CIM Algorithm for Approximating Three-Dimensional
Polygonal Curves
YONGJunhai(),HUShimin()andSUNJiaguang ( )
NationalCADEngineeringCenter,TsinghuaUniversity, Beijing100084, P.R.China
Department ofComputerScienceandTechnology,Tsinghua University,Beijing100084,P.R.China
E-mail: shimin@tsinghua.edu.cn
ReceivedNovember29,1999;revisedJuly17,2000.
Abstract The polygonal approximation problem is a primary problem in computer
graphics, pattern recognition, CAD/CAM,etc. In R 2
, the cone intersectionmethod(CIM)
isoneofthemosteÆcientalgorithmsforapproximatingpolygonalcurves. WithCIMEuand
Toussaint, byimposinganadditionalconstraintandchangingthegivenerror criteria,resolve
thethree-dimensionalweightedminimumnumberpolygonalapproximationproblemwiththe
parallel-striperrorcriterion(PS-WMN)underL2norm. Inthispaper,withoutanyadditional
constraintandchangeofthe errorcriteria,aCIMsolutiontothesameproblemwiththeline
segmenterrorcriterion(LS-WMN)ispresented,whichismorefrequentlyencounteredthanthe
PS-WMNis. Itstime complexity is O(n 3
), and the spacecomplexity isO(n 2
). An approxi-
mationalgorithmisalsopresented,whichtakesO(n 2
)timeand O(n)space. Resultsofsome
examplesaregiventoillustratetheeÆciencyofthesealgorithms.
Keywords polygonalcurve,CIM,LS-WMN,approximation,optimization
1 Introduction
Polygonalcurvesplayanimportantroleinmanyareas. Optimizationandapproximationalgorithms
for polygonal curves are widelyused. In computer graphics andimage processing, these algorithms
canbeusedtoobtainskeletonsoroutlinesofobjects [1 4]
. Inpatternrecognition,thesealgorithmscan
smoothawaynoise andhelp analyzingcharacteristic features [5]
. InCAD (Computer-Aided Design),
the polygonal curve is a basic geometric entity. In CAM (Computer-Aided Manufacturing), these
algorithmsreduce theline segmentsused in CNC(ComputerizedNumerical Control)paths soas to
reduce production cost. In communication, these algorithms can compress data and decrease the
transmission time requirements [4]
. Above all, almost all kinds of curves have to be converted into
polygonal curves before being displayed on screens, and these algorithms can be applied to reduce
computermemoryrequirementsandto speed updisplay [4]
.
An arbitrarypolygonal curve, which can be representedas P =[p
1
;p
2
;:::;p
n
], is made upof a
chain ofline segments[p
1
;p
2 ];[p
2
;p
3
];:::;[p
n 1
;p
n
]. Leteach vertex p
i
ofP havea weight"
i which
indicatestheerrortoleranceofp
i
. Thepolygonalapproximationproblemistondanapproximating
polygonalcurveP 0
=[p 0
1
;p 0
2
;:::;p 0
m
],suchthat:
i)ThesequenceofverticesofP 0
isasubsequenceofthesequenceofverticesofP,andthetwoend
vertices ofP 0
areoftencoincidentwiththoseofP,i.e.,p 0
1
=p
1 and p
0
m
=p
n
;
ii)Anylinesegment[p 0
A
;p 0
A+1 ]ofP
0
,whichsubstitutesasubchain[p
B
;p
B+1
;:::;p
C
]ofP,should
satisfyp 0
A
=p
B
;p 0
A+1
=p
C
,andthegivenerrorcriteriaEC(p
i
; p
0
A p
0
A+1 )"
i
,8i,B <i<C;
iii) misminimumover allapproximatingpolygonalcurves thatsatisfyi)andii).
