CSE 403 Computer Graphics Clipping Cohen Sutherland Algorithm (Line) Cyrus-Beck Algorithm (Line)...

CSE 403 Computer Graphics

Clipping

• Cohen Sutherland Algorithm (Line)• Cyrus-Beck Algorithm (Line)• Sutherland-Hodgeman Algorithm (Polygon)

2

(xmin , ymin )

(xmax , ymax )

x = xmin x = xmax

y = ymin

y = ymax

(x1, y1)

cliprectangle

maxmin

maxmin

yyy

xxx

For a point For a point ((x,yx,y) ) to be inside the clip rectangleto be inside the clip rectangle::

Point Clipping

3

cliprectangle

Cases for clipping linesCases for clipping lines

Line Clipping

4

Cases for clipping linesCases for clipping lines

A

B

A

B

cliprectangle

Line Clipping

5

Cases for clipping linesCases for clipping lines

D

A

BC

D'

A

BC

D'

cliprectangle

Line Clipping

6

Cases for clipping linesCases for clipping lines

D

E

F

A

BC

D'

A

BC

D'

cliprectangle

Line Clipping

7

Cases for clipping linesCases for clipping lines

D

E

F

A

BC

D'

G

H

G'

H'A

BC

D'

G'

H'

cliprectangle

Line Clipping

8

Cases for clipping linesCases for clipping lines

D

E

F

A

BC

D'

G

H

G'

H'

I

J

I'

J'

A

BC

D'

G'

H'

cliprectangle

Line Clipping

9

Clipping Lines by Solving Simultaneous EquationsClipping Lines by Solving Simultaneous Equations

cliprectangle

(xa, ya)(xb , yb)

(xc, yc) (xd , yd)

(x0, y0)

(x1, y1)

(x , y )

cliprectangle

(xa, ya)(xb , yb)

(xc, yc) (xd , yd)

(x0, y0)

(x1, y1)

(x , y )

abedgeaabedgea

lineline

yytyyxxtxx

yytyyxxtxx

,

, 010010

Line Clipping

10

The Cohen-Sutherland Line-Clipping Algorithm performs initial tests on a line to determine whether intersection calculations can be avoided.1. First, end-point pairs are checked for Trivial

Acceptance.2. If the line cannot be trivially accepted, region

checks are done for Trivial Rejection.3. If the line segment can be neither trivially

accepted or rejected, it is divided into two segments at a clip edge, so that one segment can be trivially rejected.These three steps are performed iteratively until what remains can be trivially accepted or rejected.

Cohen-Sutherland Algorithm

11

0000

1000

0100

1001 1010

0001 0010

0101 0110clip

rectangle

Region outcodesRegion outcodes

min

max

yy

yy

:1bit

: 0bit

min

max

xx

xx

:3bit

: 2bit

Cohen-Sutherland Algorithm

12

1. A line segment can be trivially accepted if the outcodes of both the endpoints are zero.

2. A line segment can be trivially rejected if the logical and of the outcodes of the endpoints is not zero.

3. A key property of the outcode is that bits that are set in nonzero outcode correspond to edges crossed.

Cohen-Sutherland Algorithm

13

cliprectangle

AB

C

D

E

An ExampleAn Example

Cohen-Sutherland Algorithm

14An ExampleAn Example

cliprectangle

B

C

D

E

Cohen-Sutherland Algorithm

15An ExampleAn Example

cliprectangle

B

C

D

Cohen-Sutherland Algorithm

16An ExampleAn Example

cliprectangle

B

C

Cohen-Sutherland Algorithm

17

(1)(1) This fundamentally different (from Cohen-This fundamentally different (from Cohen-Sutherland algorithm) and generally more Sutherland algorithm) and generally more efficient algorithm was originally published by efficient algorithm was originally published by CyrusCyrus and and BeckBeck..

(2)(2) LiangLiang and and BarskyBarsky later independently later independently developed a more efficient algorithm that is developed a more efficient algorithm that is especially fast in the special cases of upright especially fast in the special cases of upright 2D and 3D clipping regions.They also 2D and 3D clipping regions.They also introduced more efficient trivial rejection introduced more efficient trivial rejection tests for general clip regions.tests for general clip regions.

Parametric Line-Clipping

18

Inside of clip rectangleOutside of clip region

0iEi PtPN

0iEi PtPN

0iEi PtPN

iEi PtP

iEP

0P

1P

iN

iE Edge

tPPPtPPPLine 01010 :

01

0

01

0

010

010

,

0

0

0

PPDDN

PPNt

PPN

PPNt

PtPPPN

PtPPPN

PtPN

i

Ei

i

Ei

Ei

Ei

Ei

i

i

i

i

i

The Cyrus-Beck Algorithm

19

Inside of clip rectangleOutside of clip region

0iEi PtPN

0iEi PtPN

0iEi PtPN

iEi PtP

iEP

0P

1P

iN

iE Edge

DN

PPNt

i

Ei i

0

0 3

0 2

0 1

whenexists

10

DN

PPD

N

t

i

i

The Cyrus-Beck Algorithm

20

PE

PL

P 0

P 1

t = 0

t = 1

PE

PL

P 1

t = 1

P 0

t = 0

PE

PE

PLPL

P 0

t = 0

P 1

t = 1Line 2

Line 1

Line 3

Cliprectangle

90

0

Angle

PEDNi

90

0

Angle

PLDNi

PE = Potentially Entering PL = Potentially Leaving

The Cyrus-Beck Algorithm

21

The Cyrus-Beck AlgorithmPrecalculate Ni and PEi for each edgefor (each line segment to be clipped) {

if (P1 == P0)line is degenerated, so clip as a point;

else {tE = 0; tL = 1;for (each candidate intersection with a clip edge) {

if (Ni • D != 0) { /* Ignore edges parallel to line */calculate t;use sign of Ni • D to categorize as PE or PL;if (PE) tE = max(tE , t);if (PL) tL = min(tL , t);

}}if (tE > tL) return NULL;else return P(tE) and P(tL) as true clip intersection;

} }

22

ExampleExample

Polygon Clipping

23

ExampleExample

Polygon Clipping

24

ExampleExample

Polygon Clipping

25

Initial ConditionInitial Condition

Sutherland-Hodgeman Algo.

Clip Against Right Clipping Clip Against Right Clipping BoundaryBoundary

Clip Against Top Clipping Clip Against Top Clipping BoundaryBoundary

The Clipped PolygonThe Clipped PolygonClip Against Bottom Clipping Clip Against Bottom Clipping BoundaryBoundary

Clip Against Left Clipping Clip Against Left Clipping BoundaryBoundary

26

Case 1Case 1

Inside Outside

Polygonbeingclipped

Clipboundary

s

p:output

4 Cases of Polygon Clipping

Case 2Case 2

Inside Outside

s

p

i:output

Case 3Case 3

Inside Outside

s

p

(no output)

Case 4Case 4

Inside Outside

s

i:first output

p:second output

27

Reference

FV: 3.12, 3.13, 3.14