Iaoe?iiaeueia aaaionoai c ieoaiue ?ioi?iaoecaoe??

i?e I?aceaeaioia? Oe?a?ie

Aea?aeaaiee iaoeiai-aeine?aeiee ?inoeooo

?ioi?iaoe?eii? ?io?ano?oeoo?e

Eina?aae/ ?inoeneaa ss?ineaaiae/

OAeE 621.391

NO?OEOO?II-?IAA??AIOI? AEAI?EOIE

IIENO OA ?ICI?CIAAAIIss CIA?AAEAIUe

05.13.06 — Aaoiiaoeciaai? nenoaie

oi?aae?iiy oa i?ia?aneai? ?ioi?iaoe?ei? oaoiieia??

A A O I ? A O A ? A O

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa oaoi?/ieo iaoe

Euea?a -1999

eiien

?iaioa aeeiiaia o O?ceei-iaoai?/iiio ?inoeooo? ?i. A.A. Ea?iaiea IAI
Oe?a?ie

Iaoeiaee ea??aiee : aeieoi? oaoi?/ieo iaoe, noa?oee iaoeiaee
ni?a?ia?oiee

?onei Aiaaeai Iaaeiae/,

O?ceei-iaoai?/iee ?inoeooo ?i. A.A. Ea?iaiea

IAI Oe?a?ie,

caa?aeoaa/ a?aeae?eo

Io?oe?ei? iiiiaioe : aeieoi? oaoi?/ieo iaoe, aeioeaio,

?aoeaae/ TH??e Ieoaeeiae/,

Aea?aeaaiee oi?aa?neoao «Euea?anueea iie?oaoi?ea»,

caa?aeoaa/ eaoaae?e

eaiaeeaeao oaoi?/ieo iaoe, noa?oee iaoeiaee ni?a?ia?oiee,

Io?aanueeee Eaii?ae ?ai?iae/,

O?ceei-iaoai?/iee ?inoeooo ?i. A.A. Ea?iaiea

IAI Oe?a?ie,

noa?oee iaoeiaee ni?a?ia?oiee

I?ia?aeia onoaiiaa : ?inoeooo e?aa?iaoeee ?i. A.I. Aeooeiaa IAI Oe?a?ie,
a?aeae?e ea?oth/eo iaoei

Caoeno a?aeaoaeaoueny 1 eeiiy 1999 ?., i 1130 aiaeei? ia can?aeaii?
niaoe?ae?ciaaii? a/aii? ?aaee Ae.35.813.01 o Aea?aeaaiiio
iaoeiai-aeine?aeiiio ?inoeooo? ?ioi?iaoe?eii? ?io?ano?oeoo?e ca
aae?anith: 290053, i. Euea?a, aoe. Iaoeiaa 5a.

C aeena?oaoe??th iiaeia iciaeiieoeny a a?ae?ioaoe? ?inoeoooo (290053, i.
Euea?a, aoe. Iaoeiaa 5a).

Aaoi?aoa?ao ?ic?neaiee 31 o?aaiy 1999 ?.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee

aeieoi? oaoi?/ieo iaoe Aoiue ?.A.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. No/aniee ??aaiue ?icaeoeo oaoi?/iiai i?ia?ano
o?nii iia`ycaiee c aeei?enoaiiyi ??ciiiai?oieo aeaiaio?a ooo/iiai
?ioaeaeoo, aaoiiaoe/ieo nenoai a oaoiieia?/iiio i?ioean?. Oe?iea
ai?iaaaeaeaiiy oa ?o aeei?enoaiiy caeaaeeoue a?ae iiaeeeaino? cai?ieoe
iiai?noth aai /anoeiai ia aeayeiio aoai? /e ai?iaeiaae anueiai
oaoiieia?/iiai oeeeeo ethaenueeo i?aoeth. Oey iiaeeea?noue aecia/a?oueny
caeaoi?noth nenoaie ia?aeiyoe aeaye? iaeaaaeeea?o? ooieoe?? i?eoaiaii?
aeeaiio i?aai?cio. ?ici?ciaaaiiy ia?ac?a ? iaei??th ?c iniiaieo ooieoe?e
aeeaeo i?aai?ci?a.

?ici?ciaaaiiy cia?aaeaiue ia`?eo?a aeei?enoiao?oueny a aaaaoueio aaeocyo
ethaenueei? ae?yeueiino?. Cie?aia a nenoaiao oaoi?/iiai ci?o
?iaioioaoi?/ieo nenoai, i?e aiae?c? einiiaa?ioioia?ao?e aeey ciiaeoaaiiy
caiieo ?ano?n?a, a iaaeeoeei? aeey ia?iaee ?aioaaiia?ai, a oaeiae i?e
aeyaeaii? ? eeaneo?eaoe?? ia`?eo?a oa ?i.

Aeey ?ici?ciaaaiiy cia?aaeaiue neeaaeieo ia`?eo?a, ye? aeei?enoiaothoue
?ioi?iaoe?th ??cii? o?ce/ii? i?e?iaee, cai?iiiiiaaia cia/ia e?euee?noue
aeai?eoi?a. Iaeiei c aaaeeeaeo ?ioi?iaoeaieo ia?aiao??a neeaaeiiai
ia`?eoa ? eiai aaiiao?e/ia no?oeoo?a. Oiio o ?yae? aeiaaee?a o ?ic?iaoe?
aeai?eoi?a ?ici?ciaaaiiy aeei?enoiaothoue naia ?ioi?iaoe?th i?i
no?oeoo?o ia`?eoa. Cia/iee aianie a aaeoc? aeine?aeaeaiiy no?oeoo?iiai
i?aeoiaeo aeey ?ic?iaee aeai?eoi?a ?ici?ciaaaiiy ia`?eo?a c?iaeee:
A?eoeee A.A., Caaaeeoei I.A., Io/iee ?.A., ?onei A.I., N??iaeaea ?.A.,
Oeac?iaa? I.?. oa ?i. Aaaaoi i?aoeue iia`ycaieo c ?ic?iaeith ?
canoinoaaiiyi oaeeo aeai?eoi?a iaeaaeeoue O. Iaae?ae?no, I.Oa?oo, A.
A??a??noe?, Ae. Aa?no, A. Nea/a?o, O. Oaiao, E. Oo oa ?i.

A?eueo?noue cai?iiiiiaaieo aeai?eoi?a, aeey iiaoaeiae nenoaie iciae
?iaa??aioii? aei ao?iieo ia?aoai?aiue ia`?eoa, aeei?enoiao? cia/iee
ianya neeaaeieo ia?aoai?aiue oa ia?iaee aeaieo oaeeo ye: ia?aoai?aiiy
Oo?`?, iiaoaeiaa aaoi?aa?aneaieo iiaeaeae, ia/eneaiiy eiaa??aoe?eieo
iao?eoeue, neeaaei? aaiiao?e/i? ia?aoai?aiiy: ( aeae?eaiiy «neaeaoo»,
ii?oieia?/ia aeaeiiiiceoe?y, iiaoaeiaa ???a?o?/ii? i??ai?aee cia?aaeaiiy
oa ?io?. E??i neeaaeiinoae aeey ?aae?caoe??, aeei?enoaiiy oaeeo
aeai?eoi?a ia? iaiaaeaiiy a nenoaiao ?aaeueiiai ianooaao /ano. Oiio
aeooaeueiei iaaeae? caeeoa?oueny iiaoaeiaa aeai?eoi?a oi?iaeueiiai iieno
oa ?ici?ciaaaiiy ia iniia? aiae?co aaiiao?e/ii? no?oeoo?e ia`?eoa, ui
aacothoueny ia i?inoeo oa aoaeoeaieo ??oaiiyo.

O aeai?e aeena?oaoe?ei?e ?iaio? ?ica`yco?oueny caaea/a aeine?aeaeaiiy oa
?ic?iaee no?oeoo?ii-?iaa??aioieo aeai?eoi?a aeey iiaoaeiae oi?iaeueiiai
iieno oa ia?iaee ? ?ici?ciaaaiiy neeaaeieo cia?aaeaiue, a iniiao ?iaioe
yeeo iieeaaeaii aeine?aeaeaiiy eieaeueiiai aeno?aioio e?eaecie eiioo?a
ia`?eoa ia neeaaeieo cia?aaeaiiyo.