Here, "
i
can be zero, which means that p
i
is a vertex of P 0
. The formulae p 0
1
= p
1 and p
0
m
= p
n
ensure thatP 0
retainsome properties ofP,e.g., open orclose. Constraint i)isuseful forvertices of
theapproximatingcurveP 0
areexactlyonP. Thispropertycanbe usedtoreducetheaccumulative
Thisresearchwas supportedbytheNationalNaturalScienceFoundationof China(No.69902004)andNKBRSF
error. The given formula for EC(p
i
;p 0
A p
0
A+1
) determines how the error tolerance is measured, and
leadsto dierentapproximationalgorithms.
Miyaoku and Harada have classied the polygonal approximation problems [6]
. This paper dis-
cussesLS-WMNproblemwithL
2 inR
3
(thethree-dimensionalweightedminimumnumberpolygonal
approximationproblemwiththelinesegmenterrorcriterionunderL
2
norm). Abruteforcealgorithm
ofthisproblemisas follows:
1)BuildthedirectedgraphG(V;E)where V consistsoftheverticesofP,andE consistsofthedirected
edgethat satises EC(pk;pipj)"k, forallk, i<k <j. Thenumberof edgesinG(V;E) is O(n 2
). This
methodforgeneratingG(V;E)requirescheckingtherelationshipamongeverythreeverticesinP,soitrequires
O(n 3
)time.
2) Find the shortest pathfrom p1 to pn inG(V;E), which requiresO(n 2
) timeusingforwarddynamic
programming.
The CIM(Cone Intersection Method) is similar to thebrute forcealgorithm except forthe pro-
cedure of building G(V;E). The basic idea is to check as few vertices as possible. The CIM uses
thetechnique of thecone intersections to check thegiven errorcriteria EC(p
k
;p
i p
j )"
k
, forall k,
i<k<j. ThediÆcultiesare:
(a)howto representandobtain thecones,and
(b)howto representandobtain theintersections ofthesecones.
Euand Toussaint [1]
thoughtthat these two problemsin R 3
weretoo diÆcultwhile dealing with
PS-WMNproblemwithL
2 inR
3
(thethree-dimensionalweightedminimumnumberpolygonalapprox-
imationproblemwithparallel-striperrorcriterionunderL
2
norm). Thustheyimposedanadditional
constraint on the problem, i.e., P should be strictly monotonic along the x-axis, and changed the
errortoleranceregionoftheverticesfrom balls
ER(p
i
;"
i
)=fqjD(p
i
;q)"
i g;
intoplanar discs
ER
1 (p
i
;"
i
)=fqjD(p
i
;q)"
i
andX(q)=X(p
i )g;
whereD(p
i
;q)isthedistance fromp
i
toq,X(p
i
)andX(q)arethex-coordinatesofp
i
and qrespec-
tively. Then, these planar discs are projected to a common plane, and the results are still planar
discs. The intersection ofthese projectiveplanar discs is usedin place of that of thecorresponding
three-dimensioncones. Certainly,theplanar discsaremuchsmaller thantheballs,andtheresulting
polygonal curvein general containsmore vertices than that ofthe optimalresult. Furthermore, the
CIMofEuandToussaintcannotbewidelyusedbecauseoftheadditionalconstraint.
Without increasing the time, space and implementation complexities, the method in the paper
removes the additionalconstraint andthe substitution ofthe errortolerancecriteria. SincetheLS-
WMNproblemismuchmorecommonthanthePS-WMNproblem,wediscusstheLS-WMNproblem
withL
2 in R
3
. Infact the PS-WMNproblem can also be solvedand theanswers tothe above two
questionsarepresentedinthis paper.
InSection5,wealsopresentanapproximationalgorithmwhichtakesO(n 2
)timeandO(n)space.
Inthepenultimatesection,theresultsofsome examplesaregivento illustratetheeÆciency ofthese
algorithms.
2 Denitions and Notations
Theproblemdiscussedinthispaper istheLS-WMNproblemwithL
2 inR
3
. Supposep;q;s;v;st,
edand c
b 2R
3
,randr
b
2R . Somenotationsusedinthispaperaredened asfollows:
X(p),Y(p)andZ(p)arethex-coordinate,y-coordinateandz-coordinateofprespectively.
