Embed Size (px)
Citation preview
Courtesy of Andries van Dam©
Clipping Concepts, Algorithms for line clipping
1
Courtesy of Andries van Dam©Clipping 2
Assignment 3 will be out
Quiz2 next week
Courtesy of Andries van Dam©
} Inthecanonicalvolume,2operationsareeasy} Projection} Clipping
Last Time
X
Y
Z
2
2
1
(u)
(v)
(n)(P)
Courtesy of Andries van Dam©Clipping
} Easytoclip} Point?} Line?} Polygon?
4
Well, that’s a lie...
Courtesy of Andries van Dam©
} Line Clipping in 2D
5
Courtesy of Andries van Dam©
} Line Clipping in 2D
6
Courtesy of Andries van Dam©
} Line Clipping in 2D
7
Courtesy of Andries van Dam©
} ParametricformforlinesegmentParametric Line Formulation For Clipping
8
Courtesy of Andries van Dam©
} Parametricformforlinesegment
} Lineisincliprectangleifparametricvariablestlineandsedgebothin[0,1]atintersectionpointbetweenlineandedgeofcliprectangle} Slow,mustintersectlineswithalledges
Parametric Line Formulation For Clipping
9
Courtesy of Andries van Dam©
ClipRectangle} Cohen-Sutherland Line Clipping in 2D
10
Courtesy of Andries van Dam©
ClipRectangle} Cohen-Sutherland Line Clipping in 2D
11
Courtesy of Andries van Dam©
} Verysimilarto2D} Dividevolumeinto27regions(PictureaRubik’scube)
} 6-bitoutcoderecordsresultsof6boundstests} Firstbit:behindbackplane} Secondbit:infrontoffrontplane} Thirdbit:abovetopplane} Fourthbit:belowbottomplane} Fifthbit:totherightofrightplane} Sixthbit:totheleftofleftplane
} Again,lineswithOC0=0andOC1=0canbetriviallyaccepted
} Lineslyingentirelyinavolumeoutsideofaplanecanbetriviallyrejected:OC0ANDOC1≠0(i.e.,theysharean“outside”bit)
Cohen-Sutherland Line Clipping in 3D
12
Backplane000000(infront)
100000(behind)
Frontplane010000(infront)
000000(behind)
Bottomplane000000(above)
000100(below)
Leftplane000001(toleftof)
000000(torightof)
Topplane001000(above)
000000(below)
Rightplane000000(toleftof)
000010(torightof)
Courtesy of Andries van Dam©
} Ifwecanneithertriviallyaccept/reject(T/A,T/R),divideandconquer
} Subdividelineintotwosegments;thenT/AorT/Roneorbothsegments:} useaclipedgetocutline} useoutcodestochoosetheedgesthatarecrossed
} foragivenclipedge,ifaline’stwooutcodesdifferinthecorrespondingbit,thelinehasonevertexoneachsideoftheedge,thuscrosses
} pickanorderforcheckingedges:top–bottom–right–left} computetheintersectionpoint
} theclipedgefixeseitherxory} cansubstituteintothelineequation
} iterateforthenewlyshortenedline,“extra”clipsmayhappen(e.g.,E-IatH)
Cohen-Sutherland Algorithm (1/3)Cliprectangle
DC B
A
EF
GH
I
13
Courtesy of Andries van Dam©
} Cohen-Sutherland Algorithm (2/3)
ComputeOutCode(x0,y0,outcode0);ComputeOutCode(x1,y1,outcode1);
repeatcheckfortrivialrejectortrivialacceptpicktheoutsidepointandbit‘1’
ifTOPthenx=x0+(x1–x0)*(ymax–y0)/(y1–y0);y=ymax;elseifBOTTOMthenx=x0+(x1–x0)*(ymin–y0)/(y1–y0);y=ymin;
14
elseifRIGHTtheny=y0+(y1–y0)*(xmax–x0)/(x1–x0);x=xmax;elseifLEFTtheny=y0+(y1–y0)*(xmin–x0)/(x1–x0);x=xmin;
if(youpickedx0,y0)thenx0=x;y0=y;ComputeOutCode(x0,y0,outcode0)elsex1=x;y1=y;ComputeOutCode(x1,y1,outcode1)
untildone
Courtesy of Andries van Dam©
} Similaralgorithmforusing3Doutcodestoclipagainstcanonicalparallelviewvolume:Cohen-Sutherland Algorithm (3/3)
xmin=ymin=-1;xmax=ymax=1;zmin=-1;zmax=0;
ComputeOutCode(x0,y0,z0,outcode0);ComputeOutCode(x1,y1,z1,outcode1);repeat
checkfortrivialrejectortrivialacceptpickthepointthatisoutsidethecliprectangleifTOPthenx=x0+(x1–x0)*(ymax–y0)/(y1–y0);z=z0+(z1–z0)*(ymax–y0)/(y1–y0);y=ymax;elseifBOTTOMthenx=x0+(x1–x0)*(ymin–y0)/(y1–y0);z=z0+(z1–z0)*(ymin–y0)/(y1–y0);y=ymin;elseifRIGHTtheny=y0+(y1–y0)*(xmax–x0)/(x1–x0);z=z0+(z1–z0)*(xmax–x0)/(x1–x0);x=xmax;
