Геометричне моделювання та обчислення інтегральних характеристик паралельних множин: Автореф. дис… канд. техн. наук / О.М. Сівальньов, Київ. держ. т

IIIINOA?NOAI INAIOE OE?AIIE

EEIANUeEEE AeA?AEAAIEE OAOII*IEE OIIAA?NEOAO

AOAeIAIEOeOAA I A?OIOAEOO?E

OAeE 515.2

N?aaeueiueia Ieaenaiae? Ieeieaeiae/

AAIIAO?E*IA IIAeAETHAAIIss

OA IA*ENEAIIss ?IOAA?AEUeIEO OA?AEOA?ENOEE IA?AEAEUeIEO IIIAEEI

Niaoe?aeuei?noue 05.01.01 —

I?eeeaaeia aaiiao?iy, ?iaeaia?ia a?aoiea

A A O I ? A O A ? A O

aeena?oaoei? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa oaoii/ieo iaoe

Ee?a — 1998

Aeena?oaoei?th ? ?oeiien.

?iaioo aeeiiaii a Oa?eianueeiio aea?aeaaiiio iie?oaoi?/iiio
oi?aa?neoao?

Iaoeia? ea??aieee: — aeieoi? oaoi?/ieo iaoe, i?ioani?

Eooeaiei Eaiiiae Ieeieaeiae/,

O?IA IAN Oe?a?ie, canooiiee ia/aeueieea

eaoaae?e iiaeaaeii? oaoi?ee;

— eaiaeeaeao oaoii/ieo iaoe, aeioeaio

Aiaia Na?aie Aieiaeeie?iae/,

O?IA IAN Oe?a?ie, i?ioani?

eaoaae?e iiaeaaeii?
oaoi?ee;

Ioioeieii iiiiaioe: — aeieoi? oaoii/ieo iaoe, i?ioani?

Neeaeai ?aai Aiae??eiae/,

AeAeOO, caa?aeoaa/ eaoaae?e ia?enii?

aaiiao??? oa ?iaeaia?ii? a?ao?ee;

— eaiaeeaeao oaoii/ieo iaoe,
i?ioani?

Naaeeaoeueea Iaoaey Iaaiiaia

EAeOOAA, i?ioani? eaoaae?e ia?enii?

aaiiao???, ?iaeaia?ii? oa iaoeiii? a?ao?ee

I?iaiaeia onoaiiaa: Oa?e?anueeee aea?aeaaiee oaoi?/iee oi?aa?neoao

?aae?iaeaeo?ii?ee, eaoaae?a
i?eeeaaeii? aaiiao???

? eiii’thoa?ii? a?ao?ee

Caoeno aiaeaoaeaoueny 11 eenoiiaaea 1998 ?. i 13 aiaeeii ia caniaeaiii
niaoeiaeiciaaii? a/aii? ?aaee Ae 26.056.06 o Ee?anueeiio aea?aeaaiiio
oaoii/iiio oiiaa?neoaoi aoaeiaieoeoaa i a?oioaeoo?e ca aae?anith:

252037 Ee?a — 37, Iiaio?ioeionueeee i?ini., 31, aoae. 319

C aeena?oaoei NEIAIE 186 \f «Times New Roman Cyr» th iiaeia
iciaeiieoeny a aiaeiioaoei Ee?anueeiai aea?aeaaiiai oaoii/iiai
oiiaa?neoaoo aoaeiaieoeoaa i a?oioaeoo?e ca aae?anith: 252037 Ee?a — 37,
Iiaio?ioeionueeee i?ini., 31.

Aaoi?aoa?ao ?icineaii 11 aa?aniy 1998 ?.

A/aiee nae?aoa?

