Геометричне моделювання променевого енергообміну між аналітично заданими поверхнями: Автореф. дис… канд. техн. наук / О.В. Шоман, Київ. держ. техн.

I?I?NOA?NOAI INA?OE OE?A?IE

EEIANUeEEE AeA?AEAAIEE OAOII*IEE OIIAA?NEOAO

AOAeIAIEOeOAA I A?OIOAEOO?E

OAeE 515.2

Oiiai Ieueaa A?eoi??aia

AAIIAO?E*IA IIAeAETHAAIIss

I?IIAIAAIAI AIA?AIIAI?IO I?AE AIAE?OE*II CAAeAIEIE IIAA?OIssIE

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 iieioaoii/iiio
oiiaa?neoaoi I?i?noa?noaa ina?oe Oe?a?ie.

Iaoeiai ea?iaieee: — aeieoi? oaoii/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

E?aniieoonueeee Aiaoie?e Ieoaeeiae/,

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

aaiiao??? oa a?ao?ee

Ioioeieii iiiiaioe: — aeaaeai?e A?I Oe?a?ie,

aeieoi? oaoii/ieo iaoe, i?ioani?

Iaeaeeo Aieiaeeie? Ieoaeeiae/,

OAeAOA, ia?oee i?i?aeoi?;

— eaiaeeaeao oaoii/ieo iaoe,
aeioeaio

Caeaanueeee Aieiaeeie? Eineiiae/,

IOO “EI?”, eaoaae?a ia?enii? aaiiao???,

?iaeaia?ii? ? iaoeiii?
a?ao?ee

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

?aae?iaeaeo?ii?ee, eaoaae?a
?iaeaia?ii?

? 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 ?icineaiee 8 aeiaoiy 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. Iio?aaa o ?ic?aooieao i?iiaiaaiai aia?aiiai?io
i?ae iiaa?oiyie neeaaeii? i?inoi?iai? oi?ie aeieea? i?ae /an
aeine?aeaeaiue aaeeei? e?eueeino? 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 oaeeo
ai?iaaaeaeaiue oa eiie?aoieo nenoai iiaa?oiiue A ? A, ye? i?eeiathoue
o/anoue o i?iiaiaaiio aia?aiiai?i? (iicia/aoe oea aoaeaii oae A ( A),
iacaaii oai?aoe/i? ?ic?aooiee aeey: iaa??aaiiy eioe?a (oaeae iieoi’y (
iecea o?oa o ia’?i? ia/?); neeiaa?iiai ae?iaieoeoaa (aeaea?aei oaiea (
neeyia iana, ui canoeaa?); aiae?co iiaeaae (iiaa?oiy aiaith (
aoae?aaeuei? eiino?oeoe?? aai nii?yaeaeaiiy iiaeaaeieea); iaea/ee
eaca??a (ni??aeaiiae?aia eaiia ( oeee?iae?e/ia aeoeaia ?a/iaeia);
eini?/ieo aeine?aeaeaiue (niioea ( eiino?oeoe?? eini?/iiai aia?aoo)
oiui.

Aoaeoeai?noue /enaeueieo ?ic?aooie?a i?iiaiaaiai aia?aiiai?io aaaaoi a
/iio caeaaeeoue a?ae iiaeeeainoae iaoiaeo ia/eneaiiy eooiaeo
eiao?oe??io?a aei?ii?ithaaiiy (EEA) — oiaoi aaiiao?e/ieo oaeoi??a, ui
oa?aeoa?ecothoue /anoeo i?iiaiaaiai iioieo, yeee aei?ii?ith?oueny
iaei??th iiaa?oiath ? aeinyaa? ae?oai? iiaa?oi?. EEA /enaeueii
aei??aith? cia/aiith iiaa?oiaaiai ?ioaa?aeo niaoe?aeueiiai aeaeo. Aeey
?aaeueieo iiaa?oiiue ia/eneaiiy oaeeo ?ioaa?ae?a neeaaea? naiino?eio
i?iaeaio. Aiae?oe/iee ?ica’ycie ooo iiaeeeaee eeoa aeey iaiaaeaii?
e?eueeino? iiaa?oiiue ianeeaaeii? i?inoi?iai? oi?ie. Ia i?aeoeoe?
aeei?enoiaothoueny /enaeuei? ?ica’ycee c canoinoaaiiyi ia/enethaaeueii?
oaoi?ee. Aea i?e oeueiio aeai?eoi?/ia ?aae?caoe?y iia’ycaia c aiae?cii
cao?iaiiy iiaa?oi? — oiaoi c aeyaeaiiyi o?aaiaio?a iiaa?oi?, ye?
aoaeooue cae?eo? aai ?ioith iiaa?oiath, aai o??th ae iiaa?oiath (ye o
aeiaaeeo ni??aeaiiae?aii? iiaa?oi?). Aeina?ae iieaco?, ui oeth neeaaeiao
/anoeio aeai?eoi?a aaaeei oi?iae?coaaoe. Oiio aeioe?eueiei aoaea
?ic?iaea oaeiai iaoiaeo ia/eneaiiy EEA, yeee ae ia nie?aany ia aiae?c
naiicao?iaiiy iiaa?oiiue.

