AN FOR 2-D DEVELOPMENT - Computer Sciencedavid/Classes/Talks/... · FOR 2-D LINE CLIPPING ITS...

AN EFFICIENT NEW ALGORITHM FOR 2-D LINE CLIPPING ITS DEVELOPMENT AND ANALYSIS Authors: T. Nicholl, D. T. Lee, and R. Nicholl

Presentation by Marcus McCurdy

WHAT IS CLIPPING?

 Basically finding the intersection of two geometric entities.

(x1,y1)

(x2,y2)

xLeft, yBottom

xRight, yTop

Intersection

OTHER ALGORITHMS

 Cohen-Sutherland  Cyrus Beck  Liang-Barsky  SPY

  In all of these algorithms, intersection points not involved in the final output are calculated

CLIPPING INEFFICIENCIES

 SPY and CS will calculate IL before rejecting  LB will compute all 4

IB

IT IL

IR

MACHINE INDEPENDENT ANALYSIS

 Evaluated efficiency by counting the number of executions of the following operations:

 Comparisons  Addition  Subtraction  Multiplication  Division

COSTS OF OPERATIONS

 Comparison  Addition  Subtraction  Multiplication  Division

INVISIBLE

IB

IT IL

IR

VISIBLE

PARTIALLY VISIBLE

9 REGIONS

Window

Top edge

Bottom edge

Left edge Right edge

Top right corner

Bottom right corner

Bottom left corner

Top left corner

ALGORITHM

 Characterize P1’s location among the 9 regions (straightforward).

 Based on P1 characterize P2 among appropriate subdivisions (not quite so straightforward).

 Calculate intersection points according to the characterization.

 Only the intersection points required for output are calculated.

POSITIONS OF P1 AND P2

 P1 is located in the window, an edge region, or a corner region.

 P2 may be at any point on the 2D Cartesian plane

 Divide all possible positions of P2 into regions each of which corresponds to intersection points at the same boundaries (or none) to be output

 The following 3 figures show the subdivisions. All other cases are the same as one of them up to a rotation of a multiple of 90º about the origin or a reflection about the line x = -y, or a combination of the two

(~) ~ Computer Graphics, Volume 21, Number 4, July 1987

X. CONCLUSIONS AND FUTURE WORK

We have developed a new 2-dimensional line clipping algorithm. Using a machine-independent analysis, we have shown that this algorithm is faster than both the Cohen-Sutherland algorithm and the Liang-Barsky algorithm. Using program transformation techniques we showed how geometric transformations on the line to be clipped could be performed by modified program text without the overhead of actually exe- cuting statements to perform the transformation. Currently, we are investigating the possibility of extending it to polygons and to the 3-dimensional case.

Acknowledgements The authors would like to acknowledge the useful discussions with colleagues and students at the University of Western Ontario, especially Ti m Walsh. We are very grateful for the reviewers' suggestions, which we have incorporated, within the space allowed. Financial upport for this research was provided by the National Science and Engineering Research Council (NSERC) of Canada.

REFERENCES

[I] Newman, W. M. and R. F. Sproull. Principles of Interac- tive Computer Graphics, 2nd ed., McGraw-Hill, New York, 1979.

[2] Sproull, R. F. and I. E. Sutherland. A Clipping Divider, FICC 1968, Thompson Books, Washington, D.C., pp. 765-775.

[3] Cyrus, M. and I. Beck. Generalised Two- and Three - Dimensional Clipping, Computers and Graphics, Vol. 3, No. 1, 1978, pp. 23-28.

[4] Liang, Y.-D. and B. A. Barsky. A New Concept and Method for Line Clipping, ACM Transactions on Graph- ies, Vol. 3, No. 1, 1984, pp. 1-22.

[5] Sobkow, M. S., Pospisil, P. and Y.-II. Yang. A Fast Two- Dimensional Line Clipping Algorithm, University of Saskatchewan Technical Report, 86-2.

[6] Hoare, C. A. R. et al. Laws of Programming: a tutorial paper, Oxford University Programming Research Group PRG-45, 1985.

[7] Arsac, J. I. Syntactic Source to Source Transforms and Program Manipulation, Communications of the ACM, Vol. 22, No. 1, 1979, pp. 43-54.

I R ....-+

iT ~ "

Fig. I Intersection points computed b5 previous al,3orithms

top le?t corner region

le?t edge resion

bottom le?t c o rne r region

top edge

resion

window

bottom edge

resion

top right cornet- region

risht edse

resion

bottom right c o r n e r region

Fi 9. 2 Subdivision o? the plane into regions

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Fig. 3 Subdivision of the plane when Pl is in the window

261

~ , ~ SIGGRAPH '87, Anaheim, July 27-31, 1987

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Fi$. 4 Subdivision oF the plane when Pl is in an edse region

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FiS. 5 Subdivision oF the plane when P! is in a corner resion