niaoeiaeiciaaii? ?aaee Ae 26.056.06

eaiaeeaeao oaoii/ieo iaoe, aeioeaio
A.I.Ieineee

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. ?icaeoie oaoi?ee ? oaoiieia?? noeioeth?
aeine?aeaeaiiy a aaeoc? aaoa?iaaiieo nenoai ?a/iaei. Iaaaaea?ii, ui
aaoa?iaaiiith iaceaa?oueny iae?ineii?/ii — iaiaeii??aeia o?ceei-o?i?/ia
nenoaia, ui neeaaea?oueny c aeaio ??cieo ca aeanoeainoyie ?a/iaei, ye?
?iciaaeiaai? ?iciiae?eue/ith iiaa?oiath. Aaiiao?e/ia oi?ia
?iciiae?eue/i? iiaa?oi? ci?ith?oueny o i?inoi?? aiane?aeie ci?ie o /an?
o?ceei-o?i?/ieo aeanoeainoae neeaaeiaeo ?a/iaei ia e?iiee ?o ?iciiae?eo.
Aeae? aoaea ?icaeyiooi eeoa aaoa?iaaii? nenoaie, a?ao?/iei i?iyaii
?aaeoe?e yeeo ? n?i’y ia?aeaeueieo (aea?aeenoaioieo) ?iciiae?eue/eo
iiaa?oiiue. Naia ca aeiiiiiaith n?i’? ia?aeaeueieo iiaa?oiiue (o
caaaeueiiio aeiaaeeo — ia?aeaeueieo iiiaeei) aeioe?eueii aea/aoe
aeeiai?eo ?icaeoeo aaoa?iaaiieo nenoai c iiceoe?e i?eeeaaeii? aaiiao???.
Iio?aaa o ?ic?aooieao ia?aeaeueieo iiiaeei aeieea? i?ae /an
aeine?aeaeaiiy iaaiiai eeano caaea/ o aaeoc? oaoi?ee oa oaoiieia??.
?ica’ycaiiy oeeo caaea/ ia? aaaeeeaa cia/aiiy aeey noaiiaeaiiy aeiiii?ee
Oe?a?ie ye iieiaei? iacaeaaeii? aea?aeaae.

Na?aae i?eeeaae?a aaoa?iaaiieo nenoai iacaaii i?iaeooaaiiy
oaa?aeiiaeeaieo ?aeaoieo aeaeaoi?a, ?ic?aooiie iiaa?oiaai? ?aaeoe??
(ieeneaiiy, ia?aiino, ?ic/eiaiiy oiui) oaa?aeiai o?ea, iiaeaethaaiiy
?iciianthaeaeaiiy aiaith i?ae /an e?niaeo iiaeaae oiui. O ia?oiio
i?eeeaae? ?aaeoe?y aaoa?iaaiiiai oeio ei?enia — aiia ca?eueiy? aia?a?th,
yea caeaaeaoeia a?ae aaee/eie iiiaioaeueii? ieiu? ?iciiae?eue/i?
iiaa?oi?. Oiio a aeaiee iiiaio /ano aaee/eia oyae oaa?aeiiaeeaiiai
?aeaoiiai aeaeaoia caeaaeaoeia a?ae cia/aiiy iiiaioaeueii? ieiu? ai??iiy
ca?yaeo. Oeyoii aeai?o ii/aoeiai? aaiiao?e/ii? oi?ie ?iciiae?eue/i?
iiaa?oi? neooii ?aaoethaaoe caeii ci?ie a /an? aaee/eie aia?a??, yea
ca?eueiy?oueny. O inoaiiueiio c iacaaieo i?eeeaae?a ?aaeoe?y
aaoa?iaaiiiai oeio oe?aeeeaa, aaeaea aiia ca?eueiy? aia?a?th i?ae /an
e?niai? iiaeaae?. Aaee/eia aia?a?? caeaaeaoeia a?ae ia?eiao?o
?iciiae?eue/i? e?iiee iiiaioaeueii? ieiu? iiaeaae?. ?ano?ne, ye?
iaiao?aei? aeey aan?iiy iiaeaae? a aeaiee iiiaio /ano, caeaaeaoeiooue
a?ae aaee/eie iiiaioaeueiiai ia?eiao?a e?iiee ai??iiy.

Iaaaaeai? ai?iaaaeaeaiiy aeacothoue ia oa, ui i?e aea/aii? aaoa?iaaiieo
nenoai ?a/iaei aeooaeueieie ieoaiiyie aoaeooue: i) ia?aaeaa/aiiy
aaiiao?e/ii? oi?ie ?iciiae?eue/i? iiaa?oi? a iaia?aae iicia/ai? iiiaioe
/ano; oa ii) ia/eneaiiy aaee/eie ?? ieiu? a o? ae nai? iiiaioe /ano. O
aeiaaeeo ia ieiuei? aeooaeueieie ieoaiiyie aoaeooue: i) ?ic?iaea iaoiaeo
ia?aaeaa/aiiy aaiiao?e/ii? oi?ie ?iciiae?eue/i? e?iiee a iaia?aae
iicia/ai? iiiaioe /ano, a oaeiae ii) ia/eneaiiy aaee/eie ?? ia?eiao?a a
o? ae nai? iiiaioe /ano.