D(p;q)= p
(X(p) X(q)) 2
+(Y(p) Y(q)) 2
+(Z(p) Z(q)) 2
isthedistancebetweenpandq.
L(p;q)=f(1 t)p+tqjt2R gisalinepassingthroughpandq. Whenpandqarecoincident,L(p;q)is
asetcontainingonlyonepoint.
HL(p;q)=f(1 t)p+tqjt0;t2R gisaraythatstartsfrompandpassesthroughq. Whenpandqare
[p;q]=f(1 t)p+tqj0t1;t2R gisalinesegmentwithpandqastheendpoints.Whenpandqare
coincident,[p;q]isconstitutedbyonlyonepoint.
D(s;L(p;q))=minfD(s;w)jw2L(p;q)gisthedistancebetweenthepointsandthelineL(p;q).
D(s;HL(p;q))=minfD(s;w)jw2HL(p;q)gisthedistancebetweenthepointsandtherayHL(p;q).
D(s;[p;q])=minfD(s;w)jw2[p;q]gisthedistancebetweensandthelinesegment[p;q]. EC(p
k
;pipj)is
equaltoD(p
k
;[p
i
;p
j
])inthispaper.
B(p;r) =fqj(X(p) X(q)) 2
+(Y(p) Y(q)) 2
+(Z(p) Z(q)) 2
= r 2
g isthe surface ofa ballwith the
centerpointpandtheradiusr.
BS(p;r)=fqj(X(p) X(q)) 2
+(Y(p) Y(q)) 2
+(Z(p) Z(q)) 2
r 2
gisasolidballwiththecenterpoint
pandtheradiusr.
CB(p;r)=fwjD(w;HL(cb;p))=r,w2B(cb;rb),w2R 3
gisacircleontheballB(cb;rb)ifp2B(cb;rb).
ArcB(p;r;st;ed) is an arcon the ball B(c
b
;r
b
), whichlies on the circleCB(p;r), starting from st and
endingatedincounterclockwiseorderaccordingtothedirection(p c
b
),ifCB(p;r)isacircleonB(c
b
;r
b ).
DB(p;r)=fwjD(w;HL(c
b
;p))r,w2B(c
b
;r
b
),w2R 3
gisadiscontheballB(c
b
;r
b
)ifp2B(c
b
;r
b ).
OC(p;q;r)=fsjw2HL(p;s),w2BS(q;r)gis calledanopencone. Notethat OC(p;q;r)isequaltoR 3
ifD(p;q)r.
ER(p
i
;"
i
) =fwjD(p
i
;w) "
i
, w2 R 3
gis the error toleranceregion at p
i
,where p
i
is avertexof the
polygonalcurveP,and"iistheweight(i.e. theerrortolerance)ofpi.
PP(p;q)=fwjq2HL(p;w), w2B(p;1)gisthesetcontainingtheprojectionpoint ofqontheunitball
B(p;1). Notethatifpandqarecoincident,PP(p;q)isthewholeunitballB(p;1).
PR(p;q;r)=OC(p;q;r)\B(p;1)istheprojectionoftheopenconeOC(p;q;r)ontheunitballB(p;1).
3 Theories of the Algorithm
Theorem 1. Let [p
A
;p
B
;p
C
] be a subchain of P. [p
A
;p
B
;p
C
] can be approximatedby [p
A
;p
C ] if
andonlyif p
C
2OC(p
A
;p
B
;"
B )andp
A
2OC(p
C
;p
B
;"
B ).