Ca`ycie ?iaioe c iaoeiaeie i?ia?aiaie, oaiaie. Aeena?oaoe?eia ?iaioa
aeeiioaaeanue a ?aieao aea?aeathaeaeaoieo oai: ”?ic?iaea ?ioi?iaoe?eieo
oaoiieia?e oa i?ia?aiii-i???ioiaaieo cania?a ia?iaee aeaieo aeey
ae?aaiinoeee ? ioe?iee noai?a na?aaeiaeu a ?aaeueiiio /an?” (1994(1996,
Iinoaiiaa I?aceae?? IAI Oe?a?ie ?3 a?ae 22.02.94), «?ic?iaea iaoiae?a
oa aeai?eoi?a i?aeaeuaiiy yeino? iioe/ieo cia?aaeaiue noiniaii aei
caaea/ ae?aaiinoeee iaoa??ae?a oa aeaiaio?a eiino?oeoe?e » (1997(1998,
Iinoaiiaa I?aceae?? IAI Oe?a?ie ?8 a?ae 13.05.97).

Iaoa ? caaea/? aeine?aeaeaiue. Iaoith aeena?oaoe?eii? ?iaioe ?
aeine?aeaeaiiy oa ?ic?iaea aoaeoeaieo aeai?eoi?a aiae?co aaiiao?e/ii?
no?oeoo?e ia`?eo?a, iiaoaeiae ?o oi?iaeueiiai iieno ? i?iaaaeaiiy
?ici?ciaaaiiy.

O a?aeiia?aeiino? c iinoaaeaiith iaoith aeena?oaoe?eia ?iaioa aeeth/a?
?ica`ycaiiy oaeeo caaea/:

( ii??aiyeueiee aiae?c a?aeiieo iaoiae?a iiaoaeiae oi?iaeueiiai iieno oa
aeai?eoi?a aaoiiaoe/ii? eeaneo?eaoe?? ia`?eo?a;

( ?ic?iaea iaoiaeo aeae?eaiiy ia eiioo?? ia`?eoa oi/ie eieaeueiiai
aeno?aioio e?eaecie;

( iiaoaeiaa aeai?eoio iieno oa eeaneo?eaoe?? neeaaeieo iaeiica`ycieo ?
aaaaoica`ycieo ia`?eo?a ca aeiiiiiaith ?o aeaeiiiiceoe?? ia iioee?
i?aeiiiaeeie;

( iiaoaeiaa aeai?eoio cae?eniaiiy iieno oa ?ici?ciaaaiiy iioeeeo
ia`?eo?a ia iniia? ?o iiey?iiai i?aaenoaaeaiiy;

( ?ic?iaea aeai?eoi?a aeae?eaiiy ?ioi?iaoeaieo iciae oa eeaneo?eaoe??
ia`?eo?a ia iniia? aiae?co aeaieo i?i aca?iia ?icoaooaaiiy ia eiioo??
oi/ie eieaeueiiai aeno?aioio e?eaecie.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a. Ia iniia? oai?aoe/ieo oa
aenia?eiaioaeueieo aeine?aeaeaiue i?iaaaeaieo o aeena?oaoe?ei?e ?iaio?:

( ?ic?iaeaii oa aeine?aeaeaii iiaee iaoiae aeae?eaiiy oi/ie eieaeueieo
aeno?aioi?a e?eaecie ia eiioo?? ia`?eoa;

( cai?iiiiiaaii aeai?eoi iiaoaeiae iieno oa eeaneo?eaoe?? ia iniia?
i?aaenoaaeaiiy aaaaoica`yciiai ia`?eoa ?c neeaaeiith aaiiao?e/iith
no?oeoo?ith o aeaeyae? ia`?aeiaiiy iaeiica`ycieo iioeeeo i?aeiiiaeei, ui
ia ia?aoeiathoueny, yeee ia aeiaaa? ai?ieneiaoe?? eiioo?a ia`?eoa
aaaaoieooieeii;

( ?ic?iaeaii aoaeoeaiee iaoiae iieno oa ?ici?ciaaaiiy iioeeeo
iaeiica`ycieo ia`?eo?a;

( ?ic?iaeaii aeai?eoi oi?ioaaiiy iieno oa eeaneo?eaoe??, yeee
?iaa??aioiee aei ao?iieo ia?aoai?aiue neeaaeieo cia?aaeaiue, ia iniia?
ciaoiaeaeaiiy iniaeeaeo oi/ie ( iiethn?a a ?acoeueoao? ?ioa?iieyoe??
aeene?aoieo e?eaeo;

( cai?iiiiiaaii oa aeine?aeaeaii iaoaiaoe/i? iiaeae? iieno oa
eeaneo?eaoe?? ia`?eoa ia iniia? ?ioa?iieyoe?? eiai eiioo?a e?eaeie
ia?oiai oa ae?oaiai ii?yaee?a;

( ?ic?iaeaii aeai?eoi iiaoaeiae oi?iaeueiiai iieno eiioo??a aeaiaei??ieo
ia`?eo?a ia iniia? aiae?co aca?iiiai ?icoaooaaiiy oi/ie eieaeueiiai
aeno?aioio i?e ?o iaa?oaii? iaaeiei oeaio?o ian ia`?eoa.

I?aeoe/ia cia/aiiy ?acoeueoao?a. Cai?iiiiiaaiee iaoiae ? aeai?eoi
aeae?eaiiy oi/ie eieaeueiiai aeno?aioio e?eaecie ia eiioo?? nooo?ai
iie?auo? oi/i?noue aeae?eaiiy ?ioi?iaoeaieo ia?aiao??a oa aeicaiey?
ciaioeoe aia?aoi? cao?aoe ia ia?iaeo aeaieo. ?ic?iaeai? a ?iaio?
aeai?eoie aeathoue iiaeeea?noue cae?enithaaoe ia?iaeo oa ?ici?ciaaaiiy
neeaaeieo cia?aaeaiue, ye? io?eiai? a ?acoeueoao? aeenoaioe?eiiai
ciiaeoaaiiy Caie?, a oaeiae iiaeooue aoaeoeaii aeei?enoiaoaaoenue i?e
aiae?c? aeaiaei??ieo noeai. Cai?iiiiiaai? iaoiaee iiaeooue
canoiniaoaaoenue i?e ?ic?iaoe? ?aaeueieo nenoai ?ici?ciaaaiiy ia`?eo?a a
i?iieneiaino? oa ?ioeo aaeocyo.

?aae?caoe?y ? ai?iaaaeaeaiiy ?acoeueoao?a ?iaioe. Ia aac? ?ic?iaeaieo
aeai?eoi?a noai?aii i?ia?aiia caaacia/aiiy «OBJREC», yea canoiniao?oueny
aeey aiae?co aeaieo aeenoaioe?eiiai ciiaeoaaiiy Caie? c eini?/ieo
aia?ao?a eeano «N?/-1» ? ai?iaaaeaeaii a AeIAI «Eiiaen» (i. Euea?a), a
oaeiae aeei?enoiao?oueny i?e oaoi?ei-e?ei?iae?noe/ieo aeine?aeaeaiiyo
ia`?eo?a (Euea?anueeee ?inoeooo aioo??oi?o ni?aa i?e Oe?a?inuee?e
aeaaeai?? aioo??oi?o ni?aa).

Iniaenoee aianie aaoi?a. A ?iaioao iaienaieo o ni?aaaoi?noa?,
aeena?oaioo iaeaaeeoue: a [1] ( iaoiae aeaeiiiiceoe?? neeaaeieo
ia`?eo?a; a [6,7,9] ( iaoiaee oi?ioaaiiy nenoai ?ioi?iaoeaieo iciae aeey
iieno ? ?ici?ciaaaiiy ia`?eo?a.

Ai?iaaoe?y ?iaioe. Iniiai? iieiaeaiiy ? ?acoeueoaoe ?iaioe
aeiiia?aeaeenue ? iaaiai?thaaeenue ia i?aeia?iaeieo ? ?anioae?eainueeeo
eiioa?aioe?yo:

Anaoe?a?inueea i?aeia?iaeia eiioa?aioe?y “Oe?IA?AC” (i. Ee?a,
1994,1996,1998 ??.);

Ae?oaa oe?a?inueea eiioa?aioe?y c aaoiiaoe/iiai ea?oaaiiy
“Aaoiiaoeea-95” (i. Euea?a, 1995 ?.);

I?aeia?iaeia iaoeiai-oaoi?/ia eiioa?aioe?y ”No/ani? i?iaeaie
aaoiiaoeciaaii? ?ic?iaee ? ae?iaieoeoaa ?aae?iaeaeo?iiieo cania?a oa
i?aeaioiaee ?iaeaia?ieo eaae??a” (i. Euea?a, 1996 ?.);

I?aeia?iaeia iaoeiai-i?aeoe/ia eiioa?aioe?y “No/ani? oaoiieia?? a
aa?ieini?/iiio eiiieaen?” (i. AEeoiie?, 1997 ?.);

I?aeia?iaeia iaoeiaa eiioa?aioe?y “No/ani? i?iaeaie iaoai?ee ?
iaoaiaoeee” (i. Euea?a, 1998 ?.).