Na?aae a?ao?/ieo iaoiae?a ia/eneaiiy EEA oaaao i?eaa?oa? iaoiae noa?e
iaeeie/iiai ?aae?ona (iaoiae A. Ionaeueoa). Oeae iaoiae aeicaiey?
ia/enethaaoe cia/aiiy eieaeueieo EEA oeyoii iiaoaeiae iecee
?aae?aeueii-ia?aeaeueieo i?iaeoe?e iaei??? c iiaa?oiiue, yea aa?a
o/anoue o aia?aiiai?i?. Ine?eueee iiaoaeiaa i?iaeoe?e ? i?aaeiaoii
aeine?aeaeaiue i?eeeaaeii? aaiiao??? oa eiii’thoa?ii? a?ao?ee, oi
aeieeea aeoiea iiaeeo?eoaaoe iaoiae noa?e iaeeie/iiai ?aae?ona ca
aeiiiiiaith R-ooieoeie — ye iaoaiaoe/iiai aia?aoo, ui aeicaiey?
iienoaaoe eiiiiiaioe i?iaeoeithaaiiy. Aeei?enoaiiy R-ooieoeie aeaei
ciiao acyoeny ca ?ic?iaeo iaoiaeo neeaaeaiiy aeey AII aeai?eoio
ia/eneaiiy eieaeueieo oa ?ioaa?aeueiiai EEA uiaei i?iiaiaaiai
aia?aiiai?io i?ae iiaa?oiyie, ye? caaeai? aiae?oe/ii (oiaoi o aeaeyae?
iayaieo aai ia?aiao?e/ieo ??aiyiue).

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. Aeena?oaoe?eia
?iaioa aeeiiaia a ?aieao iaoeiai — aeine?aeii? i?ia?aie eaoaae?e
ia?enii? aaiiao??? oa ?iaeaia?ii? a?ao?ee OAeIO.

Iaoa ? caaea/? aeine?aeaeaiiy. Iaoith aeine?aeaeaiiy ? noai?aiiy
iaoiaeo ia/eneaiiy eooiaeo eiao?oe??io?a aei?ii?ithaaiiy aeey ?ic?aooieo
aia?aiiai?io i?ae iiaa?oiyie, ye? caaeaii aiae?oe/ii o aeaeyae?
??aiyiue, ia aac? iiaoaeiae caniaaie iaoeiii? a?ao?ee
?aae?aeueii-ia?aeaeueieo i?iaeoe?e iaei??? c iiaa?oiiue. Aeey
aeinyaiaiiy oe??? iaoe o aeena?oaoe?? iinoaaeaii oae? iniiai? caaea/?:

— aeyaeoe aaiiao?e/i? ?iaa??aioe aia?aiiai?io i?ae iiaa?oiyie, ui
aeicaieeoue ia/enethaaoe EEA iaoiaeii “iaa?iaiiai i?iiaiy”;

— ?ic?iaeoe i?eeiyoiee aeey ?iaeaia?ii? i?aeoeee iaoiae iiaoaeiae
?aae?aeueii-ia?aeaeueii? i?iaeoe?? aeaiiai ia’?eoa, eiee: a) ia’?eo
iienaii ??aiyiiyi o iayaiiio aeaeyae?; a) ia’?eo iienaii ??aiyiiyi o
ia?aiao?e/iiio aeaeyae?;

— neeanoe aeai?eoie ia/eneaiiy cia/aiue eieaeueieo oa ?ioaa?aeueiiai
EEA i?e aia?aiiai?i? i?ae eaiaeiaith iiaa?oiath c a?nnth o aeaeyae?
aaeioiai? e?i?? oa iaeiei aeoeii oe??? ae iiaa?oi? (caaea/a
naiiii?ii?ithaaiiy ni??ae?), aai aeaiaioii aeayei? ?ioi? iiaa?oi?
(caaea/a naiicao?iaiiy ni??ae?);

— ia?aa??eoe a??ia?aei?noue ?icaeyiooiai iaoiaeo oeyoii ?ica’ycaiiy
oanoiaeo i?eeeaae?a;

— ia/eneeoe cia/aiiy eieaeueieo oa ?ioaa?aeueiiai EEA i?ae /an
?ica’ycaiiy ?aaeueieo caaea/.