Proof. Accordingtothedenitionofpolygonalapproximationproblem,[p
A
;p
B
;p
C
]canbeapproxi-
matedby[p
A
;p
C
]ifandonlyifD(p
B
;[p
A
;p
C ])"
B
,i.e.,[p
A
;p
C
]\BS(p
B
;"
B
)6=. Since[p
A
;p
C ]=
HL(p
A
;p
C
)\HL(p
C
;p
A ), [p
A
;p
C
]\BS(p
B
;"
B
)6= ifand onlyifHL(p
A
;p
C
)\BS(p
B
;"
B
)6=and
HL(p
C
;p
A
)\BS(p
B
;"
B
)6=. Accordingto thedenition oftheopencone OC(p;q;r)in Section2,
HL(p
A
;p
C
)\BS(p
B
;"
B
)6=ifandonlyifp
C
2OC(p
A
;p
B
;"
B
),andHL(p
C
;p
A
)\BS(p
B
;"
B )6=if
andonlyifp
A
2OC(p
C
;p
B
;"
B
). Therefore, [p
A
;p
B
;p
C
]canbe approximatedby[p
A
;p
C
]ifandonly
ifp
C
2OC(p
A
;p
B
;"
B )andp
A
2OC(p
C
;p
B
;"
B ). 2
Corollary1. Let[p
A
;p
A+1
;:::;p
B
]beasubchainofP.[p
A
;p
A+1
;:::;p
B
] canbeapproximatedby
[p
A
;p
B
]if andonlyif p
B 2
B 1
\
i=A+1 OC(p
A
;p
i
;"
i )andp
A 2
B 1
\
i=A+1 OC(p
B
;p
i
;"
i ).
Corollary 2. Let [p
A
;p
A+1
;:::;p
B
;p
B+1
;:::;p
C
] be a subchain of P, and A < B < C. If
B
\
i=A+1 OC(p
A
;p
i
;"
i
) = or C 1
\
i=B OC(p
C
;p
i
;"
i
) = , [p
A
;p
A+1
;:::;p
C
] cannot be approximated by
[p
A
;p
C ].
Fromthetwocorollariesabove,itisveryeasytoobtaintheCIMinR 3
. However,asmentionedin
therstsection,thetwodiÆcultiesofCIMinR 3
remainunresolved. Ifweusetheopenconedirectly,
thecomputation willbe very complex. Fortunately, wedonotneedto know theexact intersections
ofthe open cones. Onlythe results of whether theintersections containingthe points arerequired.
Therefore, we can substitute the open cones with some discs on a unit ball or the unit ball itself.
Furthermore, thesediscscan beobtainedeasily,andcomputingtheintersectionsofthesediscsisnot
complex.
Theorem 2. Let [p
A
;p
B
;p
C
] be a subchain of P. [p
A
;p
B
;p
C
] can be approximated by [p
A
;p
C ] if
andonlyif PP(p
A
;p
C
)PR(p
A
;p
B
;"
B
)andPP(p
C
;p
A
)PR(p
C
;p
B
;"
B ).
Proof. PP(p
A
;p
C
) PR(p
A
;p
B
;"
B
) ifand only if HL(p
A
;p
C
) OC(p
A
;p
B
;"
B
). HL(p
A
;p
C )
OC(p
A
;p
B
;"
B
)ifandonlyifp
C
2OC(p
A
;p
B
;"
B
). SoPP(p
A
;p
C
)PR(p
A
;p
B
;"
B
)ifandonlyifp
C 2
OC(p
A
;p
B
;"
B
). Inthesame way, PP(p
C
;p
A
)PR(p
C
;p
B
;"
B
)if andonly ifp
A
2OC(p
C
;p
B
;"
B ).
Corollary3. Let[p
A
;p
A+1
;:::;p
B
]beasubchainof P.[p
A
;p
A+1
;:::;p
B
]canbeapproximatedby
[p
A
;p
B
]if andonlyif PP(p
A
;p
B )2
B 1
\
i=A+1 PR(p
A
;p
i
;"
i
)andPP(p
B
;p
A )2
B 1
\
i=A+1 PR(p
B
;p
i
;"
i ).