Ioae?eaoe?? ?acoeueoao?a aeine?aeaeaiue. Iniiaiee ci?no aeena?oaoe?eii?
?iaioe a?aeia?aaeaiee a 10 ae?oeiaaieo i?aoeyo c ieo 5 noaoae o oaoiaeo
iaoeiaeo aeaeaiiyo, 5 ?ia?o o ca??ieeao iaoa??ae?a oa i?aoeyo
eiioa?aioe?e.

No?oeoo?a ? ianya ?iaioe. Aeena?oaoe?eia ?iaioa neeaaea?oueny ?c anooio,
/ioe?ueio ?icae?e?a, aeniiae?a, nieneo aeei?enoaii? e?oa?aoo?e ?c 193
iaeiaioaaiue oa aeiaeaoe?a. ?iaioa aeeeaaeaia ia 150 noi??ieao
aeeth/ath/e 19 noi??iie aeiaeaoe?a.

INIIAIEE CI?NO ?IAIOE

O anooi? iaa?oioiaaii aeooaeuei?noue ?iaioe, iieacaii ?? ca`ycie ?c
iaoeiaeie i?ia?aiaie, ieaiaie oa oaiaie, noi?ioeueiaaii iaoo oa iniiai?
caaea/? aeine?aeaeaiue, aecia/aii iniiai? iaoiaee ae??oaiiy
noi?ioeueiaaieo caaea/, aecia/aii iaoeiao iiaecio ?iaioe ? i?aeoe/io
oe?ii?noue io?eiaieo ?acoeueoao?a, a oaeiae noenei aeeeaaeaii ei?ioeee
ci?no ?iaioe ? iniiai? ?acoeueoaoe.

O ia?oiio ?icae?e? c?iaeaii iaeyae a?aeiieo iaoiae?a oa i?aeoiae?a aei
oi?ioaaiiy ?ioi?iaoeaiiai iciaeiaiai i?inoi?o ia iniia? no?oeoo?ii?
?ioi?iaoe??, yea oa?aeoa?eco? ia`?eo, iaaaaeaii ?o eeaneo?eaoe?th,
?icaeyiooi ?aaeuei? nenoaie, ye? ?aae?cothoue oae? i?aeoiaee.

?icaeyioo? i?aeoiaee a?oioothoueny ia ia?aoai?aiiyo ia?aiao?e/iiai
i?aaenoaaeaiiy eiioo?a ia`?eoa, oiaoi eiee eiioo? caaea?oueny ca
aeiiiiiaith aeaio ooieoe?e: x=x(t) oa y=y(t), t=1,…,N, aea N (
e?euee?noue oi/ie eiioo?a. Aeinoaoi?noue aeei?enoaiiy eiioo?a ia`?eoa
aeey oi?ioaaiiy ?ioi?iaoeaieo iciae coiiaeaia iaeiicia/iith
a?aeiia?aei?noth i?ae ia`?eoii oa eiai eiioo?ii. Oaea ae i?aaenoaaeaiiy
eiioo?a aea?aii aeey ianooiieo aeine?aeaeaiue.

Ia i?aenoaa? i?iaaaeaiiai ii??aiyeueiiai aiae?co iaoiae?a iiaoaeiae
oi?iaeueiiai iieno ia`?eo?a ia iniia? a?aeiieo i?aeoiae?a ie?aneaii
canoinoaaiiy oa iaaeie?ee oeeo i?aeoiae?a aeey oi?ioaaiiy ?ioi?iaoeaieo
iciae ia`?eoa.

I?iaaaeaiee aiae?c aeai?eoi?a aaoiiaoe/ii? eeaneo?eaoe?? ia`?eo?a
iieaco?, ui oe? aeai?eoie iiaeia ?icae?eeoe ia aea? iniiai? a?oie:
aeai?eoie, ui aacothoueny ia eiia??i?ni?e no?oeoo?? aeaieo, ? aeai?eoie,
a yeeo oaea no?oeoo?a ia caaeaia. Ia?o? iathoue ia?aaaao a oiio, ui
iiaeia aeineoue eaaei ioe?ithaaoe ye?noue a?aeiia?aeieo ae??ooth/eo
ooieoe?e. Aeey aeai?eoi?a ae?oai? a?oie o caaaeueiiio i?aeoiae? oa
aeiiaao aei aeo?aeieo aeaieo oi/ia e?euee?nia ioe?iea yeino? ?
neeaaeiith. Iniiaiee iaoiae, yeee canoiniao?oueny a aeai?eoiao ia?oi?
a?oie iieyaa? a ?aeo?aioiiio aeei?enoaii? i?aaeea Aaeana, oiae? ye a
aeai?eoiao ae?oai? a?oie aeei?enoiao?oueny iaoiae, ui aaco?oueny ia
?oa?aoeai?e iioei?caoe?? aeayeiai e?eoa??th. Oea aeaa ooiaeaiaioaeuei?
iaoiaee, ui canoiniaothoueny aeey eeaneo?eaoe?? iien?a ia`?eo?a.

Aeey ?ica`yceo i?aeoe/ieo caaea/ a nenoaiao ?aaeueiiai /ano /enei
aeai?eoi?a eeaneo?eaoe?? ? cia/ii iaioei o ii??aiyii? c aaeeeith
e?euee?noth cai?iiiiiaaieo aeai?eoi?a. A i?aeoe/ieo caaea/ao ?nio?
i?iaeaia iia`ycaia ?c a?aeai?ii, ia?aoai?aiiyi ? aaia?aoe??th
aeinoaoiuei? e?eueeino? aea??eiaeo ia`?eo?a c ii/aoeiai? iiiaeeie
aeaieo. Oea ? ? aieiaiith ia?aoeiaeith a canoinoaaii? oeeo aeai?eoi?a.
Oiio aeooaeueiei ? ?ic?iaea oa aeine?aeaeaiiy aeai?eoi?a, ui aeiaaathoue
i?i?iaeueiiai ianyao ai??i?ii? ?ioi?iaoe??.

A ae?oaiio ?icae?e? ?icaeyiooi caaea/o iiaoaeiae oi?iaeueiiai iieno oa
eeaneo?eaoe?? iaeiica`ycieo oa aaaaoic`ycieo neeaaeieo ia`?eo?a ca
aeiiiiiaith ?o aeaeiiiiceoe?? ia iaiio?aei? aeaiaioe, ia iniia? aiae?co
aaiiao?e/ii? no?oeoo?e ia`?eo?a. Caaaeyee iaeiicia/i?e a?aeiia?aeiino?
i?ae ia`?eoii ? eiai eiioo?ii, a ?ie? ao?aeii? ?ioi?iaoe??
aeei?enoiaothoueny a?ia?i? cia?aaeaiiy eiioo??a ia`?eo?a.

Aeey aiae?co oi?ie cai?iiiiiaaii aeei?enoiaoaaoe oi/ee eieaeueiiai
aeno?aioio (OEA) e?eaecie ia eiioo??. I?iaaaeaii aiae?c a?aeiieo
aeai?eoi?a aeae?eaiiy oaeeo oi/ie, aeacaii ?o ia?aaaae oa iaaeie?ee.
Aeaye? c a?aeiieo aeai?eoi?a aeiaaathoue iaeiiai aai aeae?eueea ao?aeieo
ia?aiao??a, eio?? aecia/athoue aaee/eio ieieo aiae?co eiaeii? oi/ee
eiioo?a. Aeai?eoie aac ao?aeieo ia?aiao??a aeiaaathoue aeiaeaoeiaeo
ia/eneaiue. C iaoith oieeiaiiy oeeo iaaeie?e?a cai?iiiiiaaii iiaee
aeai?eoi, nooue yeiai iieyaa? a ianooiiiio.

Ine?eueee iiaa eaea i?i aeae?eaiiy OEA e?eaecie ia cia?aaeaii? oi
aoaeaii iia?oaaoe aeaiaioaie cia?aaeaiiy, a naia i?enaeaie. Aeey oiai,
uia aeae?eeoe i?enaee, ye? a?aeiia?aeathoue aeno?aioiai e?eaecie,
i?iiiio?oueny nii/aoeo ia eiioo?? ia/enethaaoe aaee/eie a?oi i?enae?a,
aeey yeeo eii?aeeiaoe x aai y iaeiaeia? (?en.1).

?en.1. I?eeeaaee a?oi i?enae?a

Ia iniia? iaeneiaeueii? aaee/eie a?oi i?enae?a aecia/a?ii ?aae?on ieieo
aiae?co oi/ie, i?eeia?ii eiai ??aiei r=dmax, aea dmax ( iaeneiaeueia
aaee/eia a?oi i?enae?a. Oi?io?ii iine?aeiai?noue ?aae?on?a V, a yeo
aeeth/a?ii /enea ii/eiath/e a?ae 1 ? aei r oa aii?yaeeiao?ii ?o ii
c?inoaiith, iai?eeeaae, yeui r =5, oi V=(1,2,3,4,5).