sse iieaco? aeina?ae, ia i?aeoeoe? oaeee iien iaea?aeaoe aaaeei, oiio ui
?iciiae?eue/a iiaa?oiy (a ia ieiuei? — ?iciiae?eue/a e?iiea) iiaea
iaaoaaoe iaaeoi “i?eioeeai?” aaiiao?e/ii? oi?ie. Oiio a ?iaio?
aeei?enoiao?oueny iaoiae iaea?aeaiiy ooieoe?? ci?ie cia/aiue ieiu?
?iciiae?eue/i? iiaa?oi? ca aeiiiiiaith ?ioi? oa?aeoa?enoeee aaoa?iaaiii?
nenoaie ?a/iaei — caeiio ci?ie cia/aiue ia’?io ?a/iaeie, ui iaiaaeaia
cacia/aiith ?iciiae?eue/ith iiaa?oiath. Ia ieiuei? iien ci?ie ia?eiao?a
e?iiee aeioe?eueii cae?enithaaoe ca aeiiiiiaith ooieoe?? ci?ie cia/aiue
ieiu?, yea iaiaaeaia a?aeiia?aeiith e?iieith.

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. Aeena?oaoe?eia
?iaioa aeeiiaia a ?aieao iaoeiai-aeine?aeii? i?ia?aie “Aaiiao?e/ia
iiaeaethaaiiy ia ae?ai? ia?niiaeueiiai eiii”thoa?a oiiia?ao?/ii? noaie
aeai?yiiy e?niai? ae?eyiee” eaoaae?e iiaeaaeii? oaoi?ee O?IA IAN Oe?a?ie
ca?aeii caaaeaiith Oi?aae?iiy aea?aeaaii? iiaeaaeii? ioi?iie IAN
Oe?a?ie.

Iaoa ?iaioe ? caaea/? aeine?aeaeaiiy. Iaoa ?iaioe iieyaa? o noai?aii?
oai?aoe/ii? iniiae oa aeai?eoi?/ii? aace aeey eiii’thoa?ieo i?ia?ai
aecia/aiiy aaiiao?e/ii? oi?ie ia?aeaeueieo iiiaeei oa ia/eneaiiy ?o
?ioaa?aeueieo oa?aeoa?enoee, a caeaaeiino? a?ae ii/aoeiai? o?ao?e ?
aeiaeaoeiaeo oiia oi?iiooai?aiiy n?i’? ia?aeaeueieo e?i?e.

Aeey aeinyaiaiiy aieiaii NEIAIE 191 \f «Times New Roman Cyr»
iaoe aeine?aeaeaiue o aeena?oaoe?? iinoaaeai? oae? iniiai? caaea/i:

— iaa?oiooaaoe iiaeeea?noue aaiiao?e/iiai iiaeaethaaiiy aaoa?iaaiieo
nenoai ia iniia? ia?aeaeueieo iiiaeei;

— i?iaiae?coaaoe iiaeeea? iaoiaee iieno n?i’? ia?aeaeueieo iiiaeei;

— ?ic?iaeoe iaoiae iieno n?i’? ia?aeaeueieo iiiaeei ca aeiiiiiaith
??aiyiiy o iayaiiio aeaeyae?;

— aeine?aeeoe caeaaeiino? i?ae ?ioaa?aeueieie oa?aeoa?enoeeaie aeayeeo
eiie?aoieo ia?aeaeueieo iiiaeei;

— neeanoe aeai?eoi ia/eneaiiy ia?eiao?o oa ieiu? aeaiaio?a n?i’?
ia?aeaeueieo iiiaeei;

— aeey ia?aa??ee aeai?eoio ?ica’ycaoe oanoia? i?eeeaaee;

— ?ic?iaeoe i?eaeaoiee aeey i?aeoe/iiai canoinoaaiiy iaoiae a?ao?/ii?
?ioa?i?aoaoe?? iane?aee?a e?niai? iiaeaae? ia i?eeeaae? iiaoaeiae
oiiia?ao?/ii? noaie e?iiee aeai?yiiy ae?eyiee e?niiai ianeao o iaia?aae
cacia/ai? iiiaioe /ano.

Iaoiaeeea aeine?aeaeaiue. A ?iaio? aeei?enoiao?oueny iaoaiaoe/iee
aia?ao R-ooieoe?e, ui aea? ciiao iienoaaoe aeaiaioe ia?aeaeueieo iiiaeei
o iayaiiio aeaeyae?. Canoiniaothoueny iieiaeaiiy i?eeeaaeii? aaiiao???,
/enaeueieo iaoiae?a, aeeoa?aioe?aeueieo ??aiyiue.