15
elseifLEFTtheny=y0+(y1–y0)*(xmin–x0)/(x1–x0);z=z0+(z1–z0)*(xmin–x0)/(x1–x0);x=xmin;elseifNEARthenx=x0+(x1–x0)*(zmax–z0)/(z1–z0);y=y0+(y1–y0)*(zmax–z0)/(z1–z0);z=zmax;elseifFARthenx=x0+(x1–x0)*(zmin–z0)/(z1–z0);y=y0+(y1–y0)*(zmin–z0)/(z1–z0);z=zmin;
if(x0,y0,z0istheouterpoint)thenx0=x;y0=y;z0=z;ComputeOutCode(x0,y0,z0,outcode0)elsex1=x;y1=y;z1=z;ComputeOutCode(x1,y1,z1,outcode1)
untildone
Courtesy of Andries van Dam©
} Scan Conversion after Clipping
B A
x=xmin
y=ymin
y=ymin–1y=ymin–1/2
Cliprectangle
16
Courtesy of Andries van Dam©
Sutherland-Hodgman Polygon Clipping
17
} The2DSutherland-Hodgmanalgorithmgeneralizestohigherdimensions} Wecanuseittoclippolygonstothe3Dviewvolumeoneplaneatatime} SearchforBALSAClippingonyoutube(0:50,3:45)
Courtesy of Andries van Dam©
} Cyrus-Beck/Liang-Barsky Parametric Line Clipping (1/3)
18
Courtesy of Andries van Dam©
} NowsolveforthevalueoftattheintersectionofP0P1withtheedgeEi:
} First,substituteforP(t):} Next,grouptermsanddistributedotproduct:} LetDbethevectorfromP0toP1=(P1–P0),andsolvefort:} notethatthisgivesavalidvalueoftonlyifthe
denominatoroftheexpressionisnonzero.} Forthistobetrue,itmustbethecasethat:
} Ni≠0(thatis,thenormalshouldnotbe0;thiscouldoccuronlyasamistake)
} D≠0(thatis,P1≠P0)} Ni•D≠0(edgeEiandlineDarenotparallel;iftheyare,
nointersection).} Thealgorithmcheckstheseconditions.
Cyrus-Beck/Liang-Barsky Parametric Line Clipping (2/3)
19
Courtesy of Andries van Dam©
} Cyrus-Beck/Liang-Barsky Parametric Line Clipping (3/3)
20
Courtesy of Andries van Dam©
Cyrus-Beck/Liang-Barsky Line Clipping AlgorithmPre-calculateNiandselectPEiforeachedge;foreachlinesegmenttobeclippedifP1=P0thenlineisdegeneratesoclipasapoint;elsebegintE=0;tL=1;foreachcandidateintersectionwithaclipedgeifNi•D≠0then{Ignoreedgesparalleltoline}begincalculatet;{oflineandclipedgeintersection}usesignofNi•DtocategorizeasPEorPL;ifPEthentE=max(tE,t);ifPLthentL=min(tL,t);endiftE>tLthenreturnnilelsereturnP(tE)andP(tL)astrueclipintersectionsend
21
Courtesy of Andries van Dam©
} D=P1–P0=(x1–x0,y1–y0)
} LeavePEiasanarbitrarypointonclipedge:it’safreevariableanddropsout
Parametric Line Clipping for Upright Clip Rectangle (1/2)
CalculationsforParametricLineClippingAlgorithm
(x0-x,y0-ymax)(x,ymax)(0,1)top:y=ymax
(x0-x,y0-ymin)(x,ymin)(0,-1)bottom:y=ymin
(x0-xmax,y0-y)(xmax,y)(1,0)right:x=xmax
(x0-xmin,y0-y)(xmin,y)(-1,0)left:x=xmin
P0-PEiPEiNormalNiClipEdgeiDiNiE
PPiNt•−
−•=
)0(
)01(
)min0(
xx
xx
−
−−
)01(
)max0(
xx
xx
−
−−
)01(
)min0(
yy
yy
−
−−
)01(
)max0(
yy
yy
−
−−
22
Courtesy of Andries van Dam©
} Examinet:} Numeratorisjustthedirecteddistancetoanedge;signcorrespondstoOC
} Denominatorisjustthehorizontalorverticalprojectionoftheline,dxordy;signdeterminesPEorPLforagivenedge
} Ratioisconstantofproportionality:“howfarover”fromP0toP1intersectionisrelativetodxordy
Parametric Line Clipping for Upright Clip Rectangle (2/2)
23
Courtesy of Andries van Dam©Clipping
} Cohen-Sutherlandisbetterwhenmanytrivialaccepts/reject} Otherwise,Liang-Barskyisbetter
24
Pros and Cons
![ICG@ [MID-1 BITS] - WordPress.com · Which algorithm not applicable for non rectangular clipping windows :->Mid point subdivision algorthm In cohen-sutherland line clipping which](https://img.dokumen.tips/doc/110x75/5ebab42c95649e427e53c177/icg-mid-1-bits-which-algorithm-not-applicable-for-non-rectangular-clipping.jpg)