Corollary4. Let[p
A
;p
A+1
;:::;p
C
]beasubchainofP,andA<B <C. If B
\
i=A+1 PR(p
A
;p
i
;"
i )=
or C 1
\
i2B PR(p
C
;p
i
;"
i
)=,then[p
A
;p
A+1
;:::;p
C
] cannotbeapproximatedby [p
A
;p
C ].
ThissolvesthepresentationproblemoftheopenconesusedbyCIMinR 3
. Thefollowingformula
illuminateshowtoobtainthese opencones.
Formula 1. PR(p
A
;p
i
;"
i
), i.e., the projection of the open cone OC(p
A
;p
i
;"
i
) on the unit ball
B(p
A
;1),can becalculated by
PR(p
A
;p
i
;"
i )=
DB(q
i
;e
i
); D(p
A
;p
i )>"
i
;
B(p
A
;1); D(p
A
;p
i )"
i
;
whereq
i
=(p
i p
A )=D(p
i
;p
A )+p
A ,e
i
="
i
=D(p
i
;p
A
),andDB(q
i
;e
i
)isa disconB(p
A
;1).
Theintersectioncanberepresentedbyaclosechainofarcsontheballs,denotedbyArcBdened
in Section 2. Gettingtheintersection oftheprojections ofthese opencones issimilar tothecase of
theplanardiscsin R 2
.
Lemma 1. The intersection of two dierent circles CB(q
i
;e
i
) and CB(q
j
;e
j
), which is on the
same ball B(p
A
;r), has no more than two vertices (i.e., the joining points of theclose chain of the
arcs).
Lemma1canbededucedfromthefactsthatthreenonlinearpointsdetermineonecircle,andthat
threedierentpointsonthesamecirclearenonlinear.
Lemma 2.
n
\
i=1 DB(q
i
;e
i
), i.e., the intersection of n discs on a ball, consists of no more than
2(n 1)vertices(i.e.,thejoiningpointsoftheclose chainofthearcs)ifno discislargerthanahalf
ball.
Themethod usedtoproveLemma 3.2in[1]canalsobe usedtoproveLemma 2in thispaper.
Lemma 1 and Lemma2 will be usedto calculate theintersections oftheprojections oftheopen
cones, i.e., B 1
\
i=A+1 PR(p
A
;p
i
;"
i ) and
B 1
\
i=A+1 PR(p
B
;p
i
;"
i
), and illuminatethe time complexity of the
algorithms.
4 Optimization Algorithm
According to Corollary 3, we can nd all edges of G(V;E); and according to Corollary 4, we
can determinewhento stop working onthesuccessive vertices whilecheckingtheerrorcriteria. We
describethealgorithmasfollows. ThedirectionofeachedgeinG(V;E)isfromthevertexwithsmaller
subscript to that with larger subscript, thus[p
i
;p
j
], which is used to indicate theline segment, can
alsobeusedto representthedirectededgewithoutconfusion.
Algorithm. CIMAlgorithmforApproximatingThree-Dimensional PolygonalCurves
Input: ArbitrarypolygonalcurveP=[p1;p2;:::;pn]inR 3
withassociatederrortolerancesf"1;"2;:::;"ng.
Output: ApproximatingcurveP 0
=[p 0
1
;p 0
2
;:::;p 0
m ].
1. fori=1ton 1dofTobuildgraphG(V;E)g
begin
IPR B(p
i
;1);
forj=i+1tonandwhile IPR6=do
begin
ifPP(pi;pj)2IPRthenG G[[pi;pj]
IPR IPR\PR(p
i
;p
j
;"
j )
endfEndofinnerfor-loopg
2. fori=nto2dofTocheckedgesingraphG(V;E)g
begin
IPR B(p
i
;1);
k minfhj[ph;pi]2Gg;
forj=i 1to1and whileIPR6=do
begin
if[pj;pi]2GandPP(pi;pj)62IPRthenG G [pj;pi];
ifkjthen
break;fJumpoutofinnerfor-loopg
IPR IPR\PR(p
i
;p
j
;"
j )
ifIPR=thenG G f[ph;pi]jh<jg
endfEndofinnerfor-loopg
endfEndofouterfor-loopg
3. Outputtheshortestpathfromp
1 top
n
accordingtoG(V;E),usingforwarddynamicprogramming.