Iaoeiao iiaecio ?iaioe ia? iiaee iaoiae ia/eneaiiy eooiaeo
eiao?oe??io?a aei?ii?ithaaiiy aeey ?ic?aooieo aia?aiiai?io i?ae
aiae?oe/ii caaeaieie iiaa?oiyie, neeaaeiaeie /anoeiaie yeiai ?:

— icia/aiiy iia?aoe??? RP-i?iaeoe?thaaiiy, yea aeicaiey? iaea?aeaoe ia
ea?oeii?e ieiuei? ??aiyiiy a iayaiiio aeaeyae? ia?eno ?aae?aeueii —
ia?aeaeueii? i?iaeoe?? i?inoi?iaiai aaiiao?e/iiai ia’?eoa, iienaiiai
??aiyiiyi o iayaiiio aeaeyae?;

— oi/iee iien RP- i?iaeoe?e aeey iiaa?oiiue iaaiiai eeano;

— iaoiae ia/eneaiiy iiaa?oiaaeo ?ioaa?ae?a niaoe?aeueiiai aeaeo.

Ai?iaiaeiinoue oa iaa?oioiaai?noue iaea?aeaieo ?acoeueoao?a
i?aeoaa?aeaeo?oueny aeiaaaeaiiyi aiae?oe/ieo caeaaeiinoae oa
iiaoaeiaaieie ca aeiiiiiaith eiii’thoa?a a?ao?/ieie cia?aaeaiiyie
?aae?aeueii — ia?aeaeueieo i?iaeoe?e aeey oanoiaeo i?eeeaae?a, a oaeiae
?ic?aooieaie ?aaeueii? caaea/? aia?aiiai?io aei?ii?ithaaiiyi a i?ioean?
ai?iaaaeaeaiiy iaoiaeo a i?aeoeeo.

I?aeoe/ia cia/aiiy aeena?oaoe?? iieyaa? a iiaeeeaino? ia i? oai?aoe/i?e
aac? ?ic?iaeoe oa ai?iaaaeeoe a ?aaeueio i?aeoeeo aeai?eoie ia/eneaiiy
eooiaeo eiao?oe??io?a aei?ii?ithaaiiy aeey ?ic?aooieo aia?aiiai?io i?ae
aeia?eueieie aiae?oe/ii caaeaieie iiaa?oiyie.

?aae?caoe?y ?iaioe a?aeaoeany a iaoeiai — aeine?aeiiio oaoiieia?/iiio
?inoeooo? i?eeaaeiaoaeoaaiiy I?iiaoi?iio Oe?a?ie — aeey ?ic?aooieo
ia?aiao??a i?eeaae?a iioe/ieo nenoai ia?aaea/? ?ioi?iaoe??; a
Oe?a?inueeiio iaoeiai-aeine?aeiiio oa eiino?oeoi?nueeiio ?inoeooo?
o?i?/iiai iaoeiiaoaeoaaiiy — aeey ?ic?aooieo aaiiao?e/ieo ia?aiao??a
o?i?/ieo oaieiiai?iiee?a; ia i?aei?e?inoa? “ssNII LTD” — aeey ?ic?aooieo
ia?aiao??a onoaoeoaaiiy aeey aeaioiaeaiiy ?cieyoe?? aeaeo?e/ieo ae?io?a.
?aae?caoe?? i?aeoaa?aeaeothoueny aeia?aeeaie i?i iiaeeea?noue
aeei?enoaiiy cai?iiiiiaaii? o ?iaio? iaoiaeeee.

Iniaenoee aianie caeiaoaa/a iieyaa? a ?ic?iaoe? oai?aoe/ii? aace oa a
neeaaeaii? aeai?eoi?a ia/eneaiiy eooiaeo eiao?oe??io?a aei?ii?ithaaiiy
aeey ?ic?aooieo aia?aiiai?io i?ae nenoaiith ia’?eo?a, iaeiei c yeeo ?
ni??aeaiiae?aia iiaa?oiy.

Iaoiaeeea aeine?aeaeaiue. A ?iaio? aeei?enoiao?oueny iaoaiaoe/iee
aia?ao R-ooieoeie, ui aea? ciiao iienoaaoe i?inoi?ia? aaiiao?e/i?
ia’?eoe oa ?o ?aae?aeueii-ia?aeaeuei? i?iaeoe?? ??aiyiiyie o iayaiiio
aeaeyae?. Canoiniaothoueny iieiaeaiiy i?eeeaaeii? aaiiao???, oai???
i?iiaiaaiai aia?aiiai?io, /enaeueieo iaoiae?a.