Aeae?eaiiy OEA iieyaa? o ?oa?aoe?eiiio a?aeai?? i?enae?a eiioo?o.
E?euee?noue ?oa?aoe?e aecia/a?oueny aaee/eiith iine?aeiaiino? V. Aeey
eiaeiiai i?enaea eiioo?a ia`?eoa ia?aa??ythoueny ianooii? oiiae:

xi-k ( xi ( xi+k , yi-k ( yi ( yi+k ,

xi-k ( xi ( xi+k , yi-k ( yi ( yi+k ,

xi-k ( xi ( xi+k , yi-k ( yi ( yi+k , (1)

xi-k ( xi ( xi+k , yi-k ( yi ( yi+k ,

xi-k ( xi ( xi+k , yi-k ( yi ( yi+k ,

xi-k ( xi ( xi+k , yi-k ( yi ( yi+k ,

aea k ( V ( aeaiaio iine?aeiaiino?.

I?enaee, aeey eii?aeeiao yeeo aeeiiothoueny oiiae (1), iicia/athoueny.
Ia ianooii?e ?oa?aoe?? ia?aa??y?oueny aeeiiaiiy oiia (1), aea aaea ?c
ianooiiei cia/aiiyi k, ? ciiao a?aeiia?aei? i?enaee iicia/athoueny.
?oa?aoe?? caaa?oothoueny, yeui ia?aiao? k iine?aeiaii iaaoa an? cia/aiiy
c aea??ee. I?enaee, ui iicia/aeenue i?ae /an eiaeii? ?oa?aoe??,
aaaaeathoueny eieaeueieie aeno?aioiaie e?eaecie eiioo?a ia`?eoa.

sse iaiio?aei? aeaiaioe aeey iiaoaeiae iieno oa eeaneo?eaoe?? ia`?eo?a
ca aeiiiiiaith ?o aeaeiiiiceoe?? aeei?enoiaothoueny iioee? i?aeiiiaeeie.
Uia iienaoe oa ?ici?ciaoe iaiio?aei? aeaiaioe ?ic?iaeaii iaoiae, yeee
aaco?oueny ia ?o iiey?iiio ia?aoai?aii?.

Iiaoaeiaa iieno iieyaa? o aecia/aii? aeiaaeeie li i?iaaaeaieo i?iiai?a
oa o?enaoe?? eooa (i, i?ae yeei aiie i?iaaaeai?. Anueiai aoaeo?oueny n
i?iiai?a c ?ioa?aaeii ((. A ?acoeueoao? oeueiai io?eio?ii
iine?aeiai?noue:

. (2)

. I?ioa, ii?ooaiiy ii?yaeeo aeaiaio?a iine?aeiaiino? (2) iiaea
i?ecaanoe aei oiai, ui iine?aeiaiino? aaee/ei li aeey ??cieo ia`?eo?a
aoaeooue iaeiaeiaeie. Oiio iio??aii ia/enethaaoe ??cieoe?
((1=((1-(n),((2=((2-(n-1), …, ye? icia/athoue aaee/eie eoo?a i?ae
iaeneiaeueieie oa i?i?iaeueieie i?iiaiyie ? caeeoathoueny iaci?iieie i?e
ao?iieo ia?aoai?aiiyo.

Cai?iiiiiaaiee aeai?eoi aeaeiiiiceoe?? ia`?eo?a aeei?enoiao? ianooii?
icia/aiiy.

Icia/aiiy 1. Oi/ea eieaeueiiai aeno?aioio e?eaecie eiioo?a iaceaa?oueny
I(oi/eith, yeui aiia ? aa?oeiith iioeei? aeoae eiioo?a ? A(oi/eith, yeui
aiia ? aa?oeiith a?aiooi? aeoae.

Icia/aiiy 2. Iineiaeiaieie aaaaea?ii aeai A(oi/ee, yeui:

a) iiae ieie ia eiioo?i iaia? aeiaeii? I(oi/ee;

a) aiae?icie, ui c`?aeith? aea? A(oi/ee, ia?oeia? eiioo? ia`?eoa A.

Icia/aiiy 3. Aeecueeith aeey A(oi/ee iaceaa?oueny oaea oi/ea eiioo?o,
yea ? iaeaeeae/ith a naini caaeaii? iao?eee ia ieiueii XY aei A(oi/ee,
na?aae oi/ie, ui ?icoaoiaaii iiae aeaiia noiiaeieie c A(oi/eith
I(oi/eaie. Noi?aeieie aei OEA iaceaaoeiaii iaeaeeu? ni?aaa ? ce?aa
OEA.

Aaaaea?ii, ui ciaeaeaii ani A(oi/ee eiioo?a ia`?eoa A (?en.2).

?ic?iaeaii ianooiiee aeai?eoi.

E?ie 1. Aeae?a?ii ia?oo A(oi/eo ? iicia/a?ii ?? oaeo/ith.

E?ie 2. sseui A(oi/ea ?aeeia , oi aeey ia? ooea?ii aeecueeo oi/eo ?
aeae? e?ie 7, ?iaeoa e?ie 3.

E?ie 3. Aeae?a?ii ianooiio A(oi/eo.

?en. 2. ?acoeueoao aeae?eaiiy oi/ie eieaeueiiai aeno?aioio e?eaecie

E?ie4. sseui aea?ai? oi/ee ia iineiaeiaii, oi niieo/a?ii ?o aiae?iceii,
iiaeoa e?ie 5.

E?ie 5. Ianooiio A(oi/eo iicia/a?ii oaeo/ith.

E?ie 6. sseui ia an? A(oi/ee aea?ai?, oiaei e?ie 3.

sseui ia?oa oa inoaiiy B(oi/ee iineiaeiaii, oi aeey inoaiiuei? ooea?ii
aeecueeo oi/eo.

sseui an? A(oi/ee aea?ai? oiae?, e?ie 7.

E?ie 7. Caaa?oaiiy aeai?eoio.

?ioeie neiaaie aeey oiai, uia aeae?eeoe iioee? i?aeiiiaeeie iio??aii
iine?aeiaii c`?aeiaoe a?ae??ceaie A(oi/ee eiioo?o ia`?eoa A. Ia oae?
a?ae??cee iaeeaaeathoueny ianooii? oiiae: 1) a?ae??cie i?ioiaeeoue
o?eueee /a?ac aioo??oi? oi/ee ia`?eoa; 2) a?ae??cie ia ia?aoeia? eiioo?
ia`?eoa; 3) a?ae??cee ia ia?aoeiathoueny i?ae niaith (?en. 3).

?en.3. ?acoeueoao aeaeiiiiceoe?? ia`?eoa

sseui ia`?eo A ? aaaaoica`yciei, oi nii/aoeo iio??aii i?aaenoaaeoe eiai
o aeaeyaei ia’?aeiaiiy iaeiic’ycieo iiaeiiiaeei. Iaoae aioo??oi?e eiioo?
i?noeoue A oa I(oi/ee. Oiae? eiaeio c oeeo oi/ie c`?aeith?ii a?ae??ceii
?c iaeaeeuith oi/eith ciai?oiueiai eiioo?o. Aeae?ey?ii i?aeiiiaeeie, ui
ooai?eeenue i?ney oeueiai. sseui aiie iaeiica`yci? oa iioee?, oi
oi?io?ii ?o iiene. sseui aiie ia ? iaeiica`ycieie, oi ciiao aeeiio?ii
aeuaiienai? ae??. ?acoeueoao canoinoaaiiy oaeiai iaoiaeo iieacaiee ia
?en.4. Oaeei /eiii io?eia?ii ?icaeooy ia`?eoa A ia eiai iioeei
i?aeiiiaeeie.

?en.4. Aeaeiiiiceoe?y aaaaoica`yciiai ia`?eoa

Ia iniia? iiaeeeaino? i?aaenoaaeoe neeaaeiee ia`?eo ye eiieaoaiaoe?th
eiai iioeeeo i?aeiiiaeei i?iiiio?oueny niin?a oi?ioaaiiy iieno ia`?eoa
o aeaeyae? caaaeaiiai a?aoa ca`yce?a iioeeeo i?aeiiiaeei oa iien?a ia
iniia? iine?aeiaiinoae aeaeo (2) aeey eiaeii? i?aeiiiaeeie.

Iieacaia ?iaa??aioi?noue cai?iiiiiaaiiai niiniao oi?ioaaiiy oi?iaeueiiai
iieno aaaaoic`ycieo oa iaeiica`ycieo iaiioeeeo ia`?eo?a ca aeiiiiiaith
?o naaiaioaoe?? ia iaiio?aei? aeaiaioe.