Oai?aoe/iith aacith aeine?aeaeaiue iineoaeeee ?iaioe a/aieo:

— a aaeoc? aaiiao?e/iiai iiaeaethaaiiy ia’?eo?a oa i?ioean?a A.?.
Ieoaeeaiea, A.A.Iaaeiaa, A.I.Iaeaeeoa, I.E.Iiaeai?iiai, ?.A.Neeaeaia,
N.I.Eiaaeueiaa, I.?.Naaeeaoeueei?, E.I.Naciiiaa, A.I.Iiaeei?eoiaa;

— a aaeoc? iieno ia?aeaeueieo iiiaeei A.E.?aa/iaa, TH.A.Noiyia,
I.A.Ni?iaeaee, TH.I.Aaaea?aa, E.I.Eooeaiea, Ae.A.Aiiaea?y.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a iieyaa? a noai?aii? oai?aoe/ii?
iniiae oa aeai?eoi?/ii? aace aeey eiii’thoa?ieo i?ia?ai ia?aaeaa/aiiy
aaiiao?e/ii? oi?ie aeaiaio?a n?i’? ia?aeaeueieo iiiaeei o iaia?aae
cacia/ai? iiiaioe /ano, neeaaeiaeie /iai ? iia?:

— iaoiae iiaoaeiae ooieoe??, a?ao?eii yei? ? iiaa?oiy ??aiiai noeeo, ui
aeicaiey? iaea?aeaoe ia ea?oeii?e ieiuei? n?i’th i?iaeoe?e e?i?e ??aiy,
ye? ca?aaoeiooueny c aeaiaioaie n?i’? ia?aeaeueieo iiiaeei;

— aaiiao?e/ia oeoia/aiiy aieeao ia?aoeiae ia oi?iiooai?aiiy aeaiaio?a
n?i’? ia?aeaeueieo e?i?e ;

— aeai?eoie ia/eneaiiy ?ioaa?aeueieo oa?aeoa?enoee n?i’? ia?aeaeueieo
iiiaeei (ia?eiao?o e?i?? oa ieiu?, yeo oa iaiaaeo?).

A??ia?aei?noue oa iaa?oioiaai?noue iaea?aeaieo ?acoeueoaoia
i?aeoaa?aeaeo?oueny oeyoii aeiaaaeaiiy oaa?aeaeaiue, a oaeiae
aaiiao?e/iiai iiaeaethaaiiy ia ae?ai? eiii’thoa?a aeaiaio?a eiie?aoieo
ia?aeaeueieo iiiaeei noiniaii e?iiie aeai?aiiy e?niaeo iiaeaae.

I?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoao?a aeena?oaoe?? iieyaa? a
ni?iiiaeiino? ia i? oai?aoe/i?e aac? ai?iaaaeaeoaaoe a ?aaeueio i?aeoeeo
iaoiae eiii’thoa?iiai ia?aaeaa/aiiy aaiiao?e/ii? oi?ie oa ?ioaa?aeueieo
oa?aeoa?enoee oiiia?ao?/ii? noaie eiioo?o aeai?yiiy ae?eyiee e?niaiai
ianeao o iaia?aae cacia/ai? iiiaioe /ano. Oey iia?aoeaia a?ao?/ia
?ioi?iaoe?y aeiiiiiaea i?eeiaoe iaa?oioiaai? ??oaiiy i?e ?iciiae?e? nee
oa ?ano?n?a uiaei aan?iiy iiaeaae?, a oaeiae iiaeaethaaoe a??ooaeuei?
neooaoe?? ia aeiaaeie aeieeiaiiy iiaeaae. ?aae?caoe?y aeeiiaia a OIAN
Oa?e?anueei? iaeano?, ui i?aeoaa?aeaeo?oueny aeia?aeeith i?i
aeei?enoaiiy cai?iiiiiaaii? o ?iaio? iaoiaeeee.

Iniaenoee aianie caeiaoaa/a. Iniaenoi aaoi?ii ?ic?iaeaia oai?aoe/ia
iniiaa oa neeaaeaii aeai?eoie iiaoaeiae n?i’? ia?aeaeueieo e?i?e ia
i?eeeaae? aaiiao?e/iiai iiaeaethaaiiy e?iiie aeai?yiiy aeey ?ic?aooieo
?ioaa?aeueieo oa?aeoa?enoee i?ioeano e?niai? iiaeaae?

