135

1 INTRODUCTION

A ship domain (Coldwell, 1983; Davis 1982) is

generally thought as the space around the ship, 

which the navigator wants to keep clear of other

objects(includingships).Shipdomainmodelsarein

abundanceandthenewonesarebeingcontinuously

proposed,eitherbasedontheoreticalanalysesorreal

data (Hansenetal.,2013).Theyareusedinma

rine

trafficengineering,e.g.fordeterminingthecapacity

of traffic lanes and assessing collision risk

(Pietrzykowski,2008;Montewkaetal.,2011;Xianget

al., 2013), as well as in collision avoidance for

determining safe manoeuvres (Śmierzchalski, 2000).

The shape and size of a ship domain is usually

dependent on ship’s lengt

h and speed (Fuji and

Tanaka,1971),thoughparametersofothershipsmay

alsobetakenintoaccount(PietrzykowskiandUriasz,

2009). Ship domains are often given explicitly as

geometrical figures but (especially in case of

restrictedwaters)theymayalsobegivenasfunct

ions

proposedonthebasisofsafetyparametersdefinedin

the ECDIS (Pietrzykowski and Wielgosz, 2011;

Weintrit, 2006; Weintrit, 2009). This paper abstracts

fromthemoregeneraltrafficengineeringissuesand

focusesonthepractical impactofadomain’s shape

on determining course and speed alteration

manoeuvres in encounter situations (Szłap

czyński,

2007).Whilethecollisionavoidancemanoeuvresare

often strongly dependent on the domain’s size (the

larger the domain, the larger the manoeuvres), the

precise impact of its shape has not been researched

before. Various researchers develop their domain

models based on empirical data (distances between

ships for different relative courses and bea

rings),

collision risk assessments, COLREGS and good

marine practice. The differences between the

proposed shapes are sometimes very subtle, hence

the present authors’ idea to check, whether those

small differences actually translate to different

navigational decisions (collision avoidance

manoeuvres)andtowhatextent.Obviously,inma

ny

real situations navigational decisions depend on

reasons other than assumed domain shapes and

sizes.Actualcourse alterations may be much larger

than those  determined to avoid a ship domain

A Simulative Comparison of Ship Domains and Their

Polygonal Approximations

R.Szlapczynski

GdanskUniversityofTechnology,Gdansk,Poland

.Szlapczynska

GdyniaMaritimeUniversity,Gdynia,Poland

ABSTRACT:Thepaperinvestigatestheimpactofapreciseshipdomainshapeonthesizeofcollisionavoidance

manoeuvres.The considered collision avoidance manoeuvres include both course and speed alterations.

Variousshipdomainsarecomparedwiththeirpolygonalapproximations,whichvaryinthenumberofpoints

ofadomaincontourandplacementofthesepoints. Thebestofallconsideredapproximat

ionsisdeterminedin

thecourseofsimulationexperimentsperformedforhead‐on,crossingandovertakingsituations.Thechosen

number and placement of contour points combine precision of domain approximation with reasonable

computationaltime.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 9

Number 1

March 2015

DOI:10.12716/1001.09.01.17

136

violation.Insuchcasesshipdomainsarepractically

irrelevantandthereforethesesituations areofnouse

forthisresearchandarenottakenintoaccountinthe

simulation experiments. For the purpose of this

research it is assumed that we are dealing with

isolatedcaseswhencoursealterationmanoeuvres

are

based solely on ship domains and that avoiding

domain violations is both necessary and sufficient

conditionforthealterationtobesafe.

Because of their various sizes, the ship domains

cannot be compared directly with each other.

Thereforeintheexperimentspresentedinthepaper

eachof theconsidered

domainsis comparedwith a

series ofitspolygonal approximations containing 8,

16 or 32 points placed uniformly or non‐uniformly

(constant or varying angular distances between

pointslyingonthedomain’sboundary).Thenumber

ofpointsisofkeyimportancebecauseitaffectsboth

theprecision of resultsandthe

computationaltime.

In a decision support system potential domain

violationsareusuallycheckedwithinmultipleloops,

thusbeingthemostfrequentoperationsandlargely

contributing to the overall computational time.

Doubling the number of points in a ship domain

modelpracticallydoublesthecomputationaltimeof

determiningasafetrajectory

oracollisionavoidance

manoeuvre.Therefore, thepaperaimstoinvestigate

the impact of approximation points and their

placementonthesizeofcourseandspeedalterations

performedtoavoidcollisions.