I?e eeaneo?eaoe?? ia?aa??y?oueny ?ciii?oi?noue io?eiaiiai a?aoa oa a?aoa
aeayeiai eeano. sseui a?aoe ?ciii?oi?, oi ii??aiththoueny iiene iioeeeo
i?aeiiiaeei. I?e ?o ni?aiaae?ii? ia`?eo a?aeiineoueny aei oeueiai eeano.

Aenia?aiaioaeuei? aeine?aeaeaiiy iieacaee iiaeeea?noue iiaoaeiae iieno
oa ?ici?ciaaaiiy neeaaeieo ia`?eo?a ia iniia? cai?iiiiiaaieo aeai?eoi?a.

O?ao?e ?icae?e i?enay/aii ?ic?iaoe? no?oeoo?ii-?iaa??aioieo aeai?eoi?a
iiaoaeiae oi?iaeueiiai iieno iaeiica`ycieo ia`?eo?a ia iniia? aiae?co
?ioi?iaoe?? i?i aca?iia ?icoaooaaiiy ia eiioo?? oi/ie eieaeueiiai
aeno?aioio e?eaecie oa eeaneo?eaoe?? oaeei /eiii ooai?aieo iien?a.

?ic?iaeaii aeai?eoi iiaoaeiae nenoaie iciae ia`?eoo ia iniia?
ciaoiaeaeaiiy niaoe?aeueieo oi/ie ( iiethn?a i?e ?ioa?iieyoe?? eiioo?a.
Eiioo? i?aaenoaaey?oueny ca aeiiiiiaith ia?aiao?e/ieo ooieoe?e x(t)=xt,
y(t)=yt, t=1,…,N, aea N ( e?euee?noue oi/ie eiioo?a.

Aeey ia/eneaiiy cia/aiue iiethn?a aeei?enoiao?oueny ia?aiao?e/ia
i?aaenoaaeaiiy ?ioa?iieyoe?eiiai iie?iiia o?aouei? noaiai? o oi?i?
Eaa?aiaea. Aeey ooieoe?e x(t) oa y(t) ( oeae iie?iii ia? aeaeyae

(3)

aea ?ttt ( cia/aiiy iiethna, ?000 , ?111 , ?222 ( cia/aiiy o aoceiaeo
oi/eao ?ioa?iieyoe??, ti ( aeaiaioe ianiaaeath/i? iine?aeiaiino?,
noi?iiaaii? c eii?aeeiao aoceiaeo oi/ie.

I?aenoaaeyth/e a iie?iii (3) aeaiaioe iine?aeiaiino? io?eia?ii cia/aiiy
iiethn?a. Iiethne ?ttt ( ?icoaoiaothoueny iaiiaeae?e a?ae aaiiao?e/iiai
ia`?eoa ?, ye aeieeaa? c ae?aco (2), aeia?eueia e?i?eia ci?ia ia?aiao?a
t oeio t=aT+b, aea a,b( R, ia aieeaa? ia aaiiao?e/ia ?icoaooaaiiy
iiethn?a. Oaea cai?ia icia/a? ia ui ?ioa ye eiiiiceoe?th ia?aoai?aiiy
ia?aiino oa ia?aoai?aiiy iiae?aiino?. Ioaea, aaee/eie a?aenoaiae
iiethn?a aei e?eaeo x(t) oa y(t) iiaeia aeei?enoaoe ye iien ia`?eoa. Ia
?en. 5a iieacaii aeayeee ia’?eo oa iiaoaeiao iiethn?a aeey ooieoe?e x(t)
(?en. 5a) oa y(t) (?en. 5a).

Iaeiicia/ia a?aeiia?aei?noue i?ae iiethnaie e?eai? ? naiith e?eaith,
aeieeaa? c ?aeeiico? ?ica`yceo caaea/? ?ioa?iieyoe??. ?iaa??aioi?
aeanoeaino? iiethn?a e?eaeo a?aeiinii ia?aoai?aiiy iiae?aiino? oa
ia?ai?uaiiy aeicaieythoue canoinoaaoe ?o i?e iiaoaeia? iien?a aeey
?ici?ciaaaiiy ia`?eo?a.

?en. 5. ?acoeueoaoe ia/eneaiiy iiethn?a ia`?eoa

Ianooiiee aeai?eoi ?aae?co? niin?a oi?ioaaiiy ???a?o?/ii? no?oeoo?e
iciae ia`?eoa, ye? iienothoue ye aaiiao?e/io oi?io ia`?eoa, oae ?
e?euee?nii ae?aaeathoue eiai oa?aeoa?enoeee.

A ?acoeueoao? aiae?co eiioo? ia`?eoa i?aaenoaaey?oueny ye
iine?aeiai?noue c?o/ieo aeey eieueeiniiai iieno aeaiaioia, a naia
a?ae??ce?a i?yieo, aeoa e?e oa aeoa ae?in?a. Aeey oeueiai, nii/aoeo,
aecia/athoueny OEA e?eaecie eiioo?a. A ?acoeueoao? aeae?eaiiy OEA
aaaaea?ii, ui eiioo? neeaaea?oueny c /anoei, e?eaecia ia yeeo ?
iino?eiith. Oe? /anoeie noaiiaeyoue iaiio?aei? aeaiaioe eiioo?a.

Aeey oiai, uia aecia/eoe aeae eiaeii? /anoeie eiioo?a ?icaeyaeathoueny
??aiyiiy aeioe/ieo i?iaaaeaieo a OEA. sseui aeey i?iaaaeaieo aeioe/ieo
aeeiio?oueny ianooiia oiiaa

,

aea k, k1, k2 ( eooia? eiao?oe??ioe, a?aeiia?aeii, i?yii?, i?iaaaeaii?
/a?ac OEA, oa aeioe/ieo, i?iaaaeaieo a oeeo oi/eao, oi i?eeia?oueny, ui
a?aeiia?aeia /anoeia eiioo?a ( a?ae??cie i?yii?. sseui oaea oiiaa ia
aeeiio?oueny, oi ciaoiaeeoueny oi/ea ia?aoeio ii?iaeae aei aeioe/ieo,
i?iaaaeaieo a OEA, ? a?aenoai? a?ae oe??? oi/ee aei OEA. O aeiaaeeo,
eiee aeey aaee/ei oeeo a?aenoaiae aeeiio?oueny oiiaa

,

aea r1 , r2 ( ia/eneai? aaee/eie, i?eeia?ii, ui /anoeia eiioo?a ( aeoaa
eiea, a i?ioeeaaeiiio aeiaaeeo /anoeia eiioo?a ( aeoaa ae?ina.

A eiaeiiio aeiaaeeo /anoei? eiioo?a noaaeoueny o a?aeiia?aei?noue
neiaie: l (yeui /anoeia eiioo?a ( a?ae??cie i?yii?, c ( yeui /anoeia
eiioo?a ( aeoaa eiea, e ( yeui /anoeia eiioo?a ( aeoaa ae?ina.

Iieacaii, ui oaeei /eiii noi?iiaaiee iien ? ?iaa??aioiei a?aeiinii
ao?iieo ia?oai?aiue. Iaaaaeaii aeai?eoi eeaneo?eaoe?? ia`?eo?a ca oaeei
iienii.

?ic?iaeaii oa aeine?aeaeaii aeai?eoi aeey iiaoaeiae nenoaie iciae, ia
iniia? aca?iiiai ?icoaooaaiiy OEA i?e iaa?oaii? ia`?eoa iaaeiei eiai
oeaio?o ian.

Aeey iiaoaeiae nenoaie iciae c`?aeith?ii a?ae??ceaie an? OEA ?c
oeaio?ii ieiu? ia`?eoa ? a?aene?aeeiao?ii ci?io aaee/ei i?iaeoe?e l(t)
oeeo a?ae??ce?a ia ai?eciioaeueio a?nue i?e iaa?oaii? eiioo?a iaaeiei
oeaio?o ian (?en. 6).

?en. 6. Ci?ia aaee/ei i?iaeoe?e aeey /ioe?eeooieea

Aeey /ioe?eeooieea, iai?eeeaae, io?eia?ii /ioe?e a?ao?ee. Eiaeai a?ao?e
ia ?en.6 ae?aaea?oueny aiae?oe/ii ca aeiiiiiaith ooieoe?e cos(t) aai
sin(t). I?ney oeueiai ii?yaeie ooieoe?e ?o ia?aiao?e oa ni?aa?aeiioaiiy
i?ae ieie aeei?enoiaothoueny aeey iiaoaeiae nenoaie iciae ia`?eoa.

Iieacaii, ui oaea nenoaia iciae aieiae?? ?iaa??aioieie aeanoeainoyie i?e
ao?iieo ia?aoai?aiiyo ia`?eoa.