EndofCIMAlgorithm
SinceIPRconsistsofatmostO(n)discsaccordingtoLemma2,testingPP(p
i
;p
j
)2IPRrequires
O(n)time,andobtainingIPR\PR(p
i
;p
j
;"
j
)alsorequiresO(n)time. Hence, Step1hasO(n 3
)time
complexity. The edges of G(V;E) are obtained from Step 1, so they can be ordered. Therefore,
determiningmin fhj[p
h
;p
i
]2GgrequiresonlyO(n)time,andobtainingf[p
h
;p
i
]jh<jgrequiresO(n)
time. Hence,Step2hasO(n 3
)timecomplexity. SinceG(V;E)hasO(n 2
)edges,thetimecomplexity
ofStep3is O(n 2
).
In practice, the minimization of the vertices is not always mandatory, especially if it is at the
expense of requiring much more time or space. An approximationalgorithm is givenbelow; it only
requiresO(n 2
)timeandO(n)space.
5 Approximation Algorithm
In order to nd a fast algorithm, we substitute Theorem 2, which species the suÆcient and
necessaryconditions,withTheorem3,which speciesthesuÆcientconditions.
Theorem 3. Let [p
A
;p
B
;p
C
] be asubchain of P. [p
A
;p
B
;p
C
] can be approximated by [p
A
;p
C ] if
PP(p
A
;p
C
)PR(p
A
;p
B
;"
B
)andD 2
(p
A
;p
C )D
2
(p
A
;p
B ) "
2
B .
Firstly,accordingtoTheorem3,wecan omitStep2oftheCIMalgorithmin Section4. Secondly,
wedonotneedto searchforeveryapproximatingline segment. Therefore,thegraph G(V;E)is not
neededintheapproximationalgorithm. Theapproximationalgorithmisdescribed asfollows.
Algorithm. ApproximationAlgorithm
Input: ArbitrarypolygonalcurveP=[p
1
;p
2
;:::;p
n ]inR
3
withassociatederrortolerancesf"
1
;"
2
;:::;"
n g.
Output: ApproximatingcurveP 0
=[p 0
1
;p 0
2
;:::;p 0
m ].
i=1;
outputp1;
whilei<ndo
begin
IPR B(pi;1);
forj=i+1ton 1do
begin
rt=(p
j p
i )(p
j p
i ) "
j
"
j
;
ifj=i+1then
rr=rt;fTosettheinitialvalueofrrg
else
begin
if(p
j p
i )(p
j p
i
)<rr orPP(p
i
;p
j
)62IPR,then begin
outputpj 1;
break;fJumpoutoffor-loopg
ifrr<rt,thenrr=rt;
endfEndofelseg
IPR IPR\PR(p
i
;p
j
;"
j )
endfEndoffor-loopg
i j 1;
endfEndofwhileloopg
outputpn;
EndofApproximationAlgorithm
TheboundaryofIPRconsistsofO(n)ArcBs,sothetimecomplexityofcalculatingIPR\PR(p
i
;p
j ,
"
j
)isO(n). Althoughtheapproximationalgorithmhastwonestedloops,itstimecomplexityisO(n 2
).
BecauseIPRrequiresO(n)space, thespacecomplexityofthealgorithmisO(n).
