1. 2D Viewing & Clipping
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2. Contents The viewing pipeline Viewing coordinate reference frame
Window-to-viewport coordinate transformation
Clipping operations
Point clipping
Line clipping

3. The viewing pipeline [1/5]
window
a world-coordinate area selected for display
define what is to be viewed
view port
an area on a display device to which a window is mapped
define where it is to be displayed
windows & viewport
be rectangles in standard position, with the rectangle edges parallel to the coordinate axes
other geometries : take longer to process

4. The viewing pipeline [2/5]
viewing transformation
the mapping of a part of a world-coordinate scene to device coordinates
2D viewing transformation = window-to-viewport, windowing transformation

5. The viewing pipeline [3/5]
viewing-transformation in several steps
construct the world-coordinate scene using modeling transformations
transform descriptions in world coordinates to viewing coordinates
map the viewing-coordinate description of the scene to normalized coordinates
transfer to device coordinates

6. The viewing pipeline [4/5]
viewing-transformation
by changing the position of the viewport
can view objects at different positions on the display area of an output device
by varying the size of viewports
can change the size and proportions of displayed objects
zooming effects
= panning effects

7. The viewing pipeline [5/5]
viewport
define within the unit square
device-independent
⇒ separating the viewing & other transformations from specific output-device requirements
the unit square is mapped to the display area for the particular output device in use at that time
viewport clipping
perform in normalized coordinates or in device coordinates
to reduce computations by concatenating the various transformation matrices
provide for specifying the world-coordinate window

8. Viewing coordinate reference frame
view up vector
define the viewing yv direction
unit vectors v=(vx, vy) and u=(ux, uy) for the viewing yv and xv axes
The composite 2D transformation to convert world coordinates to viewing coordinates
MWC,VC = R • T

9. Window-to-viewport coordinate transformation
transfer to the viewing reference frame
choose the window extents in viewing coordinate
select the viewport limits in normalized coordinate
to maintain the same relative placement in the viewport as in the window

10. Window-to-viewport coordinate transformation
Relative proportions of objects
sx=xvmax-xvmin/xwmax-xwmin, sy=yvmax-yvmin/ywmax-ywmin
sx=sy : maintain
sx≠sy : stretch or contract in either the x or y direction when displayed on the output device
workstation transformation
select a window area in normalized space and a viewport area in the coordinates of the display device

11. Clipping operations clipping algorithm(simply clipping) clip window
identify those portions of picture that are either inside or outside of a specified region of space
clip window
the region against which an object is to clipped
clipping application
extracting part of a defined scene for viewing
identifying visible surfaces in 3D views
antialiasing line segments or object boundaries
creating objects using solid-modeling procedures
displaying a multiwindow environment
drawing & painting operations that allow parts of a picture to be selected for copying,moving,erasing, or duplicating

12. Clipping operations world-coordinate clipping viewport clipping
remove those primitives outside the window from further consideration
eliminating the processing necessary to transform those primitives to device space
viewport clipping
reduce calculations by allowing concatenation of viewing and geometric transformation matrices
require that the transformation to device coordinates be performed for all objects, including those outside the window area

13. Point clipping
Assuming that the clip window is a rectangle in standard position
P=(x, y)
the edges of the clip window(xwmin, xwmax, ywmin, ywmax) can be either the world-coordinate window boundaries or viewport boundaries
ex. apply to scenes involving explosions or sea foam

14. Line clipping Line clipping procedure
test a given line segment to determine whether it lies completely inside the clipping window
if it doesn’t, we try to determine whether it lies completely outside the window
if we can’t identify a line as completely inside or completely outside, we must perform intersection calculations with one or more clipping boundaries

15. Line clipping Checking the line endpoints ⇒ inside-outside test
Cohen-Sutherland line clipping
Liang-Barsky line clipping

16. Line clipping Cohen-Sutherland line clipping
one of the oldest and most popular line-clipping procedures
speed up the processing of line segments by performing initial tests that reduce the number of intersections that must be calculated
region code : four-digit binary code
⇒ bit 1:left, bit 2:right, bit 3:below, bit 4:above
for languages in which bit manipulation is possible

17. Line clipping region-code bit values determined
calculate difference between endpoint coordinates and clipping boundaries
use the resultant sign bit of each difference calculation to set the corresponding value in the region code
bit 1 : the sign bit of x – xwmin
bit 2 : the sign bit of xwmax – x
bit 3 : the sign bit of y – ywmin
bit 4 : the sign bit of ywmax – y

18. Line clipping
Determine which lines are completely inside the clip window and which line are clearly outside
completely inside the clip window
region code of 0000 for both endpoints
trivially accept these lines
clearly outside the clip window
any line that a 1 in the same bit position in region code
trivially reject these lines
by using logical and operation with both region codes

19. Line clipping check for intersection with the window boundaries
using the slope-intercept form of the line equation
x value Is set either to xwmin or to xwmax
y value Is set either to ywmin or to ywmax