The remaining part of the paper is organized as

follows. In Section 2 all considered domains are

briefly described. The

 prepared encounter scenarios

and polygonal approximations of domains are

presented in Section 3 and the results of the

simulationexperiments areshownanddiscussedin

Section 4. Based on these results, a summary and

conclusionsaregiveninSection5.

2 SHIPDOMAINSTAKENINTOACCOUNTIN

THECOMPUTERSIMULATION

EXPERIMENTS

Theshapesusedintheexperimentsarethoseofship

domains according to Fuji (1971), Davis (1982),

Coldwell (1983) and Pietrzykowski (2009). Two

slightlydifferingshapeshavebeenappliedincaseof

Coldwell’s domain (one for head‐on and crossing 

encounters and another one for overtaking) and six

shapesin

caseofPietrzykowski(threededicatedfor

fuzzy domains and three for the crisp ones). The

shapes of ship domains accordingto Fuji, Coldwell

and Davis are commonly known and therefore are

not reproduced here. Examples of more interesting,

irregular ship domain shapes according to

Pietrzykowski (2009)areshown in Figures

1 and 2.

Thefuzzydomainutilizesanavigationalsafety level

γ. The bipolar grid in all figures features circles,

whoseradiusesare0.5,1,1.5and2nauticalmiles.



Figure1. Fuzzy domain according to Pietrzykowski and

Uriasz withγparameter (navigational safety level) set to

0.5.



Figure2. Fuzzy domain according to Pietrzykowski and

Uriaszdedicatedtohead‐onencounters.

3 SIMULATIONEXPERIMENTS(ENCOUNTER

SCENARIOSANDPOLYGONAL

APPROXIMATIONSOFDOMAINS)

Three types of one‐on‐one encounter scenarios are

used throughout the simulation experiments: head‐

on,crossingandovertaking.Thedifferencesbetween

shipcourses are 180, 90and0degrees  respectively,

and the speeds and initial positions are such that

ships

wouldcollide if neither of themmanoeuvred.

The distances  between ships vary in the scenarios,

depending on a  particular domain, which is used. 

Thedistancesaredeliberatelysetinsuchaway,that

the necessary safe course alteration manoeuvre

wouldbebetween15and20degrees and(incaseof

crossing encounters) speed reduction manoeuvres

wouldbeinthesamerange(speedreducedby4to6

knots)forallshipdomains.

Ifweassumethatacoursealterationismadesoas

toavoidviolatingatarget’sdomain,thanclearlyboth

shape and size of domain is of importance. It

 is

illustratedinFigures  3 and 4, where courses which

lead to violating ship domains are shown for a

circulardomain(safedistanceoftenusedincollision

avoidance systems) and elliptical domain

respectively. Forbidden (collision) course sectors

varyinbothcasesduetousingdifferentdomains.

137



Figure3.Collisioncoursesectorforagivensafedistance.



Figure4. Collision course sector for a given target’s 

domain.

The algorithm for determining the minimal

acceptable course alterations (and consequently‐

collisioncoursesectors)hasbeendescribedindetail

in(Sz łapczyński,2007).Itisbrieflyrecalledherefor

thereadersconvenience.Inthealgorithm(Figure5),

the course alteration is modified iteratively until

target’s domain is not violated (the

 approach factor

minislargerthan1)andtheaccuracyissufficient(the

error is smaller than given value of the parameter

δѱ).



Figure5.Algorithmdeterminingminimalcoursealteration

foragiveshipdomainbasedonapproachfactorf

min.

For each domain five polygonal approximations

arecompared.Ineachcasetheapproximationisdone

automatically: the coordinates of points on the

originaldomainboundariesaredeterminedfirstand

then are joined by straight segments so as to form

polygons. The approximations vary in the number

pointsandtheirplacement:8

‐point,16‐pointand32‐

pointapproximationsare used.The8‐point and16‐

point approximations appear in two variants each:

either uniform angular distances or non‐uniform

onesareused.Fortheuniformangulardistancesthe

points on the approximated domain boundary are

placed in either 45 degree intervals

(8‐point

approximation) or 22.5 degree intervals (16 degree

intervals).Forthenon‐uniformangulardistances,the

points on the approximated domain boundary are

placedasfollows.For8points:0,30,90,150,180,210,

270and330degrees. For16points:0,15,30,60,90,

120,150,165,

180,195,210,240,270,300,330and345

degrees.Thereasonfortestingnon‐uniformangular

distancesaswellasuniformonesisthattheleftand

right sides of a ship domain are often easier to

approximatethantheir foreand aftandtherefore it

makessenseto

saveonpointsatthesidesandusethe

extra points in the front and back. Examples of

uniformandnon‐uniformplacementof points in8‐

pointpolygonalapproximationsofshipdomainsare

giveninFigures6to9.



Figure6.An8‐pointapproximationofashipdomainfrom

Fig.1.(uniformplacementofpoints)



Figure7.An8‐pointapproximationofashipdomainfrom

Fig.1.(non‐uniformplacementofpoints)

138



Figure8.An8‐pointapproximationofashipdomainfrom

Fig.2.(uniformplacementofpoints)



Figure9.An8‐pointapproximationofashipdomainfrom

Fig.2.(non‐uniformplacementofpoints)

4 RESULTSOFTHEEXPERIMENTS

In Tables 1 to 3 the resulting course alteration

manoeuvres are shown for each scenario and each

approximation of a given ship domain. Collision

coursesectorsaregiveninTable1andtheerrorsof

approximation (the differences between values

obtained for original domains and the

 polygonal 

approximations) are given in Tables 2 (manoeuvres

tostarboard)and3(manoeuvrestoport).InTables4

and5thevaluesofspeedreductionandtheerrorsof

approximation (the differences between values

obtained for original domains and the polygonal

approximations)areshownrespectively.Someofthe

original domains

are also polygons with vertices

placed uniformly and as a result polygonal

approximationswillreturnnoerrors.Thereforetwo

separate averages are computed, with one of them

ignoring those polygonal domains (last rows of

Tables2to5).

As can be seen in the Tables 2 to 5, the 8‐point



approximations give imprecise results. For course

alterationmanoeuvressometimestheerrorisnearly

4 degrees (over 24%), which is unacceptable. The

volumeoferrors variesbetween scenarios, withthe

averagecoursealterationerrorsbeingabout10%for

manoeuvrestoport(Table2)and7%formanoeuvres

tostarboard(Table3).

Thevaluesofcoursealteration

errors are similar for uniform and non‐uniform

placementofapproximationpoints.Incaseofspeed

reduction manoeuvres  the uniform placement of

points again returns significant errors, the average

errorbeingabout7%(Table5).Howevertheresults

fornon‐uniformapproximationaremuchbetter:the



averageerror is under 4% evenwhenthe relatively

easycases(polygonaldomains)areexcluded.



Table1.Collisioncoursesectorsfororiginaldomainsandtheirautomaticpolygonalapproximations

__________________________________________________________________________________________________

Domain  EncounterOriginal  8points   8points16points  16points  32points

type  domain  (uniform)  (non‐uniform) (uniform)  (non‐uniform) (non‐uniform)

__________________________________________________________________________________________________

Fujihead‐on (‐20.05;20.05)(‐19.52;19.52) (‐19.20;19.20) (‐19.82;19.82) (‐19.78;19.78) (‐19.97;19.97)

crossing (‐20.18;16,17)(‐18.17;14.29) (‐18.67;14.81) (‐18.84;15.69) (‐19.39;16.09) (‐19.90;16.02)

overtaking(‐20.49;20.49)(‐17.34;17.34) (‐19.42;19.42)

 (‐19.82;19.82) (‐19.76;19.76) (‐20.24;20.24)



Davishead‐on (‐23.90;19.30)(‐20.91;17.66) (‐20.81;16.61) (‐23.85;19.17) (‐23.70;18.96) (‐23.74;19.17)

crossing (‐22.09;18.37)(‐20.30;18.16) (‐21.84;17.87) (‐21.61;18.19) (‐21.96;17.96) (‐22.04;18.30)

overtaking(‐20,53,16,31)(‐18,75,15,15) (‐18,39,14,69)

 (‐20,47,16,20) (‐19,94,16,11) (‐20,48,16,26)