6 Examples
First, we use the points evenly distributed on some Bezier curves as examples, and the results
can be checked by readers. The rst Bezier curve has eight control points, (0;0;0), (40;240;50),
(100;300;100),(120;40;0),(160;100;0),(220;30;0),(240;300;300)and(300;250;250),andisdenoted
byC1. Thesecondonehasvecontrolpoints,(0;0;350),(100;400;50),(200;350;0),(300;0;400)and
(400;300;100), and is denoted by C2. The third one has nine control points, (0;0;0), (200;0;0),
(400;0;0), (400;200;100), (400;400;200), (200;400;200), (0;400;200), (0;200;100) and (0;0;0), and
is denoted byC3,which cannot be approximatedbythe algorithmof Euand Toussaint. The num-
ber of sampled pointsis denoted by M. All of the associated error tolerances havethe same value,
although theymightbe dierent. Wehaveimplemented fouralgorithms: thebruteforce algorithm,
the algorithm of Eu and Toussaint, the optimization algorithm using CIM and the approximation
algorithmusingCIM.AllthesealgorithmsareimplementedwithVisualC++5.0under Windows98
Table1. TheEÆciencyoftheOptimization
AlgorithmUsingCIM
Curve M "i Result(vertices) Time(ms)
C1 21 10 7 9
C1 101 1 20 130
C1 101 1e 1 78 25
C1 301 1e 2 240 70
C1 10011e 3 772 255
C1 50011e 4 2089 2620
C2 21 10 7 8
C2 101 1 20 125
C2 151 1e 1 68 69
C2 401 1e 2 245 145
C2 10011e 3 777 265
C2 50011e 4 2071 2570
Table2. TheEÆciencyofthe
BruteForceAlgorithm
Curve M "
I
Result(vertices) Time(ms)
C1 21 10 7 2
C1 101 1 20 70
C1 101 1e 1 78 45
C1 301 1e 2 240 385
C1 10011e 3 772 4191
C1 50011e 4 2089 105030
C2 21 10 7 3
C2 101 1 20 70
C2 151 1e 1 68 105
C2 401 1e 2 245 688
C2 10011e 3 777 4089
C2 50011e 4 2071 104980
Table3. TheEÆciencyoftheAlgorithm
ofEuandToussaint
Curve M "
i
Result(vertices) Time(ms)
C1 21 10 10 5
C1 101 1 27 80
C1 101 1e 1 88 15
C1 301 1e 2 267 40
C1 10011e 3 872 151
C1 50011e 4 3030 1525
C2 21 10 10 5
C2 101 1 27 80
C2 151 1e 1 107 45
C2 401 1e 2 322 80
C2 10011e 3 882 155
Table4. TheEÆciencyoftheApproximation
AlgorithmUsingCIM
Curve M "
i
Result(vertices) Time(ms)
C1 21 10 7 0
C1 101 1 20 10
C1 101 1e 1 78 5
C1 301 1e 2 240 15
C1 10011e 3 772 60
C1 5001 1e 4 2089 370
C2 21 10 7 0
C2 101 1 20 10
C2 151 1e 1 68 10
C2 401 1e 2 245 28
C2 1001 1e 3 777 60
Fig.1. ApproximateC1(M =101;"
i
=10). (a)ResultoftheoptimizationalgorithmusingCIMorthebruteforce
algorithm(5segments). (b)ResultofthealgorithmofEuandToussaint(7segments). (c)Resultoftheapproximation
algorithmusingCIM(6segments).
Fig.2. ApproximateC2 (M =21;"
i
=10). (a)Resultoftheoptimizationalgorithmusing CIMorthebruteforce
algorithm(6segments). (b)ResultofthealgorithmofEuandToussaint(9segments). (c)Resultoftheapproximation
algorithmusingCIM(6segments).
on a personal computer with Pentium II 350
CPUchipand64Mmemory(Tables1{4,Figs.1{
3). Theresults ofthebruteforcealgorithm and
the optimization algorithm using CIM are the
same, butwithdierenteÆciency.