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??. Iniiai? iieiaeaiiy
aeena?oaoe?eii? ?iaioe aeiiia?aeaeeny oa iaaiai?thaaeeny ia 4 e
I?aeia?iaei?e iaoeiai-i?aeoe/i?e eiioa?aioe?? «No/ani? i?iaeaie
aaiiao?e/iiai iiaeaethaaiiy» (i.Iaeioiiieue; 1997 ?.), ia I?aeia?iaei?e
iaoeiai-i?aeoe/i?e eiioa?aioe?? «No/ani? i?iaeaie aaiiao?e/iiai
iiaeaethaaiiy» (i. Oa?e?a; 1998 ?.), a oaeiae ia nai?ia?ao i?ia?aeieo
eaoaae? a?ao?/iiai i?io?eth aeueo oaoi?/ieo iaa/aeueieo caeeaae?a
Oe?a?ie oa o i?aai?caoe?yo iiaeaaeii? aaciaee IAN Oe?a?ie.

E?euee?noue ioae?eaoe?e ca oaiith ?iaioe neeaaea? 5 iaeiaioaaiue.

No?oeoo?a i ianya ?iaioe. Aeena?oaoe?y neeaaea?oueny ?c anooio,
/ioe?ueio ?icae?e?a, aeniiaeo, nieneo e?oa?aoo?e ?c 143 iaeiaioaaiue oa
aeiaeaoeo. ?iaioa i?noeoue 85 noi??iie iaoeiiieniiai oaenoo oa 46
?enoie?a, iiaoaeiaaieo ca aeiiiiiaith eiii’thoa?a.

CI?NO ?IAIOE

O anooi? ?ice?eaa?oueny nooi?noue ? noai iaoeiai? i?iaeaie oa ??
cia/ou?noue, iaa?oioiao?oueny aeooaeuei?noue oaie, noi?ioeueiaai? iaoa ?
caaea/? aeine?aeaeaiiy, a?aeaeo? iaoeiaa iiaecia ? i?aeoe/ia cia/aiiy
iaea?aeaieo ?acoeueoao?a, cacia/aii eiie?aoiee iniaenoee aianie
caeiaoaa/a, iaaaaeaii ia?ae?e iniiaieo iaoeiaeo ?acoeueoao?a, ui
aeiinyoueny ia caoeno.

A ia?oiio ?icae?e? aeena?oaoe?? oi?ioeth?oueny aaiiao?e/ia caaea/a
aeine?aeaeaiue o aaeoc? aaoa?iaaiieo nenoai ?a/iaei. Aeaco?oueny ia
iiaeeea?noue ?? ?ica’ycaiiy oeyoii aea/aiiy aeanoeainoae n?i’?
ia?aeaeueieo iiiaeei oi/ie. C?iaeaii e?eoe/iee iaeyae iiaeeeaeo iaoiae?a
iieno n?i’? ia?aeaeueieo iiiaeei oa ia/eneaiiy ?ioaa?aeueieo
oa?aeoa?enoee ?? aeaiaio?a.

? neeaaea?oueny c a?ae??ce?a i?yieo e?i?e oa aeoa e?e.

iaceaathoueny ia?aeaeueieie e?eaeie, ye? a?aenoiyoue i?ae niaith ia
a?aenoai? t.

} aoaeaii iaceaaoe cia/aiiy aaee/ei L(t) ? U(t), ye? ia/eneai? aeey
aeayeiai o?eniaaiiai cia/aiiy t.

ca aeiiiiiaith:

— iaoiaeo iioi/eiaiai ia?aianaiiy (?en.1)

x = ( + t cos (; y = ( + t sin (;

— iaoiaeo “aaiethoe — aaieueaaioe” (?en. 2);

;

— oaiaaioe?aeueiiai ??aiyiiy (?en. 4)

x cos ( +y sin ( — ((() + t = 0;

— noie I?ieianueeiai (?en.5).

?en.1.

?en.3

?en.4

.