Aeey eeaneo?eaoe?? ia`?eo?a cia/aiiy eiaeii? ooieoe?? ao?aeiiai ia`?eoo
ii??aiththoueny ?c cia/aiiyie a?aeiia?aeii? ooieoe?? i?aaenoaaieea
eeano. Ca e?eoa??e aa?aoueny i?i?ioi na?aaeiueieaaae?aoe/ii? iioeaee.
sseui aaee/eia iioeaee i?eeia? i?i?iaeueia cia/aiiy i?e ii??aiyii? ?c
i?aaenoaaieeii eeano, oi aaaaea?ii, ui ao?aeiee ia`?eo iaeaaeeoue aei
oiai ae eeano, ui ? i?aaenoaaiee. Oaea canoinoaaiiy e?eoa??th i?i?ioio
na?aaeiueieaaae?aoe/ii? iioeaee iiaeeeaa caaaeyee ia??iaee/iino?
ooieoe?e.

*aoaa?oee ?icae?e i?enay/aii i?eeeaaeiei aniaeoai canoinoaaiiy
?ic?iaeaieo no?oeoo?ii-?iaa??aioieo aeai?eoi?a oi?iaeueiiai iieno oa
?ici?ciaaaiiy ia`?eo?a.

Iaaaaeaii ooieoe?iiaeueio noaio nenoaie ?ici?ciaaaiiy cia?aaeaiue
eiioo??a ia`?eo?a ia iniia? cai?iiiiiaaieo no?oeoo?ii-?iaa??aioieo
aeai?eoi?a oi?iaeueiiai iieno oa eeaneo?eaoe?? (?en. 7). ?ic?iaeaii
i?ia?aiia caaacia/aiiy OBJREC, yea ?aae?co? iaoaiaoe/io iiaeaeue
cai?iiiiiaaii? nenoaie, iienai? eiai no?oeoo?a ? neeaaeia?.

Ca nai?ie aeanoeainoyie cai?iiiiiaaia nenoaia a?aeiineoueny aei
eiia?iiaaieo nenoai ?ici?ciaaaiiy aac iaa/aiiy.

Nenoaia ?ici?ciaaaiiy i?aoeth? ianooiiei /eiii. Caaeaia iiiaeeia
ia`?eo?a, ye? iio??aii ?ici?ciaoe . Ai??i?ia ?ioi?iaoe?y i?i oe? ia`?eoe
aeicaiey? iiae?eeoe ?o ia ?yae eean?a, ye? ia ia?aoeiathoueny c oi/ee
ci?o aea?aiiai i?eioeeia eeaneo?eaoe??, oiaoi neeanoe aeoaa?o eean?a.
Noi?iiaaiee neiaiee iciae, ca aeiiiiiaith yeiai iieno?oueny eiaeai eean
ia`?eo?a. Eiee ia ao?ae nenoaie iinooia? iaa?aeiiee ia`?eo ( oaoi?/i?
i?eno?i? (OI) aecia/athoue eiai iciaee (a iaoiio aeiaaeeo iiiaeeio
eiioo??a ia`?eoa c aeae?eaieie OEA).

?en. 7. Ooieoe?iiaeueia noaia nenoaie ?ici?ciaaaiiy

Ia iniia? oe??? aiinoa??i?ii? ?ioi?iaoe?? i?eeia?oueny ??oaiiy i?i
canoinoaaiiy ??cieo aeae?a aeai?eoi?a eeaneo?eaoe?? (AE1, AE2, AE3,
AE4). sseui ia?eo neeaaeiee, i?noeoue aeeaaeai? eiioo?e,
canoiniao?oueny AE1 ( oea aeaeiiiiceoe?y ia`?eoa ia iaiio?aei? aeaiaioe,
iiaoaeiaa a?aoa cayce?a oeeo aeaiaio?a ? iiaeaeueoee ?o iien. Aeai?eoi
eeaneo?eaoe?? ni?anoaaey? aiinoa??i?io ?ioi?iaoe?th i?i iciaee ia`?eoa,
ui ?ici?cia?oueny, c ai??i?iei iienii eean?a, yeee i?noeoueny a aeioe?
ai??i?ii? ?ioi?iaoe?? (AA?), ? a?aeiineoue ia`?eo aei iaeiiai ?c eean?a,
oiaoi ?ici?cia? eiai. sseui ia`?eo ia i?noeoue aeeaaeaieo eiioo??a, oi
canoiniaothoueny iaeii/anii aeai?eoie AE2 (oi?ioaaiiy iciae ?
eeaneo?eaoe?y ca aeiiiiiaith ciaoiaeaeaiiy iiethn?a), AE3 (oi?ioaaiiy
iciae ? eeaneo?eaoe?y ca aeiiiiiaith ?ioa?iieyoe?? e?eaeie ia?oiai oa
ae?oaiai ii?yaee?a) oa AE4 (oi?ioaaiiy iciae ? eeaneo?eaoe?y ca
aeiiiiiaith iaa?oaiiy iaaeiei in?). Eiaeai c aeai?eoi?a eeaneo?eaoe??
ni?anoaaey? aiinoa??i?io ?ioi?iaoe?th i?i iciaee ia`?eoa, ui
?ici?cia?oueny, c ai??i?iei iienii eean?a, yeee i?noeoueny a aeioe? AA?,
? a?aeiineoue ia`?eo aei iaeiiai ?c eean?a. ?acoeueoaoe eeaneo?eaoe??
ia`?eoa oeeie aeai?eoiaie ia?aaeathoueny ia aeie aieinoaaiiy (AA), aea
i?eeia?oueny inoaoi/ia ??oaiiy i?i a?aeianaiiy ia`?eoa aei iaaiiai eeano
ia iniia? a?eueoino? aiein?a.

Aeey aecia/aiiy aoaeoeaiino? iiaoaeiaaii? nenoaie ?ici?ciaaaiiy
caaia?iaaii oanoiao iiiaeeio ia`?eo?a. ?acoeueoaoe i?iaaaeaiiai
oanooaaiiy ia ia`?eoao oanoiai? iiiaeeie iieacaee aenieo aoaoeai?noue
oe??? nenoaie.

A aeiaeaoeao iiaeaii ?acoeueoaoe eiii`thoa?ii? ?aae?caoe?? iienaieo a
?iaio? iaoiae?a, aeai?eoi?a oa eiiieaeno i?ia?ai, a oaeiae eii??
aeieoiaio?a i?i ai?iaaaeaeaiiy ?acoeueoao?a aeena?oaoe?eieo
aeine?aeaeaiue.

INIIAI? ?ACOEUeOAOE ?IAIOE OA AENIIAEE

O aeena?oaoe?ei?e ?iaio? ia aac? oai?aoe/ieo aeine?aeaeaiue ?ic?iaeaii
aeai?eoi?/ia oa i?ia?aiia caaacia/aiiy aeey iiaeaethaaiiy i?ioeano
?ici?ciaaaiiy cia?aaeaiue ia`?eo?a ia iniia? no?oeoo?ii-?iaa??aioieo
iaoiae?a. ?acoeueoaoe aeiaaaeai? aei i?aeoe/ii? ?aae?caoe?? oa
ai?iaaaeaeai? a aeaio onoaiiaao c iaoith aaeineiiaeaiiy i?ioeano
aeaoeo?oaaiiy cia?aaeaiue.

A aeena?oaoe?ei?e ?iaio? iaea?aeaii oae? iniiai? ?acoeueoaoe.

1. I?iaaaeaiaii aiae?c a?aeiieo i?aeoiae?a iiaoaeiae iieno neeaaeieo
ia`?eo?a ia iniia? ?ioi?iaoe?? i?i ?o aaiiao?e/io oi?io, yeee aeaco? ia
iaiao?aei?noue ?ic?iaee iiaeo iaoiae?a, ui aeicaieythoue i?iaiaeeoe
iacaeaaeii a?ae ao?iieo ia?aoai?aiue eeaneo?eaoe?th cia?aaeaiue.

2. ?ic?iaeaii iiaee i?aeo?ae aei ciaoiaeaeaiiy oi/ie eieaeueiiai
aeno?aioio e?eaecie ia eiioo?? ia`?eoa, yeee ia aeiaaa? ia/eneaiiy
cia/aiue e?eaecie a eiaei?e oi/oe? eiioo?a, ui aeicaiey? ciaioeoe ia`?i
ia/eneaiue oa aia?aoi? ?ano?ne, ia aeiaaa? ao?aeieo ia?aiao??a, ?iaeoue
aeai?eoi aaeaioeaiei aei aeaeo eiioo?a.

3. I?aaenoaaeaii iaoiae iiaoaeiae oi?iaeueiiai iieno oa eeaneo?eaoe??
ia`?eo?a ?c neeaaeiith aaiiao?e/iith oi?iith ia iniia? ?o aeaeiiiiceoe??
ia iioee? iaeiica`yci? i?aeiiiaeeie. Iaoiae, ia a?aei?io a?ae a?aeiieo,
ia aeiaaa? eoni/ii-e?i?eii? ai?ieneiaoe?? eiioo?a ia`?eoa.