Coldwell head‐on (‐21.90;10.85)(‐20.85;10.83) (‐20.34;10.47) (‐21.57; 10.83) (‐21.58;10.83) (‐21.71;10.83)

crossing (‐24.55;15.15)(‐20.21;14.72) (‐21.27;14.75) (‐23.15;14.79) (‐24.20;14.83) (‐24.07;14.85)

overtaking(‐16.24;16.24)(‐12.33;12.33) (‐13.77;13.77)

 (‐14.93;14.93) (‐16.12;16.12) (‐15.72;15.72)



Pietrzykowskihead‐on (‐18.00;16.97)(‐17.01;16.60) (‐15.46;15.46) (‐17.24;16.66) (‐17.87; 16.86) (‐17.86;16.94)

fuzzy,γ=0.5 crossing (‐21.04;19.02)(‐18.05;17.24) (‐20.51;18.65) (‐21.04;18.25) (‐20.54;18.68) (‐21.04;18.95)

overtaking(‐19.06;17.82)(‐18.07;17.39) (‐

16.08;16.08) (‐18.13; 17.44) (‐19.02;17.66) (‐18.87;17.79)



Pietrzykowskihead‐on (‐21.92;20.93)(‐19.91;18.25) (‐17.49;17.39) (‐21.70;20.57) (‐21.51; 20.63) (‐21.71;20.60)

fuzzy,γ=0.1 crossing (‐25.37;22.07)(‐23.15;22.03) (‐25.22;21.68) (‐24.88;22.06) (‐25.25;21.76) (‐25.37;22.07)

overtaking(‐18.09;17.16)(‐15.93;14.63) (‐

14.40;14.34) (‐17.91; 16.92) (‐17.70;16.82) (‐17.92;16.93)



Pietrzykowskihead‐on (‐19.11;19.11)(‐19.08;19.08) (‐17.24;17.24) (‐19.11;19.11) (‐18.30; 18.30) (‐19.08;19.08)

crisp,Dmin crossing (‐19.36;17.92)(‐19.36;17.91) (‐19.36;16.82) (‐19.36;17.92) (‐19.36;16.91) (‐19.37;17.91)

overtaking(‐19.87;19.87)(‐19.86;19.86) (‐

19.19;19.19) (‐19.87; 19.87) (‐19.78;19.78) (‐19.83;19.83)



Pietrzykowskihead‐on (‐16.00;16.00)(‐16.00;16.00) (‐15.22;15.22) (‐16.00;16.00) (‐15.81; 15.81) (‐15.96;15.96)

crisp,Dmean crossing (‐19.05;19.45)(‐19.05;19.45) (‐18.80;17.20) (‐19.05;19.45) (‐18.80;17.20) (‐19.04;19.42)

overtaking(‐16.80;16.80)(‐16.80;16.80) (‐

15.53;15.53) (‐16.80; 16.80) (‐16.46;16.46) (‐16.75;16.75)



139

Pietrzykowskihead‐on (‐20.05;20.05)(‐20.05;20.05) (‐20.00;20.00) (‐20.05;20.05) (‐20.05; 20.05) (‐20.04;20.04)

crisp,Dmax crossing (‐23.57;23.44)(‐23.33;23.44) (‐23.53;20.99) (‐23.54;23.44) (‐23.54;20.99) (‐23.55;23.43)

overtaking(‐16.63;16.63)(‐16.63;16.63) (‐

16.09;16.07) (‐16.62; 16.62) (‐16.63;16.63) (‐16.62;16.62)



Pietrzykowskihead‐on (‐16.17;15.27)(‐15.22;14.79) (‐16.02;14.70) (‐15.49;15.05) (‐16.06; 15.17) (‐15.81;15.25)

(head‐on

dedicateddomain)

__________________________________________________________________________________________________

Table2. Differences between manoeuvres to port for original domains and their automatic polygonal approximations

(absoluteandrelativevalues)

__________________________________________________________________________________________________

DomainEncounter 8points   8points16points  16points  32points

type(uniform)  (non‐uniform) (uniform)  (non‐uniform) (non‐uniform)