O ?iaio? iaaaaeaii cia?aaeaiiy a?ao?e?a aeayeeo ii?iaeueieo ooieoe?e
z=f(x,y), ye? iathoue aeaeyae iiaa?oiiue ??aiiai noeeo c eooii iaoeeo
oa??ieo 45 SYMBOL 176 \f «Symbol» \s 14 . Iaaieioaii, ui aeey
aecia/aiiy ia?aeaeueieo o?ao? aeioe?eueii canoiniaoaaoe ii?iaeuei?
ooieoe??. O oeueiio aeiaaeeo ia?aeaeuei? e?ea? ia eii?aeeiaoi?e ieiuei?
iiaeia ?icaeyaeaoe ye noi?uai? i?iaeoe?? ia?aoeio a?ao?eo ii?iaeueii?
ooieoe?? n?i’?th ia?aeaeueieo ieiuei ??aiy. Aea a?aeii? aiae?oe/i?
iaoiaee iiaoaeiae ii?iaeueii? ooieoe?? aeey aeaiiai e?aneaiiy ia ?
oi?aa?naeueieie. O ?iaio? i?e/eia oeueiai iieacaia ca aeiiiiiaith
a?ao?/ieo iiaoaeia.

.

. A?aenooi?noue aeai?noiai oaa?aeaeaiiy ia aeicaiey? noai?thaaoe
caaaeuei? aeai?eoie iiaoaeiae ii?iaeueii? ooieoe?? aeey aeaiiai
e?aneaiiy. Oiio ia i?aeoeoe? aeei?enoiaothoue ooo/i? niiniae iiaoaeiae.

A ?iaio? iaaaaeaii iaeyae iaoiae?a iiaoaeiae ii?iaeueieo ooieoe?e aeey
aaiiao?e/ieo ia’?eo?a, ui neeaaeathoueny c a?ae??ce?a oa aeoa e?e.

Cacia/aii, ui iaoiaee iiaoaeiae ii?iaeueieo ooieoe?e iiaeooue
aacoaaoeny ia ?ica’ycaii? aeeoa?aioe?aeueiiai ??aiyiiy. Ae?enii, ca?aeii
oai?aie A. Iiiaea (aeea. Iiiae A. I?eeiaeaiea aiaeeca e aaiiao?ee.-
I.-E.: IIOE, 1936) ooieoe?y z = z(x,y), a?ao?e yei? ? iiaa?oiy ??aiiai
noeeo c eooii iaoeeo 45( ? nie?a?oueny ia e?i?th L, caaeiaieueiy?
aeeoa?aioe?aeueiiio ??aiyiith aeeiiaea

c a?aie/iith oiiaith z(x,y)(L = 0. Oiaoi ii?iaeueia ooieoe?y
caaeiaieueiy? ??aiyiith aeeiiaea. Ca?aene ia?ii, ui ii?iaeueio ooieoe?th
iiaeia iaea?aeaoe ye ?ica”ycie aeeoa?aioe?aeueiiai ??aiyiiy aeeiiaea.

A ae?oaiio ?icae?e? iaaaaeaii oai?aoe/i? iniiae iieno ia?aeaeueieo
iiiaeei oi/ie.

Nii/aoeo ?icaeyiooi aeai?ei iiaoaeiae ia?aeaeueieo e?eaeo, ye? caaeaii
a ia?aiao?e/iiio aeaeyae?. Iaoae caieiaia e?eaa G0 caaeaia
ia?aiao?e/ieie ??aiyiiyie x = x((), y = y((). Iicia/eii /a?ac L iiaio
aeiaaeeio e?eai? G0, /a?ac, /a?ac U — ieiuo o?ao?e, ui iaiaaeaia e?eaith
G0. ?icaeyiaii e?eao Gt, yea aoaea ia?aeaeueiith ii/aoeia?e e?ea?e G0.
Oiae? ia?aiao?e/ia ??aiyiiy e?eai? Gt ia? aeaeyae

, (1)

Aeai?eoi neeaaeaii aeey i?ia?aiiiai i?ioeani?a Maple V. A ?iaio?
iieacai? iaaeie?ee iieno ia?aeaeueieo e?i?e ca aeiiiiiaith ia?aiao?e/ieo
??aiyiue. Iaaieioaii, ui a aeayeeo aeiaaeeao ia?aeaeuei? e?ea? c?o/ii
iienoaaoe ??aiyiiyie o iayaiiio aeaeyae?. Cie?aia — o aeaeyae?
ii?iaeueieo ??aiyiue.