4. Ia iniia? ciaoiaeaeaiiy niaoe?aeueieo oi/ie ( iiethn?a ( ca
aeiiiiiaith ?ioa?iieyoe?? ooieoe?e ia?aiao?e/iiai i?aaenoaaeaiiy eiioo?a
ia`?eoa, cai?iiiiiaaii ?iaa??aioiee aei ia?aoai?aiue ia?ai?uaiiy,
iiai?ioo oa iiae?aiino? iien ia`?eoa. Oaeee iien ia aeiaaa? aeiaeaoeiaeo
ia/eneaiue iciae ia`?eoa ia iniia? ai?ieneiaoe?? eiai eiioo?a.

5. ?ic?iaeaii iiaee niin?a anoaiiaeaiiy a?aeiia?aeiino? i?ae ae?eyieith
eiioo?a ia`?eoa oa a?ae??ceii aai aeoaith, yeee ia caeaaeeoue a?ae
aaee/eie ae?eyiee eiioo?a, ui aeicaiey? ciaioeoe ia/enethaaeuei?
cao?aoe. Ia iniia? oaeiai niiniao cai?iiiiiaaii iaoiae iiaoaeiae iieno
oa eeaneo?eaoe?? ia`?eoa.

6. Cai?iiiiiaaii ?iaa??aioiee aei ao?iieo ia?aoai?aiue eiioo?a ia`?eoa
iaoiae iiaoaeiae nenoaie iciae oa eeaneo?eaoe?? ia iniia? aiae?co
aca?iiiai ?icoaooaaiiy oi/ie eieaeueiiai aeno?aioio e?eaecie. Iaoiae
aeicaiey? aoaeoaaoe iciaee ia`?eoa o?eueee ca aeiiiiiaith oi/ie
eieaeueiiai aeno?aioio e?eaecie.

7. ?ic?iaeaii i?ia?aiia caaacia/aiiy, ui iiaeaeth? nenoaio ?ici?ciaaaiiy
cia?aaeaiue ia`?eo?a ia iniia? cai?iiiiiaaieo no?oeoo?ii-?iaa??aioieo
aeai?eoi?a.

NIENIE IIOAE?EIAAIEO I?AOeUe

Eina?aae/ ?.ss., ?onei A.I. Iniaeeaino? aeei?enoaiiy aiae?oe/ieo ?
no?oeoo?ii-neioaene/ieo iaoiae?a aeey iieno oa ?ici?ciaaaiiy ia`?eo?a //
A?aea?? ? ia?iaea ?ioi?iaoe??. — 1997. — Aei. 11(87). — N. 86-90.

Eina?aae/ ?.ss. Aeae?eaiiy oi/ie eieaeueieo aeno?aioi?a e?eaecie ia
eiioo?? // ?ioi?iaoe?ei? oaoiieia?? ? nenoaie. — 1998. — O1. — ?1/2.-
N. 163-167.

Eina?aae/ ?.ss. Ai?ieneiaoe?y eiioo??a ia?ac?a aeey ?o eeaneo?eaoe?? i?e
aeenoaioe?eiiio ciiaeoaaii? Caie? // Eini?/ia iaoea ? oaoiieia?y .-
1998.- O.4, ? 4.- N. 124-127.

Eina?aae/ ?.ss. Maoiae iiaoaeiae nenoaie iciae ia`?eo?a ca aeiiiiiaith
i?inoeo iiethn?a // A?aea?? ? ia?iaea ?ioi?iaoe??. — 1998.- Aei. 12(88).
— N. 110-113.

Eina?aae/ ?.ss. Iiaoaeiaa i?ia?aiiiai caaacia/aiiy «OBJREC» aeey
iiaeaethaaiiy nenoaie ?ici?ciaaaiiy ia`?eo?a, eiai no?oeoo?a ? neeaaeia?
// A?niee Aea?aeaaiiai oi?aa?neoaoo «Euea?anueea iie?oaoi?ea».
Eiii`thoa?ia ?iaeaia??y oa ?ioi?iaoe?ei? oaoiieia??. — 1998. — ? 349.-
N. 71-74.

Eina?aae/ ?.ss., ?onei A.I., Ei?iethe E.N. Iaoiae eeaneo?eaoe?? iioeeeo
iaeiica`ycieo iaeanoae // I?aoe? ?? Anaoe?a?inueei? i?aeia?iaeii?
eiioa?aioe?? «Oe?IA?AC-94». — Ee?a. — 1994. — C. 173-176.

Kosarevich R., Rusyn B. The combined method of classification of areas
// I?aoe? ae?oai? oe?a?inueei? eiioa?aioe?? c aaoiiaoe/iiai ea?oaaiiy
«Aaoiiaoeea-95». — Euea?a. — 1995. — C. 31-32.

Eina?aae/ ?.ss. Ii?aeiaiiy aiae?oe/iiai oa no?oeoo?iiai i?aeoiae?a aei
iieno neeaaeieo ia`?eo?a // I?aoe? i?aeia?iaeii? iaoeiai-oaoi?/ii?
eiioa?aioe?? ”No/ani? i?iaeaie aaoiiaoeciaaii? ?ic?iaee ? ae?iaieoeoaa
?aae?iaeaeo?iiieo cania?a oa i?aeaioiaee ?iaeaia?ieo eaae??a”. — *.2.
-Euea?a. — 1996.- N. 137-138.

Eina?aae/ ?.ss., ?onei A.I. Naaiaioaoe?y eiioo??a aeaiaei??ieo ia`?eo?a
aeey iiaoaeiae oi?iaeueiiai iieno // I?aoe? ??? Anaoe?a?inueei?
i?aeia?iaeii? eiioa?aioe?? «Oe?IA?AC-96». — Ee?a. — 1996. — C. 213-216.

Eina?aae/ ?.ss. Iiaoaeiaa iciae ca aeiiiiiaith iaa?oaiiy ia`?eoa iaaeiei
in? // I?aoe? ?V anaoe?a?inueei? i?aeia?iaeii? eiioa?aioe??
«Oe?IA?AC-98». — Ee?a. — 1998. — C. 151-152.

Eina?aae/ ?.ss. No?oeoo?ii-?iaa??aioi? aeai?eoie iieno ? ?ici?ciaaaiiy
cia?aaeaiue. ( ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa oaoi?/ieo iaoe
ca niaoe?aeuei?noth 05.13.06 ( aaoiiaoeciaai? nenoaie oi?aae?iiy oa
i?ia?aneai? ?ioi?iaoe?ei? oaoiieia??. ( Aea?aeaaiee iaoeiai-aeine?aeiee
?inoeooo ?ioi?iaoe?eii? ?io?ano?oeoo?e, Euea?a, 1999.

Oai?aoe/ii oa aenia?aiaioaeueii aeine?aeaeaii iaoiaee iiaoaeiae
oi?iaeueiiai iieno oa eeaneo?eaoe?? cia?aaeaiue ia iniia? aaiiao?e/ii?
oi?ie ia`?eo?a. Cai?iiiiiaai? aeai?eoie iaeaaeaoue aei eeano
no?oeoo?ii-?iaa??aioieo iaoiae?a, ye? aeei?enoiaothoue ?ioi?iaoe?th i?i
oi/ee eieaeueieo aeno?aioi?a e?eaecie ia eiioo?? ia`?eoa. ?ic?iaeai?
iaoiaee a?oioothoueny ia ?iaa??aioieo aeanoeainoyo oaeeo oi/ie ?
oi?iothoue iiene ia`?eo?a oa i?iaiaeyoue eeaneo?eaoe?th a ?acoeueoao?
iaaieo ia?aoai?aiue canoiniaaieo aei oeeo oi/ie.

Eeth/ia? neiaa: ?ici?ciaaaiiy cia?aaeaiue, eeaneo?eaoe?y cia?aaeaiue,
no?oeoo?ii-?iaa??aioiee aeai?eoi, eieaeueiee aeno?aioi e?eaecie,
i?ia?aiia caaacia/aiiy.

Eina?aae/ ?.ss. No?oeoo?ii-eiaa?eaioiua aeai?eoiu iienaiey e
?aniiciaaaiy ecia?aaeaiee. ( ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa oaoie/aneeo iaoe
ii niaoeeaeueiinoe 05.13.06 ( aaoiiaoece?iaaiiua nenoaiu oi?aaeaiey e
i?ia?anneaiua eioi?iaoeeiiiua oaoiieiaee. ( Ainoaea?noaaiiue
iao/ii-enneaaeiaaoaeueneee einoeooo eioi?iaoeeiiiie eio?ano?oeoo?u,
Eueaia, 1999.