__________________________________________________________________________________________________

Fujihead‐on  0.53/2.63%0.85/4.22%0.23/1.15% 0.27/1.37%0.08/0.38%

crossing  2.01/9.96%1.52/7.51% 1.34/6.64%0.79/3.92%0.29/1.42%

overtaking 3.15/15.39% 1.07/5.20% 0.67/3.27%0.73/3.54%0.25/1.23%



Davishead‐on  2.99/12.51% 3.09/12.92% 0.04/0.18% 0.20/0.83%0.15/0.64%

crossing  1.79/8.11%0.25/1.14% 0.48/2.19%0.13/0.60%0.05/0.25%

overtaking 1.78/8.67% 2.14/10.43% 0.07/0.32%0.59/2.89%0.05/0.27%



Coldwell  head‐on  1.04/4.77%1.56/7.12%0.33/1.51%0.32/1.46% 0.19/0.85%

crossing  4.34/17.67% 3.28/13.38% 1.41/5.73%0.35/1.43% 0.48/1.97%

overtaking 3.91/24.09% 2.47/15.22% 1.31/8.05% 0.12/0.74%0.52/3.18%



Pietrzykowski head‐on  0.99/5.49% 2.54/14.10% 0.76/4.21%0.12/0.67%0.13/0.73%

fuzzy,γ=0.5 crossing  2.99/14.20% 0.53/2.51%0.00/0.00%0.49/2.35% 0.00/0.00%

overtaking 0.99/5.19% 2.98/15.62% 0.93/4.90%0.04/0.23%0.19

/0.98%



Pietrzykowski head‐on  2.01/9.17%4.43/20.20% 0.22/1.00%0.41/1.85% 0.21/0.95%

fuzzy,γ=0.1 crossing  2.22/8.75%0.14/0.56%0.48/1.91% 0.12/0.48%0.00/0.00%

overtaking 2.16/11.96% 3.69/20.40% 0.19/1.03% 0.40/2.19%0.18

/0.97%



Pietrzykowski head‐on  0.02/0.12%1.87/9.78%0.00/0.00% 0.80/4.20%0.02/0.12%

crisp,Dmin crossing  0.00/0.00%0.00/0.00%0.00/0.00% 0.00/0.00%0.01/0.06%

overtaking 0.01/0.06% 0.68/3.43%0.00/0.00%0.10/0.50% 0.04

/0.22%



Pietrzykowski head‐on  0.00/0.00% 0.78/4.88%0.00/0.00%0.19/1.17% 0.03/0.21%

crisp,Dmean crossing  0.00/0.00%0.25/1.33% 0.00/0.00%0.25/1.33%0.01/0.06%

overtaking 0.00/0.00% 1.26/7.52%0.00/0.00%0.34/2.03% 0.04

/0.26%



Pietrzykowski head‐on  0.00/0.00%0.05/0.27%0.00/0.00% 0.00/0.00%0.01/0.05%

crisp,Dmax crossing  0.23/0.98% 0.03/0.14%0.02/0.09%0.02/0.09%0.01/0.05%

overtaking 0.00/0.00% 0.54/3.24%0.01/0.07%0.00/0.00% 0.01

/0.07%

Pietrzykowski head‐on  0.96/5.91% 0.15/0.95%0.68/4.21%0.11/0.68% 0.36/2.24%

(head‐on

dedicateddomain)

__________________________________________________________________________________________________

Averagedifference1.3648/6.61%  1.446/7.28% 0.3668/1.86% 0.2756/1.33% 0.1324/0.69%

__________________________________________________________________________________________________

Averagedifference

withoutcrisppolygonal 2.12/10.26% 1.92/9.46% 0.57/2.89% 0.32/1.58% 0.20/1.00%

domainsbyPietrzykowski

__________________________________________________________________________________________________

Table3.Differencesbetweenmanoeuvrestostarboardfororiginaldomainsandtheirautomaticpolygonalapproximations

(absoluteandrelativevalues)

__________________________________________________________________________________________________

DomainEncounter 8points   8points16points  16points  32points

type(uniform)  (non‐uniform) (uniform)  (non‐uniform) (non‐uniform)

