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Introduction to Computer Graphics

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Introduction to Computer Graphics

Chapter 6 â€“ 2D Viewing Pt 2

- Procedures that identifies those portions of a picture that are either inside or outside a specified region of a space is referred to as clippingalgorithm or clipping
- The region against which an object is to
clipped is called clip window.

after

before

- Note: different graphic elements may require different clipping techniques. Character, need to include all or completely omit depending on whether or not its center lies within the window.

- Application of clipping include:
- Extracting part of defined scene for viewing.
- Identifying visible surfaces in three-dimensional views.
- Anti-aliasing line segments or object boundaries.
- Creating objects using solid-modeling procedures.
- Displaying a multiwindow environment.
- Drawing and painting operations that allow parts of picture to be selected for copying, moving, erasing, or duplication.

- Clipping algorithm can be applied in both world coordinate and normalized device coordinates
- In world coordinate:
- only the contents of the window interior are mapped to device coordinates
- world coordinate clipping removes those primitives outside the window from further consideration. Eliminating the unnecessary processing to transform

- In normalized device coordinate:
- mapping is fully done from world to normalized device.
- clipping will reduce calculations by allowing concatenation of viewing and geometric transformation matrices.

- Algorithms have been developed for clipping for the following primitives types:
1. Point Clipping.

2. Line Clipping (straight-line

segments)

3. Area Clipping (polygons)

4. Curve Clipping

5. Text Clipping.

- Assuming clip window is standard rectangle position, point P-(x,y) will be save for display if its satisfied:
xwmin <= x <= xwmax

ywmin <= y <= ywmax

- where the edges of the clip window (xwmin, xwmax, ywmin, ywmax) can be either the world coordinate window boundaries or viewport boundaries.

- Cohen-Sutherland Outcode Algorithm
- This algorithm quickly removes lines which entirely to one side of the clipping where both endpoints above, below, right or left.
- The advantage of the algorithm is, it uses bit operations for testing.

E

I

K

1000

1010

1001

G

B

L

H

S

M

A

0010

0001

0000

F

T

J

D

0110

0101

C

0100

- Bit convention 1
- First bity > ymax
- Second bity < ymin
- Third bitx > xmax
- Fourth bitx < xmin

- bit 1 : left
- bit 2 : right
- bit 3 : below
- bit 4 : above

- Segment endpoints are each given 4-bit binary codes.
- The high order bit is set to 1 if the point above window
- The next bit set to 1 if point is below window
- The third and fourth indicate right or left of window.

- 0000 if both endpoints are inside window.
- if the line segment lies entirely on one side of the window, then both endpoints will have a 1 in the outcode bit position for that side.

- By taking AND operation. If NONZERO then line segment may be rejected. Ex: AB and CD

- Certain lines cannot be identified as completely inside or outside a clip window by doing AND test. Lines which completely outside the clip window can also produce AND 0000. So, process of discarding is required and intersection with boundary must be calculated (divide and conquer).
- Difficulty is when a line crossed one or more of the lines which contain the clipping boundary. Ex: EF and DJ

- The point of intersection between the line segment and clipping lines may be used to break up the line segment.
- Intersection points with a clipping boundary can be calculated using the slope-intercept form of the line equation.

- For a line with endpoint (x1,y1) to (x2,y2)
- Y coordinate of the intersection point with vertical boundary is obtained with
y = y1 + m(x-x1)

- where x value is set either xwmin or xwmax and the slope of the line is m = (y2 â€“ y1) / (x2 â€“ x1)
- X coordinate of the intersection point with horizontal boundary is calculated by
x = x1 + (y-y1)/m

- where y is set either to ywmin or to ywmax

- Y coordinate of the intersection point with vertical boundary is obtained with

- Brief outline of the algorithm.
- First compute the outcodes for the two endpoints (p1 and p2).
- Enter the loop, check whether both outcodes are zero
- Both zero, then enter segment in display list.
- If not both zero, perform the AND function. If nonzero, reject the line. If zero, subdivide the line segment. Repeat the loop.

- Parametric equation of a line segment (x1,y1) and (x2,y2)
- x = x1 + uâˆ†x
- y = y1 + uâˆ†y

- Where 0 <= u <=1
- âˆ†x = x2 â€“ x1
- âˆ†y = y2 â€“y1
- From point clipping, we have
- xwmin < = x <= xwmax
- Ywmin <= y <= ywmax

- For line clipping, we derived
- xwmin < = x1 + uâˆ†x <= xwmax
- ywmin <= y1 + uâˆ†y <= ywmax

- Based on these four equations, the following rules are introduced
upk <= qk

- where k = 1 (left boundary)
2 (right boundary)

3 (bottom boundary)

4 (top boundary)

parameters p and q are defined as:

p1 = -âˆ†x,q1 = x1 - xwmin

p2 = âˆ†x,q2 = xwmax â€“ x1

p3= - âˆ†y,q3 = y1 - ywmin

p4=âˆ†y,q4 = ywmax â€“ y1

- where k = 1 (left boundary)

- If any line that is parallel to one of the clipping boundaries, pk = 0. If qk < 0 then line is completely outside the boundary.
- When pk < 0, the infinite extension of the line proceeds from the outside to the inside.
- If pk > 0, the line proceeds from the inside to the outside.
- For nonzero value of pk , we can calculate the value of u corresponds to the point where infinitely extended line intersects the extension boundary k as
u = qk/ pk

- u1(outside to inside) where its value ranges from 0 to r
- u2 (inside to outside) where its value ranges from 1 to r.
- If u1 > u2, the line is completely outside the clip window and can be rejected

- Liang-Barsky algorithm is more efficient than Cohen since intersection calculations are reduced.
- Liang-Barsky require only one division and window intersections of the line are computed only once.
- Cohen required repeated calculation even though the line may be completely outside the clip window.