Eaaeaeue iauaeo ?aaeueiiai ie?a eiaao aaiiao?e/aneoth oi?io. E?iia
ae?oaeo oa?aeoa?enoee, oaeeo eae, iai?eia?, oeaao, ianna, caiao, ( oi?ia
iauaeoa e anoue i?eciae, eioi?ue aiciiaeii yaeyaony iaeaieaa
eioi?iaoeaiui. Iniiaiie caaea/ae ai iiiaeo nenoaiao iaoeiiiai c?aiey
yaeyaony ?aniiciaaaiea oi?iu iauaeoa ia ecia?aaeaiee e iienaiea anaai
ecia?aaeaiey. ?aniiciaaaiea oi?iu i?ienoiaeeo, eae i?aaeei, iooai
iino?iaiey aa iienaiey, o.a. oi?ia iauaeoa noaaeony a niioaaonoaea
iaeioi?ia iiiaeanoai /enae, a iioii i?ienoiaeeo eeanneoeeaoeey, a
niioaaonoaee n aaee/eiaie yoeo /enae. Oaeaea, oi?ia iauaeoa iiaeao auoue
i?aaenoaaeaia eae iiiaeanoai i?eieoeaia. Oiaaea aeey ?aniiciaaaiey oi?iu
eniieuecothony no?oeoo?iua iaoiaeu. Aiiaua ana iaoiaeu iienaiey
?acaeaeythony ia aiaoiea e aioo?aiiea aeey oiai, /oiau ?acaeaeeoue oa,
eioi?ua iino?iaiu ia iniiaaiee enneaaeiaaiey eiioo?ia e oa, eioi?ua
enneaaeotho anth ieiuaaeue oi?iu iauaeoa. A ?aaioa i?iaaaeai
n?aaieoaeueiue aiaeec iaoiaeia iino?iaiey oi?iaeueiiai iienaiey
iauaeoia. Oeacaiu ioaeaeueiua iaaeinoaoee e ia?aie/aiey i?e i?eiaiaiee
oaeeo iaoiaeia.

I?e iienaiee oi?iu iauaeoa aieueoia cia/aiea eiaao enneaaeiaaiea oaeia,
aunooiia, aiaaeei e ae?oaeo oi/ae n aunieei cia/aieai e?eaeciu ia aai
eiioo?a. Ia iniiaaiee oaeeo yeaiaioia iiaeii no?ieoue eioi?iaoeaiua
nenoaiu i?eciaeia. N oeaeueth yooaeoeaiiai auaeaeaiey yoeo yeaiaioia
?ac?aaioai e yenia?aiaioaeueii enneaaeiaai aeai?eoi auaeaeaiey oi/ae
eieaeueiuo yeno?aioiia e?eaeciu ia eiioo?a.

Oe?ieiai i?eiaiaiey i?eia?aee iaoiaeu ?aniiciaaaiey iauaeoia ia
iniiaaiee eo aeaeiiiiceoeee ia iai?iecaiaeiua yeaiaiou. A aieueoeinoaa
oaeeo iaoiaeia iaiaoiaeeiui oneiaeai yaeyaony iieeaiiaeueiay
aii?ieneiaoeey eiioo?a iauaeoa, eioi?ay i?eaiaeeo e aeiiieieoaeueiui
au/eneeoaeueiui cao?aoai. Iiyoiio, i?aaeeiaeai iaoiae aeaeiiiiceoeee
iauaeoia, eioi?ue ia o?aaoao oaeiai oneiaey. Iai?iecaiaeiui yeaiaioii
i?e yoii aunooiaao auioeeia iaeiinayciia iiaeiiiaeanoai iauaeoa,
iienaiea eioi?iai oi?ie?oaony ia iniiaaiee aai iiey?iiai i?aia?aciaaiey.
?acoeueoaoii aeaeiiiiceoeee yaeyaony a?ao naycae auioeeuo iiaeiiiaeanoa
iauaeoa.

Aeaoia?iua iiiaeanoaa, eae ia?acu iauaeoia, /anoi aunooiatho eae
i?iaeoeee o?ueioia?iuo iauaeoia. Eiioo?u oaeeo iiiaeanoa i?aaenoaaeytho
niaie caieiooua e?eaua aac oi/ae naiiia?ana/aiey. Aii?ieneiaoeey eee
eioa?iieyoeey oaeeo e?eauo ( yoi eiiiaeoiia iienaiea, eioi?ia o?aaoao
iaaieueoeo cao?ao iaiyoe, iicaieyao oi?inoeoue aiaeec yoeo iiiaeanoa
aeaaiaea?y cia/eoaeueiiio oiaiueoaieth eiee/anoaa aeaiiuo, nio?aiyy a oi
aea a?aiy iaiaoiaeeioth eioi?iaoeeth ia iaeanoe, iicaieyao ono?aieoue
ooi. Ia iniiaaiee iaoiaeaeaiey niaoeeaeueiuo oi/ae ( iiethnia i?e iiiiue
eioa?iieyoeee ooieoeee ia?aiao?e/aneiai i?aaenoaaeaiey eiioo?a iauaeoa,
i?aaeeiaeaii eiaa?eaioiue e i?aia?aciaaieyi ia?aiauaiey, iiai?ioa e
iiaeiaey iienaiea iauaeoa.

A caaea/ao ?aniiciaaaiey eiioo?ia iauaeoia, eae i?eciaee, eniieuecotho
ia?aiao?u iaeioi?uo noaiaea?oiuo, eco/aiiuo a aaiiao?ee oeao?, oaeeo
eae: yeeein, eaaae?ao, o?aoaieueiee, e?oa, i?yiioaieueiee e ae?. oi?ioi
aii?ieneie?othueo iauaeou, eioi?ua ioaeii eeanneoeoee?iaaoue. Oaeia
i?aaenoaaeaiea ioee/aaony eiiiaeoiinoueth e iicaieyao i?eiaieoue aiia?ao
aiaeeoe/aneie aaiiao?ee. Aeey iino?iaiey oi?iaeueiiai iienaiey
i?aaeeiaeaii /anoe eiioo?a n iinoiyiiui cia/aieai e?eaeciu
eioa?iiee?iaaoue e?eauie ia?aiai e aoi?iai ii?yaeeia. ?ac?aaioai
e?eoa?ee eeanneoeeaoeee /anoae eiioo?a eae io?aceia i?yiuo, aeoa e?oaia
e aeoa yeeeinia. Nenoaia i?eciaeia iieo/aia ii oaeiio iaoiaeo yaeyaony
eiaa?eaioiie e aooeiiui i?aia?aciaaieyi iauaeoa.

Iaeaieaa ?ani?ino?aiaiiui i?aia?aciaaieai iauaeoa, eioi?ia ano?a/aaony a
?aaeueiuo oneiaeyo anoue iiai?io aai ia iaeioi?ue oaie. A aieueoeinoaa
iaoiaeia oi?ie?iaaiey i?eciaeia aeey aeinoeaeaiey eiaa?eaioiinoe
iienaiey ioiineoaeueii oaeiai i?aia?aciaaiey i?eiaiythony
aeiiieieoaeueiua ia?aie/aiey. A ?aaioa iieacaii, /oi eiaaea
oi?ie?iaaoue iienaiea iauaeoa i?e aai a?auaiee aie?oa nianoaaiiiai
oeaio?a iann, oi oaeea ia?aie/aiey iaioaeiu.

?ac?aaioaii i?ia?aiiiia iaania/aiea, eioi?ia iiaeaee?oao nenoaio
?aniiciaaaiey ia iniiaaiee i?aaeeiaeaiiuo a aeenna?oaoeeiiiie ?aaioa
iaoiaeia iino?iaiey oi?iaeueiiai iienaiey e ?aniiciaaaiey ecia?aaeaiee
iauaeoia, iieo/aiiuo a ?aaeueiuo oneiaeyo.

Eeth/aaua neiaa: ?aniiciaaaiea ecia?aaeaiee, eeanneoeeaoeey
ecia?aaeaiee, no?oeoo?ii-eiaa?eaioiue aeai?eoi, eieaeueiue yeno?aioi
e?eaeciu, i?ia?aiiia iaania/aiea.

Kosarevych R. Ja. Structural-invariant algorithms for description and
recognition of the images. ( Manuscript.

Thesis for the candidate`s degree in technical sciences on a speciality
05.13.06 ( automatic control systems and advanced information
technologies. ( State Scientific and Research Institute of Information
Infrastructure, Lviv, 1999.

The theoretical and experimental investigations of methods of
construction of the formal description and classifications on the basis
of the geometrical form of objects are presented. The proposed
algorithms belong to a class of structural-invariant methods, which use
the information on points local extreme of curvature on a contour of
object. The developed methods are based on invariant properties of such
points and form the descriptions of objects and also will carry out
classification in result of some transformations, which are applied to
these points.

Key words: recognition of the images, classification of the images,
structural-invariant algorithm, local extreme of curvature, software.

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