__________________________________________________________________________________________________

Fujihead‐on  0.53/2.63%0.85/4.22%0.23/1.15% 0.27/1.37%0.08/0.38%

crossing  1.88/11.62% 1.36/8.42%0.48/2.99% 0.08/0.48%0.15/0.95%

overtaking 3.15/15.39% 1.07/5.20% 0.67/3.27%0.73/3.54%0.25/1.23%



Davishead‐on  1.65/8.54%2.69/13.94% 0.13/0.68%0.34/1.76% 0.13/0.68%

crossing  0.21/1.14%0.49/2.69% 0.18/0.96%0.41/2.21%0.07/0.36%

overtaking 1.16/7.14% 1.63/9.97%0.11/0.67%0.21/1.28% 0.05/0.34%



Coldwell  head‐on  0.02/0.20%0.38/3.54%0.02/0.20%0.02/0.20% 0.02/0.20%

crossing  0.43/2.83%0.40/2.61% 0.36/2.39%0.32/2.10%0.30/1.96%

overtaking 3.91/24.09% 2.47/15.22% 1.31/8.05% 0.12/0.74%0.52/3.18%



Pietrzykowski head‐on  0.37/2.20% 1.52/8.93%0.32/1.88%0.11/0.65% 0.03/0.19%

fuzzy,γ=0.5 crossing  1.78/9.36%0.36/1.91%0.77/4.04% 0.34/1.79%0.07/0.35%

overtaking 0.43/2.40% 1.74/9.74%0.38/2.16%0.16/0.92% 0.03

/0.18%



Pietrzykowski head‐on  2.68/12.81% 3.54/16.90% 0.36/1.73% 0.30/1.42%0.33/1.57%

fuzzy,γ=0.1 crossing  0.04/0.20%0.40/1.79%0.01/0.05% 0.31/1.39%0.00/0.00%

overtaking 2.53/14.72% 2.82/16.45% 0.24/1.41% 0.34/1.98%0.23

/1.34%



Pietrzykowski head‐on  0.02/0.12% 1.87/9.78%0.00/0.00%0.80/4.20% 0.02/0.12%

crisp,Dmin crossing  0.01/0.06%1.10/6.13%0.00/0.00% 1.01/5.64%0.01/0.06%

overtaking 0.01/0.06% 0.68/3.43%0.00/0.00%0.10/0.50% 0.04

/0.22%



140

Pietrzykowski head‐on  0.00/0.00% 0.78/4.88%0.00/0.00%0.19/1.17% 0.03/0.21%

crisp,Dmean crossing  0.00/0.00%2.24/11.53% 0.00/0.00% 2.24/11.53% 0.02/0.11%

overtaking 0.00/0.00% 1.26/7.52%0.00/0.00%0.34/2.03% 0.04

/0.26%



Pietrzykowski head‐on  0.00/0.00% 0.05/0.27%0.00/0.00%0.00/0.00% 0.01/0.05%

crisp,Dmax crossing  0.00/0.00% 2.45/10.45% 0.00/0.00% 2.45/10.45% 0.01/0.05%

overtaking 0.00/0.00% 0.56/3.37%0.01/0.07%0.00/0.00% 0.01

/0.07%



Pietrzykowski head‐on  0.48/3.17% 0.57/3.74%0.22/1.44%0.10/0.65% 0.02/0.14%

(head‐on

dedicateddomain)

__________________________________________________________________________________________________

Averagedifference0.8516/4.74% 1.3312/7.30%  0.232/1.33% 0.4516/2.32% 0.0988/0.57%

__________________________________________________________________________________________________

Averagedifference

withoutcrisppolygonal 1.3281/7.22% 1.3931/7.56%  0.3618/2.00% 0.26/1.32% 0.1425/0.79%

domainsbyPietrzykowski

__________________________________________________________________________________________________

Table4. Speed reduction allowing for safe passage for original domains and their automatic polygonal approximations

(crossingencountersonly)

__________________________________________________________________________________________________

DomainEncounter 8points  8points16points 16points  32points

type(uniform) (non‐uniform) (uniform) (non‐uniform) (non‐uniform)

__________________________________________________________________________________________________

Fuji4.394.144.154.174.244.33

Davis4.364.014.314.344.334.36

Coldwell4.364.334.344.344.364.36

Pietrzykowski,fuzzy,γ=0.5  4.624.024.524.624.534.62

Pietrzykowski,fuzzy,γ=0.1  5.284.815.225.195.225.19

Pietrzykowski,crisp, Dmin  4.334.334.334.334.334.33

Pietrzykowski,crisp, Dmean

4.264.26  4.214.264.214.26

Pietrzykowski,crisp, Dmax  5.095.095.085.095.085.08

__________________________________________________________________________________________________

Table5. Difference between safe speed reduction values computed for original domains and their automatic polygonal

approximations(absoluteandrelativevalues)

