1. Two-Dimensional Viewing
Chapter 6
Two-Dimensional Viewing

2. The Viewing Pipeline Window
A world-coordinate area selected for display.
defines what is to be viewed
Viewport
An area on a display device to which a window is mapped.
defines where it is to be displayed
Viewing transformation
The mapping of a part of a world-coordinate scene to device coordinates.
A window could be a rectangle to have any orientation.

3. Two-Dimensional Viewing

4. The Viewing Pipeline

5. The Viewing Pipeline

6. Viewing Effects Zooming effects Panning effects Device independent
Successively mapping different-sized windows on a fixed-sized viewports.
Panning effects
Moving a fixed-sized window across the various objects in a scene.
Device independent
Viewports are typically defined within the unit square (normalized coordinates)

7. Viewing Coordinate Reference Frame
The reference frame for specifying the world-coordinate window.
Viewing-coordinate origin: P0 = (x0, y0)
View up vector V: Define the viewing yv direction

8. Window-to-Viewport Coordinate Transformation

9. Window-to-Viewport Coordinate Transformation

10. Workstation transformtion

11. Clipping Operations Clipping Clip window Applications of clipping
Identify those portions of a picture that are either inside or outside of a specified region of space.
Clip window
The region against which an object is to be clipped.
The shape of clip window
Applications of clipping
World-coordinate clipping

12. Clipping Operations Viewport clipping Types of clipping
It can reduce calculations by allowing concatenation of viewing and geometric transformation matrices.
Types of clipping
Point clipping
Line clipping
Area (Polygon) clipping
Curve clipping
Text clipping
Point clipping (Rectangular clip window)

13. Line Clipping
Possible relationships between line positions and a standard rectangular clipping region
Before clipping
after clipping

14. Line Clipping Possible relationships
Completely inside the clipping window
Completely outside the window
Partially inside the window
Parametric representation of a line
x = x1 + u(x2 - x1)
y = y1 + u(y2 - y1)
The value of u for an intersection with a rectangle boundary edge
Outside the range 0 to 1
Within the range from 0 to 1

15. Cohen-Sutherland Line Clipping
Region code
A four-digit binary code assigned to every line endpoint in a picture.
Numbering the bit positions in the region code as 1 through 4 from right to left.

16. Cohen-Sutherland Line Clipping
Bit values in the region code
Determined by comparing endpoint coordinates to the clip boundaries
A value of 1 in any bit position: The point is in that relative position.
Determined by the following steps:
Calculate differences between endpoint coordinates and clipping boundaries.
Use the resultant sign bit of each difference calculation to set the corresponding bit value.

17. Cohen-Sutherland Line Clipping
The possible relationships:
Completely contained within the window
0000 for both endpoints.
Completely outside the window
Logical and the region codes of both endpoints, its result is not 0000.
Partially

18. Liang-Barsky Line Clipping
Rewrite the line parametric equation as follows:
Point-clipping conditions:
Each of these four inequalities can be expressed as:

19. Liang-Barsky Line Clipping
Parameters p and q are defined as

20. Liang-Barsky Line Clipping
pk = 0, parallel to one of the clipping boundary
qk < 0, outside the boundary
qk >= 0, inside the parallel clipping boundary
pk < 0, the line proceeds from outside to the inside
pk > 0, the line proceeds from inside to outside

21. Liang-Barsky Line Clipping

22. Liang-Barsky Line Clipping
Parameters u1 and u2 that define that part of the line lies within the clip rectangle
The value of u1 : From outside to inside (pk < 0)
The value of u2 : From inside to outside (pk > 0)
If u1 > u2, the line is completely outside the clip window.

23. Liang-Barsky Line Clipping

24. Nicholl-Lee-Nicholl Line Clipping
Compared to C-S and L-B algorithms
NLN algorithm performs fewer comparisons and divisions.
NLN can only be applied to 2D clipping.
The NLN algorithm
Clip a line with endpoints P1 and P2
First determine the position of P1 for the nine possible regions.
Only three regions need be considered
The other regions using a symmetry transformation
Next determine the position of P2 relative to P1.

25. Nicholl-Lee-Nicholl Line Clipping

26. Nicholl-Lee-Nicholl Line Clipping

27. Nicholl-Lee-Nicholl Line Clipping

28. Nicholl-Lee-Nicholl Line Clipping
To determine the region in which P2 is located
Compare the slope of the line to the slopes of the boundaries of the clip region.
Example: P1 is left of the clipping rectangle, P2 is in region LT.

29. Nonrectangular Clip Windows
Algorithms based on parametric line equations can be extended
Liang-Barsky method
Modify the algorithm to include the parametric equations for the boundaries of the clip region
Concave polygon-clipping regions
Split into a set of convex polygons
Apply parametric clipping procedures
Circle or other curved-boundary clipping regions
Lines can be clipped against the bounding rectangle

30. Splitting Concave Polygons
Identify a concave polygon
Calculating the cross product of successive edge vectors.
If the z component of some cross product is positive while others have a negative, it is concave.

31. Splitting Concave Polygons
Vector method
Calculate the edge-vector cross product in a counterclockwise order.
If any z component turns out to be negative
The polygon is concave.
Split it along the line of the first edge vector in the cross-product pair.

32. Splitting Concave Polygons
Rotational method
Proceeding counterclockwise around the polygon edges.
Translate each polygon vertex Vk to the coordinate origin.
Rotate in a clockwise direction so that the next vertex Vk+1 is on the x axis.
If the next vertex, Vk+2, is below the x axis, the polygon is concave.
Split the polygon along the x axis.

33. Splitting Concave Polygons

34. Polygon Clipping

35. Sutherland-Hodgeman Polygon Clipping
Processing the polygon boundary as a whole against each window edge
Processing all polygon vertices against each clip rectangle boundary in turn

36. Sutherland-Hodgeman Polygon Clipping
Pass each pair of adjacent polygon vertices to a window boundary clipper
There are four cases:

37. Sutherland-Hodgeman Polygon Clipping
Intermediate output vertex list
Once all vertices have been processed for one clip window boundary, it is generated.
The output list of vertices is clipped against the next window boundary.
It can be eliminated by a pipeline of clipping routine.
Convex polygons are correctly clipped.
If the clipped polygon is concave
Split the concave polygon

38. Sutherland-Hodgeman Polygon Clippingv3 v2 v1

39. Weiler-Atherton Polygon Clipping
Developed as a method for identifying visible surfaces
It can be applied with arbitrary polygon-clipping region.
Not always proceeding around polygon edges
Sometimes follows the window boundaries
For clockwise processing of polygon vertices
For an outside-to-inside pair of vertices, follow the polygon boundary.
For an inside-to-outside pair of vertices, follow the window boundary in clockwise direction.

40. Weiler-Atherton Polygon Clipping

41. Other Clipping Curve clipping Text clipping
Use bounding rectangle to test for overlap with a rectangular clip window.
Text clipping
All-or-none string-clipping
All-or-none character-clipping
Clip the components of individual characters

42. Exterior Clipping Save the outside region Applications
Multiple window systems
The design of page layouts in advertising or publishing
Adding labels or design patterns to a picture
Procedures for clipping objects to the interior of concave polygon windows

43. Exterior Clipping