Oaa?aeaeaiiy. Ii?iaeueia ??aiyiiy a?ae??cea i?yii?, ui niieo/a? oi/ee c
eii?aeeiaoaie A(xa,ya,za) ? B(xb,yb,zb), ia? aeaeyae

, (2)

— ciae R1 — eii”thieoe??, ( — ciae R1 — aeec”thieoe??,

; (3)

; (4)

. (5)

Aeae? ?icaeyiooi iaoiae iieno e?eai?, yea ? ia?aeaeueiith i?inoiio
iiiaieooieeo. Iiiaieooiee ? i?inoei, yeui ?) c eiaeii? eiai aa?oeie
aeoiaeyoue o?eueee aea? noi?iie; ??) noi?iie ia iathoue ni?eueieo oi/ie;
???) aa?oeie ia eaaeaoue ia eiai noi?iiao.

C oai?aie A.E.?aa/iaa ne?aeo? oaea oaa?aeaeaiiy.

Oaa?aeaeaiiy. Ii?iaeueia ??aiyiiy i?inoiai iiiaieooieea (, yeee
iaiaaeaii N a?ae??ceaie {(i} i?yieo, ia? aeaeyae

, (6)

aea fi (x,y) — ii?iaeuei? ooieoe?? a?ae??ce?a i?yieo aeaeyaeo (2).

Ia ?en.6 cia?aaeaii i?eeeaaee e?eaeo, ia?aeaeueieo i?inoiio
iiiaieooieeo, a oaeiae a?ao?ee ii?iaeueieo ooieoe?e. Ooo ia?aeaeuei?
e?ea? ca?aathoueny c i?iaeoe?yie ia?aoeio a?ao?ea n?i”?th ieiuei ??aiy.

?en.6.

Aeae? ?icaeyiooi iaoiae aecia/aiiy ia ieiuei? aaiiao?e/iiai oeaio?a
iiiaieooieea — oiaoi oeaio?a eiea i?i?iaeueiiai ?aae?ona, yea iiai?noth
ioiieth? oeae iiiaieooiee.

Oaa?aeaeaiiy. Eii?aeeiaoe aaiiao?e/iiai oeaio?a N-eooieea ( ca?aathoueny
c eii?aeeiaoaie oi/ee, a ye?e ooieoe?y

(8)

aeinyaa? i?i?ioia. Ooo ( — ciae R1 -aeec”thieoe??, a /a?ac fi (x,y)
(i=1…N-1) iicia/aii ii?iaeuei? ooieoe?? eiai aa?oei. Cie?aia,
ii?iaeueia ooieoe?y aa?oeie P(xP,yP) ia? aeaeyae

. (9)

Aeey aeayeeo i?inoeo iiiaieooiee?a ia ?en. 6a cia?aaeaii a?ao?ee oa
e?i?? ??aiy a?aeiia?aeieo ooieoe?e aeaeyaeo (8). I?e oeueiio
aaiiao?e/iee oeaio? iiiaieooieea ciaoiaeeoueny o iaaeao e?i?? ??aiy
i?i?iaeueii? aeiaaeeie.

Neeaaeai? aeai?eoie iiaoaeiae e?eaeo, ye? ia?aeaeuei? iiiaieooieeo, oa
ciaoiaeaeaiiy aaiiao?e/iiai oeaio?a oeueiai iiiaieooieea iiaeia aaaaeaoe
oi?aa?naeueieie. Aaeaea o ieo a?aenooi? iaaeie?ee, iia”ycai? c
iiia?aaei?i a?ao?/iei cia?aaeaiiyi n?i”? ia?aeaeueieo e?eaeo.

Iaaieioaii, ui o?aaeeoe?eiei iienii n?i’? e?eaeo ? iayaiee aeaeyae
F(x,y,t)=0. Cai?iiiiiaai? aeai?eoie aacothoueny ia iieno ia?aeaeueieo
e?eaeo o aeaeyae? t=O(o,o), ye? ca?aathoueny c ii?iaeueieie ooieoe?yie.
Cacia/aio caaea/o iiaeia oaeiae ?ica”ycaoe, yeui ia?aeaeuei? e?ea?
aaaaeaoe ?ica’yceii aeeoa?aioe?aeueiiai ??aiyiiy aeeiiaea.

?en.6a

Iaoae ia ?ioa?aae? a ( x ( b ia?ii e?eao y = f(x). Iaiao?aeii iienaoe
n?i”th e?eaeo, aeaiaioe yei? aoee a ia?aeaeueieie aeai?e e?ea?e.

Oaa?aeaeaiiy. Iine?aeiai?noue ooieoe?e

, (10)

aea xi = [(N-i)a + bi] / N, i?e n ( ( ca?aa?oueny aei ooioe?? z =
u(x,y), ui ? oi/iei ?ica”yceii aeeoa?aioe?aeueiiai ??aiyiiy aeeiiaea

?en.7

(11)

c a?aie/iith oiiaith u(y = f(x) =0.