Anotherexampleisfromamilitaryprojectus-
ingtheniteelementanalysisonsomelarge-scale
landingcrafts. Thesampledpointsonthecrafts
arepreprocessedbeforethemodelsofthesecrafts
arereconstructedincomputers. Fig.4showsthe
shape of the boundary curve of the large-scale
landing craft. Table 5 shows the preprocessing
results fortheboundarycurvefrom AtoB.
From theabove tables, wecan nd that the
time required by the brute force algorithm in-
creasessharplywhenthenumberofthesampled
Fig.3. ApproximateC3 (M = 101;"
i
=10). Note that
thiscurvecannotbeapproximatedbythealgorithmofEu
andTousssaint. (a)Resultoftheoptimizationalgorithm
usingCIMorthebruteforcealgorithm(9segments). (b)
ResultoftheapproximationalgorithmusingCIM(9seg-
ments).
Fig.4.Theboundarycurveofalarge-scalelandingcraft.
Table5. ApproximationResultsoftheBoundaryCurve
Optimizationalgorithm Bruteforce Algorithmof Approximationalgorithm
usingCIM algorithm EuandToussaint usingCIM
Result(vertices) 201 201 251 201
Time(ms) 135 300 145 35
points and the number of the resulting points increase. The results of the algorithm of Eu and
Toussainthavemore pointsthantheotherthree algorithms,sincethediscusedbyEuandToussaint
astheerrortoleranceregionofanyvertexissmallerthantheballusedbytheotherthreealgorithms.
Additionally, C3 cannot be approximated by thealgorithm of Eu and Toussaint. According to the
tablesabove,theresultsoftheapproximationalgorithmareveryclosetotheoptimizationalgorithm,
anditscostismuch lower.
7 Conclusion
Without any additionalconstraint, wepresent thesolution to theLS-WMN problemwith L
2 in
R 3
. If the polygonal curve is used to approximate a G 1
curve, and if the minimization of vertices
is not mandatory, the approximation algorithm may be preferable, for it has much lower cost and
producesonlyafewmorevertices thanthatoftheoptimizationalgorithm.
In thispaper,wehavemadethe CIMfeasiblein R 3
, which meansthat notonlya more eÆcient
algorithm is realized, but also theCIM has been proved to be applicable in both R 2
and R 3
. The
dierencesonlyconsistof:
1. representationsofpoints,vectorsandopencones,
2. methodsofdecidingwhetheravertexliesintheopencone,and
3. methodsofcomputingtheintersectionsoftheopencones(ortheprojectionsoftheopencones).
Thispropertytswellwithobject-orientedsoftware,thusthecodesofCIMcanbesharedinboth
R 2
andR 3
.
Acknowledgements Theauthors thankMr. Zhu Xiang forproviding theoriginal dataof the
last example in thepaper. The authors acknowledgewith thanks thecommentsand suggestions of
thereviewers.
References
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YONG Junhaireceivedhis Ph.D. degree from theDepartment of ComputerScience and Technology,
Tsinghua University, Beijing, P.R. China in 2000. He received the B.S. degree from the Department of
Computer Science and Technology, Tsinghua University, in1996. From Marchto Junein 2000, hewas a
visitingresearcherintheHongKongUniversityofScienceandTechnology. Hisresearchinterestsarecomputer
aideddesign,computergraphicsandstandardizationofproductdata.
HUShiminisanassociateprofessorintheDepartmentofComputerScienceandTechnology,Tsinghua
University,Beijing,P.R.China. HereceivedthePh.D.degreein1996fromZhejiangUniversity,P.R.China,
andnishedpostdoctoralresearchworkin1998atTsinghuaUniversity. Hisresearchinterestsarecomputer
aideddesign,computergraphics,fractalgeometryanditsapplications.
SUN Jiaguangis a professorintheDepartment of ComputerScience andTechnology, TsinghuaUni-
versity,Beijing,P.R.China. Hisresearchinterestsarecomputergraphics,computeraided design,computer
aidedmanufacturing,productdatamanagementandsoftwareengineering.
