the form
x=x0+u Δx
y=y0+u Δy, // 0<=u<=1
Δx=x1-x0 &
Δy=y1-y0
More efficient
Both Cohen-Sutherland and Liang-Barsky algorithm can be extended to 3-D clipping

23. Liang-Barsky line clipping (cont..)
Algorithm:-
1. Set umin = 0 and umax = 1.
2. Calculate the u values:
3. If u < umin or u > umax ignore it.
Otherwise classify the u values as entering or exiting.
4. If umin < umax then draw a line from: ( x0 + Δx · umin, y0 + Δy · umin ) to
( x0 + Δx · umax, y0 + Δy · umax ).

24. Liang-Barsky line clipping (cont..)
Example:-
Let P1 (-1, -2), P2 (2, 4)
=> XL = 0, XR = 1, YB = 0, YT = 1
=> dx = 2 - (-1) = 3; dy = 4 - (-2) = 6
P1 = -dx = -3; Q1 = x1 - XL = = -1; So, Q1 / P1 = 1/3
P2 = dx = 3; Q2 = XR - x1 = 1 - (-1) = 2; So, Q2 / P2 = 2/3
P3 = -dy = -6; Q3 = y1 - YR = -2; So, Q3 / P3 = 1/3
P4 = dy = 6; Q4 = YT - y1 = 3; So, Q4 / P4 = 1/2
for (Pi < 0) t1="MAX" ( 1 / 3, 1 / 3, 0 )= 1 / 3
for (Pi > 0) t2 = MIN ( 2 / 3, 1 / 2, 1 ) = 1 / 2
Since t1 < t2 there is a visible section

25. Liang-Barsky line clipping (cont..)
Example cont.. :-
Now, Compute new endpoints:
t1 = 1 / 3 x1' = x1 + dx . t1 = -1 + (3 * 1 / 3) = 0
y1' = y1 + dy . t1 = -2 + (6 * 1 / 3) = 0
t2 = 1 / 2 x2' = x1 + dx . t2 = -1 + (3 * 1 / 2) = 1 / 2
y2' = y1 + dy . t2 = -1 + (6 * 1 / 2) = 1
These are the new end point after performing
clipping using Liang-Barsky line clipping algorithm.

26. NLN clipping Algorithm
NLN method uses more regions testing in the xy plane to reduce intersection calculations. The extra intersection calculation eliminated in the NLN algorithm by carrying out more region testing before intersection position is calculated.
It uses symmetry to categorize endpoints into one of 3 regions (in the clip window, edge & corner).
Then according to end point category it checks several clip cases.

27. NLN clipping Algorithm (cont..)
Fig: NLN clipping window.
Totally, we can see 9 regions in each image.
Only the three region pointed above need be considered
If P1 lies in any one of the other six region, we can move it to one of the three regions using a symmetry transformation.

28. Clipping against Concave Clipping Windows
If the clipping polygon is of concave topology, then if the line to be clipped intersects the clipping polygon then it does so at even number of points
In this case, in order to find the clipped line, it is necessary to sort the points of intersection, pair the intersection points and then join each pair by a line segment. Sorting can be performed based on either x or y coordinates

29. POINT CLIPPING Consider a point (x,y)
If wxmin ≤ x ≤ wxmax AND wymin ≤ y ≤ wymax
It is not clipped!
Otherwise
It is clipped!

30. 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 Below are few examples:

31. AREA CLIPPING (cont..)
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

32. Sutherland-Hodgman Area Clipping Algorithm (cont…)
AREA CLIPPING (cont..)
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

33. CURVE CLIPPING
Curve clipping procedures will involve non-linear equations (so requires more processing than for objects with linear boundaries In general, methods depend on how characters are represented).
Clipping curves requires more work
For circles we must find the two intersection points on the window boundary

34. CURVE CLIPPING (cont..) Preliminary test (Test for overlapping)
The bounding rectangle for a circle or other curved object is used to test for overlap with a rectangular clip window.
If the bounding rectangle is completely inside (save object), completely outside (discard the object)
Both cases- no computation is necessary.
If bounding rectangle test fails, use computation-saving approaches
Circle – coordinate extents of individual quadrants & then octants are used for preliminary testing before calculating curve–window intersections
Ellipse – coordinate extents of individual quadrants are used.
If 2 regions overlap, solve the simultaneous line-curve equations to obtain the clipping intersection points.

35. TEXT CLIPPING It is used to clip character strings.
In general, methods depend on how characters are represented.
Based on the method used to generate the original characters, three strategies can be followed:
1) All-or-none string clipping
use a bounding rectangle for the string
2) All-or-none character clipping
use a bounding rectangle for the character
3) Individual character clipping or component clipping
like line/curve clipping (outlined char’s)
compare individual pixels (bit-mapped)

36. TEXT CLIPPING (cont..) All-or-none string clipping : Example:
It is used to clip the strings relative to the window boundary. Just consider a bounding rectangle around the string.
If all of the string is inside the window
Save it!
Otherwise discard it!
Example:

37. TEXT CLIPPING (cont..) Example: All-or-none character clipping:
It is used to clip the characters relative to the window boundary.
If all of the character is inside the window
Save it!
Otherwise discard it!
Example:

38. TEXT CLIPPING (cont..) Example: Component clipping :
Clip the components of individual characters.
If an individual character crosses or overlaps clip window boundary
Discard that component of the character!
Otherwise saveit!
Example:

39. EXTERIOR CLIPPING Clip a picture to the exterior of a specified region
The picture parts to be saved are those that are outside the region.
Ex: Multiple window systems, applications that require overlapping pictures.

40. Polygon Clipping
A polygon boundary processed with a line clipper may be displayed as a series of unconnected line segments, depending on the orientation of the polygon to the clipping window.
What we really want to display is a bounded area after clipping. For polygon clipping, we require an algorithm that will generate one or more closed areas that are then scan converted for the appropriate area fill.
The output of a polygon clipper should be a sequence of vertices that defines the clipped polygon boundaries.

41. Polygon Fill-Area Clipping

42. Steps involved in Polygon Clipping
STEP 1: Original polygon
STEP 2: After clipping RIGHT Boundary
STEP 3: After clipping RIGHT & BOTTOM Boundaries
STEP 4: After clipping RIGHT & BOTTOM & LEFT Boundaries
STEP 5: After clipping ALL Boundaries

43. Weiler- Atherton Polygon Clipping
In this algorithm, the vertex –processing procedures for window boundaries are modified so that concave polygons are displayed correctly. This clipping procedure was developed as a method for identifying visible surfaces, and so it can be applied with arbitrary polygon-clipping regions.
The basic idea in this algorithm is that instead of always proceeding around the polygon edges as vertices are processed,we sometimes want to follow the window boundaries .Which path we follow depend on the polygon-processing direction(clockwise or counterclockwise)

44. Weiler-Atherton Polygon Clipping

45. REFERENCES