__________________________________________________________________________________________________

Domain8points   8points16points  16points  32points

(uniform)  (non‐uniform) (uniform)  (non‐uniform) (non‐uniform)

__________________________________________________________________________________________________

Fuji0.25/5.68%0.23/5.34%0.22/5.01% 0.15/3.34%0.06/1.34%

Davis0.35/8.07%0.04/1.01%0.01/0.34%0.03/0.67% 0.00/0.00%

Coldwell0.03/0.67%0.01/0.34%0.01/0.34% 0.00/0.00%0.00/0.00%

Pietrzykowski,fuzzy,γ=0.5

0.60/13.00% 0.10/2.22%0.00/0.00%0.09/1.90% 0.00/0.00%

Pietrzykowski,fuzzy,γ=0.10.47/8.88%0.06/1.11%0.09/1.66% 0.06/1.11%0.09/1.66%

Pietrzykowski,crisp, Dmin0.00/0.00%0.00/0.00%0.00/0.00% 0.00/0.00%0.00/

0.00%

Pietrzykowski,crisp, Dmean0.00/0.00%0.04/1.03%0.00/0.00% 0.04/1.03%0.00/0.00%

Pietrzykowski,crisp, Dmax0.00/0.00%0.01/0.29% 0.00/0.00%0.01/0.29%0.01/0.29%

__________________________________________________________________________________________________

Averagedifference0.2125/4.54% 0.0612/2.55% 0.0412/0.92% 0.0475/1.04% 0.0187/0.38%

Averagedifferencewithout0.34/7.26% 0.088/3.82% 0.066/1.47% 0.066/1.40% 0.03/0.6%

crispdomainsbyPietrzykowski 

__________________________________________________________________________________________________



Both 16‐point approximations fare much better

than their 8‐point equivalents: for course alterations

the average error is about 3 to 4 times smaller, for

speedreduction2 to3timessmaller.Forbothkinds

ofmanoeuvresthe16‐pointapproximationwithnon‐

uniform placement of points is

considerably better

thantheonewithpointsplaceduniformly(especially,

if polygonal domains are excluded). The average

errors diminish to about 1.5%‐2% for course

alterationmanoeuvres(Tables2and3)andto1.4%‐

1.5%forspeedreductionmanoeuvres(Table5).

Unsurprisingly, applying 32‐point approximation

(withuniform placementofpoints)

giveseven more

precise results: one can observe further

diminishments in both course alteration errors and

speed reduction errors. However, the benefits of

using the extra 16 points are relatively insignificant

this time. The average error diminishes from 1.5%‐

2% (16 point approximation with non‐uniform

placement of points) to

0.8%‐1% for course

alterations(Tables2and3)andfrom1.4%to0.6%for

speedreduction(Table5).

5 SUMMARYANDCONCLUSIONS

In the course of simulation experiments that have

been carried out and presented in this paper five

polygonalapproximationsofshipdomainshavebeen

tested. The tests included

 eight ship domain shapes

and three types of encounter scenarios (head‐on,

crossingandovertaking).Theresultshaveshownthat

8‐point polygonal approximations can only give a

rough idea of the collision avoidance manoeuvres

performed for the original domains. The rise in

precisionoftheapproximationhasbeensignificant

if

thenumber ofpolygon’sverticeshas beenincreased

from8to16.Theresults thathavebeenobtainedfor

16 points can be considered to be satisfactory, with

theaverageerrorsbeingbelow0.5degreeforcourse

alteration manoeuvres and below0.1knotforspeed

reduction manoeuvres. The results have

 also

indicatedthesuperiorityofnon‐uniformplacementof

approximationpoints(usingmorepointsonthefore

andaftthanonthesides).Theprogressthathasbeen

made when further increasing the number of points

(from 16 to 32) was considerably smaller and

therefore a 16‐point solution has

been considered to

141

be the best choice for decision support systems

applying ship domains of polygonal shapes or

polygonal approximations of other shapes. Another

conclusionisthatanyirregulardomainshape,which

usesmorethanthose16pointsmayberedundantto

someextent.Theextrapointsdonotnecessarilycarry

significant value

 in terms of their impact on the

collision avoidance manoeuvres, though they

certainlycontributetoahigheroverallcomputational

time.

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