Ia ?en. 7 cia?aaeaii a?ao?e ? e?i?? ??aiy ooieoe?? — iaaeeaeaiiai
?icayceo ??aiyiiy (11) c a?aie/iith e?eaith — ia?aaieith y = x2 ia
?ioa?aae? 0.. 1,8. Aeey ii??aiyiiy ia ?en. 7a cia?aaeaii a?ao?e ? e?i??
??aiy oi/iiai ?ica”yceo aeey ia?aaiee ia ?ioa?aae? -3…3

?en.7a

O?ao?e ?icae?e i?enay/aii iaoiaeo ia/eneaiiy ia?eiao??a aeaiaio?a n?i’?
ia?aeaeueieo e?eaeo. ?aeath iaoiaeo iiynieii ia i?eeeaae?. Iaoae ia?ii
e?oa ?aae?ona R, yeee “caai??any” ii e?iioe? eiea. Oiae? a iiiaio /ano t
= t0 ( R e?iieo ai??iiy, a oaeiae ieiuo, yea ua ia “cai??ea”, iiaeia
ioe?ieoe ca oi?ioeaie L(t)=2(t oa U(t)=(t2. I?e oeueiio ia? i?noea
ooiaeaiaioaeueia ni?aa?aeiioaiiy L(t) = dU(t)/dt.

. Ca ?? aeiiiiiaith iiaoaeo?ii ?ano?ia? cia?aaeaiiy n?i”? o?ao? a aeey
aeene?aoiiai ?yaeo i?ii?aee?a /ano. I?ae?aooieii /enea i?enae?a ioe?ieii
ieiu? eiaeii? c o?ao? n?i”?. A ?acoeueoao? iaea?aeeii a?aee?ee ooieoe??
U(t). Ni?aa?aeiioaiiy L(t) = dU(t)/dt aeicaiey? ciaeoe cia/aiiy ooieoe??
L(t) ca oi?ioeaie /enaeueiiai aeeoa?aioe?thaaiiy a?aeiinii oeeo
a?aee?e?a.

Oaa?aeaeaiiy. sseui e?i?? ??aiy ooieoe?? f(x,y) ? i?inoeie caieiaieie
e?eaeie ? iaeanoue S(u,v) iaiaaeaia e?i?yie f(x,y) = u ? f(x,y) = v, oi

(12)

aea F(z) — ieiua, yea iaiaaeaia e?eaeie f(x,y) = u ? f(x,y) = v.

Oea oaa?aeaeaiiy aea? iiyniaiiy aeeoa?aioe?aeueiiio ni?aa?aeiioaiith
aeey ooieoe?e, ui iienothoue ci?io ieiu ? ia?eiao??a n?i’? ia?aeaeueieo
o?ao? .

Oaa?aeaeaiiy. Iaoae a iiiaio /ano t cia/aiiy ooieoe?e U(t) e L(t)
ni?aiaaeathoue c? cia/aiiyie ieiu? o?ao?e Gt ? aeiaaeeiith eiioo?o St,
a?aeiia?aeii. Oiae? ia? i?noea ni?aa?aeiioaiiy

(U(t)/(t = L(t).
(13)

A ?iaio? iaea?aeaii ooieoe?th, yea aeicaiey? iienaoe caeii ci?ie ieiu,
ui iaiaaeai? aeaiaioaie n?i’? ia?aeaeueieo e?i?e

. (14)

Ooo L0 — aeiaaeeia e?eai? S, U0- ieiua, ui iaiaaeaia e?eaith S0. E??i
oiai, iieacaii, ui

. (15)

Ooieoe?y L a ?iaio? ciaoiaeeoueny oeyoii /enaeueiiai aeeoa?aioe?thaaiiy.

o?ao?e G.

Ca?aeii iaoiaeo ?ica’ycaiiy caaea/? Athoiia ia ieiuei? iaiao?aeii
caaeaoe iiiaeeio ia?aeaeueieo i?yieo c e?ieii a. Oiae? cia/aiiy ae?aco
p=2d/ SYMBOL 112 \f «Symbol» \s 14 a aei??aith? ?iia??iino? oiai, ui
?ici?uaiee ia ieiuei? aeiaaeeiaei /eiii a?ae??cie aeiaaeeie d

Добавить комментарий

Ваш e-mail не будет опубликован. Обязательные поля помечены *