I CS LB NLN 25.5 20 8

I 1.5 2 1 4.5 6 4

l 1.5 2 2 1.5 4 1

COMPARISONS +

/

CS LB NLN 21 20 7.5 1 2 1 3 6 3 1 2 1 1 4 1

CS LB NLN P1 i0 20 8 0 0 0 0 6 0 0 0 0 0 4 0

J

FIG.6 Analysis of the three algorithms when P1 is in the window

L

CS LB NLN 9.5 10.5 3.5 0 0 0 0 4 0 0 0 0 0 2.5 0

COMPARISONS +

/

/ /

/

CS LB / NLN i0 10.5 3.5 0 01 0 0 4 0 0 ,0 0 0 /2.5 0

/ / /_ - n

CS NLN F ~ .75 " NLN LB CS LB ~. 5 21 i~.75 ~.5 0 14.75 1 1 0 3 I/3 55 i/ CS LB NLN 32 /20 8.5 2/ 4 2 6 6 4 /~ 4 2

4 2

CS LB NLN 21 20 8 1 2 1 3 6 3 1 2 1 1 4 l

1 3.5 0

CS LB NLN 3b.5 20 g.5 2.5 4 2 7.5 6 5 2.5 4 3 2.5 4 2

CS LB NLN 32 20 8 2 4 2 6 6 4 2 4 2 2 4 2

FIG.7 Analysis of the three algorithms when P1 is in an edge region

"Further subdivision is needed. Average shown is for the largest unbounded subregion."

CS LB NLN 9 6 2.5 0 0 0 0 2.5 0 0 0 0 0 1.5 0

1 [ CS LB NLN CS LB NLN 9.5 10.5 3.75 9 10.5 3.75

~ 0 0 0 0 '-" -° °- ! I o o,o I \ \ o ~ o o o o \ \ 2.5 . 0 2.5 0

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+ 2.5 4 2.5 4 2 7.5 6 7.5 6 5

* 2.5 4 2.5 4 3 / 2.5\4 2.5 4 2

CS LB NLN 41 20 i0 3 4 2 9 6 6 3 4 4 3 4 2

CS LB NLN 25 14.5 8 1.5 0 0 4.5 5.5 5 1.5 0 3 1.5 3.5 0

FIG.8 Analysis of the three algorithms when P1 is in a corner region

262

GEOMETRIC TRANSFORMATIONS

 P1 may be in any one of 9 regions  Different procedure could be written to handle

each case  However, writing similar but not identical code

may be error-prone  Exploit symmetry to prevent this

ALGORITHM

 Find where P1 is in relation to the Left and Right boundaries.

 May be beyond left, beyond right, or between the two.

 Beyond left is symmetrical to beyond right up to a rotation of 180º about the origin and they can be handled together

P1 IS BEYOND LEFT BOUNDARY

  If P2 is also beyond LB then line is invisible stop  Now find if P1 is beyond top boundary, bottom

boundary, or between the two.  Again, beyond top is symmetrical with beyond

bottom, reflection about the x-axis.

P1 IS IN TOP LEFT CORNER

  If P2 is above top stop line is invisible  Compare P2 against the line joining P1 and the

top-left corner  The case where P2 is on one side of this line is

symmetrical to the case in which it is on the other side. Handle one of the two cases.

P1 TOP LEFT CORNER

 P2 not beyond the left and to the right of vector from P1 to the top left corner

  If P2 is above the bottom then no more oblique boundaries need to be computed

 However, if P2 is below bottom 3 possibilities  Boundary from line joining P1 to bottom left has

to be computed no matter which side of right boundary P2 happens to be on. Do this comparison first

  If left no more oblique boundaries, else more comparison to decide which subdivision P2 is in

P1 IN LEFT EDGE

  If P2 is beyond left then line is invisible  P2 may be above top, below bottom, or between

top and bottom.  Again above top is symmetric to below bottom

P1 IN LEFT EDGE & P2 BEYOND BOTTOM

 Comparing boundary joining P1 to the bottom left corner cannot be avoided so do this one first.

P1 BTW LEFT AND RIGHT

  If P1 above top or below bottom case is symmetrical to that of P1 in the left edge region so use the same procedure

P1 IN THE WINDOW

  If P2 is in an edge region then the intersection point can be seen from figure 3

  If P2 is in a corner region then comparison with one oblique boundary is necessary before appropriate intersection calculation is possible

 Use symmetry to reduce different cases

(~) ~ Computer Graphics, Volume 21, Number 4, July 1987

X. CONCLUSIONS AND FUTURE WORK

We have developed a new 2-dimensional line clipping algorithm. Using a machine-independent analysis, we have shown that this algorithm is faster than both the Cohen-Sutherland algorithm and the Liang-Barsky algorithm. Using program transformation techniques we showed how geometric transformations on the line to be clipped could be performed by modified program text without the overhead of actually exe- cuting statements to perform the transformation. Currently, we are investigating the possibility of extending it to polygons and to the 3-dimensional case.

Acknowledgements The authors would like to acknowledge the useful discussions with colleagues and students at the University of Western Ontario, especially Ti m Walsh. We are very grateful for the reviewers' suggestions, which we have incorporated, within the space allowed. Financial upport for this research was provided by the National Science and Engineering Research Council (NSERC) of Canada.

REFERENCES

[I] Newman, W. M. and R. F. Sproull. Principles of Interac- tive Computer Graphics, 2nd ed., McGraw-Hill, New York, 1979.

[2] Sproull, R. F. and I. E. Sutherland. A Clipping Divider, FICC 1968, Thompson Books, Washington, D.C., pp. 765-775.

[3] Cyrus, M. and I. Beck. Generalised Two- and Three - Dimensional Clipping, Computers and Graphics, Vol. 3, No. 1, 1978, pp. 23-28.

[4] Liang, Y.-D. and B. A. Barsky. A New Concept and Method for Line Clipping, ACM Transactions on Graph- ies, Vol. 3, No. 1, 1984, pp. 1-22.

[5] Sobkow, M. S., Pospisil, P. and Y.-II. Yang. A Fast Two- Dimensional Line Clipping Algorithm, University of Saskatchewan Technical Report, 86-2.

[6] Hoare, C. A. R. et al. Laws of Programming: a tutorial paper, Oxford University Programming Research Group PRG-45, 1985.

[7] Arsac, J. I. Syntactic Source to Source Transforms and Program Manipulation, Communications of the ACM, Vol. 22, No. 1, 1979, pp. 43-54.

I R ....-+

iT ~ "

Fig. I Intersection points computed b5 previous al,3orithms

top le?t corner region

le?t edge resion

bottom le?t c o rne r region

top edge

resion

window

bottom edge

resion

top right cornet- region

risht edse

resion

bottom right c o r n e r region

Fi 9. 2 Subdivision o? the plane into regions

• -l~-l~r]q i i | i i i 1 t 11 ITi ~ | I I I I I | I I I I I I I I I 1 1~ I 1 1 l-r~ l 1111111 ~ 71~Iq I I I I I I I I I I I I-1"I~ I I l i i i I I I i | I I l'l']'~Tl~lq I [ I I i I I J~'~R~ [~|'I IIIII tlll III I I~ tl II lllllll I III I I I I I~ I I III III I I I k~RRR~ ['FI~[TIq I I I L I I I I I FIt I I I I I I II II II Ill I II I |I I I I I I I I I I I~RRRRR~

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Fig. 3 Subdivision of the plane when Pl is in the window

261

ANALYSIS

 Fewest divisions among the algorithms. Equal to the number of intersection points for output.

 Fewest comparisons. About 1/3 CS and about 1/2 LB.

 CS requires largest number of comparison in most cases

 With large window dominating case is when P2 is in window. CS does few set operations, and so LB is slowest

 Small window dominating cases are the four corner regions. CS is slower in this case,


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262


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I 1.5 2 1 4.5 6 4

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CS LB NLN 21 20 7.5 1 2 1 3 6 3 1 2 1 1 4 1

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~ 0 0 0 0 '-" -° °- ! I o o,o I \ \ o ~ o o o o \ \ 2.5 . 0 2.5 0

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+ 2.5 4 2.5 4 2 7.5 6 7.5 6 5

* 2.5 4 2.5 4 3 / 2.5\4 2.5 4 2

CS LB NLN 41 20 i0 3 4 2 9 6 6 3 4 4 3 4 2

CS LB NLN 25 14.5 8 1.5 0 0 4.5 5.5 5 1.5 0 3 1.5 3.5 0


262


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I CS LB NLN 25.5 20 8

I 1.5 2 1 4.5 6 4

l 1.5 2 2 1.5 4 1

COMPARISONS +

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CS LB NLN 21 20 7.5 1 2 1 3 6 3 1 2 1 1 4 1

CS LB NLN P1 i0 20 8 0 0 0 0 6 0 0 0 0 0 4 0

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CS LB NLN 9.5 10.5 3.5 0 0 0 0 4 0 0 0 0 0 2.5 0

COMPARISONS +

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CS LB / NLN i0 10.5 3.5 0 01 0 0 4 0 0 ,0 0 0 /2.5 0

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4 2

CS LB NLN 21 20 8 1 2 1 3 6 3 1 2 1 1 4 l

1 3.5 0

CS LB NLN 3b.5 20 g.5 2.5 4 2 7.5 6 5 2.5 4 3 2.5 4 2

CS LB NLN 32 20 8 2 4 2 6 6 4 2 4 2 2 4 2



CS LB NLN 9 6 2.5 0 0 0 0 2.5 0 0 0 0 0 1.5 0

1 [ CS LB NLN CS LB NLN 9.5 10.5 3.75 9 10.5 3.75

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CS LB NLN 41 20 i0 3 4 2 9 6 6 3 4 4 3 4 2

CS LB NLN 25 14.5 8 1.5 0 0 4.5 5.5 5 1.5 0 3 1.5 3.5 0


262

AN FOR 2-D DEVELOPMENT - Computer Sciencedavid/Classes/Talks/... · FOR 2-D LINE CLIPPING ITS...

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Transcript of AN FOR 2-D DEVELOPMENT - Computer Sciencedavid/Classes/Talks/... · FOR 2-D LINE CLIPPING ITS...