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

— a aaeoc? aiae?oe/iiai iiaeaethaaiiy ia’?eoia — A.?. Ieoaeeaiea,
A.A.Iaaeiaa, A.I.Iaeaeeoa, I.E.Iiaeai?iiai, A.I.Iiaeei?eoiaa,
N.I.Eiaaeueiaa, I.A.Neeaeaia, TH.I.Aaaea?aa, A.E.?aa/iaa, TH.A.Noiyia;

— a aaeoc? i?iiaiaaiai aia?aiiai?io — A.A.A?aiaea?/oea, I.A.?oaoeiaa,
TH.A.No??iiaa, A.A.Eaaaae?aa, I.I.Iaeueiaia, A.I. I?ia?aa,
N.E.?aoaoiyea, A.Oai??i.

Ia caoeno aeiinyoueny iieiaeaiiy, ye? neeaaeathoue iaoeiao iiaecio
?acoeueoao?a aeine?aeaeaiue, a oaeiae i?ia?aiia caaacia/aiiy aeey
ia/eneaiiy EEA i?e aia?aiiai?i? i?ae aiae?oe/ii caaeaieie iiaa?oiyie.

Ai?iaaoe?y ?iaioe. Iniiai? iieiaeaiiy aeena?oaoe?eii? ?iaioe
aeiiia?aeaeenue oa iaaiai?thaaeenue ia 3-e oa 4-e I?aeia?iaeieo
iaoeiai-i?aeoe/ieo eiioa?aioe?yo «No/ani? i?iaeaie aaiiao?e/iiai
iiaeaethaaiiy» (i.Iaeioiiieue; 1996 oa 1997 ??.), ia iaoeiai-iaoiaee/i?e
eiioa?aioe?? “?iaeaia?ia a?ao?ea oa aaiiao?e/ia iiaeaethaaiiy ?c
canoinoaaiiyi eiii’thoa?ii? oaoiieia??” (i. ??aia; 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.

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 aeei?enoaieo aeaea?ae ?c 164
iaeiaioaaiue oa aeiaeaoeo. ?iaioa i?noeoue 145 noi??iie iaoeiiieniiai
oaenoo oa 56 ?enoie?a, iiaoaeiaaieo ca aeiiiiiaith eiii’thoa?a.

CIINO ?IAIOE

O anooii ?ice?eaa?oueny nooi?noue ? noai iaoeiai? i?iaeaie oa ??
cia/ou?noue, iaa?oioiao?oueny aeooaeueiinoue oaie aeineiaeaeaiue,
i?iaaaeaii e?eoe/iee iaeyae eioa?aoo?ieo aeaea?ae, noi?ioeueiaaii iaoo
oa caaea/i aeena?oaoeieiii ?iaioe, ?? iaoeiao iiaecio oa i?aeoe/ia
cia/aiiy.

O ia?oiio ?icae?e? aeaii icia/aiiy eooiaeo eiao?oe??io?a
aei?ii?ithaaiiy (EEA) oa ?icaeyiooi ?nioth/? a?ao?/i? iaoiaee ?o
ia/eneaiiy.

?en.1. Noaia aia?aiiai?io

i?ae iiaa?oiyie

Ia ?en.1 cia?aaeaii noaio aia?aiiai?io i?ae aeaiia iiaa?oiyie —
aei?ii?ithth/ith (A) ? ni?eeiath/ith (A). Aeaa?aii ia iiaa?oiyo A ? A
aeaiaioa?i? ae?eyiee dA ? dA. Iicia/eii /a?ac ( ? ( aino?? eooe i?ae
ii?iaeyie aei ae?eyiie dA ? dA ? iai?yiii aei?ii?ithaaiiy, a /a?ac rMN —
a?aenoaiue i?ae oeaio?aeueieie oi/eaie M ? N ae?eyiie dA ? dA
a?aeiia?aeii.

I?e aea/aii? ?aae?aoe?eiiai aia?aiiai?io i?ae ?aaeueieie iiaa?oiyie
aaeeea cia/aiiy ia? oaeoi? aca?iii? aeaeiino? oi/ie M ? N ia iiaa?oiyo A
? A. I?eionea?oueny, ui oi/ee M ? N ? aca?iii aeaeei?; oiaoi a?ae??cie
MN ia iiaeiai iaoe c iiaa?oiyie A ? A ?ioeo oi/ie, a?aei?iieo a?ae M ?
N.

sse a?aeiii, a?aeiioaiiy iioieo aei?ii?ithaaiiy, ui aeoiaeeoue c
ae?eyiee dA ? iaaea? ia iiaa?oith A, aei anueiai iioieo, ui
aei?ii?ith?oueny ae?eyieith dA o i?anoa?o, caaoueny eieaeueiei EEA ?
aecia/a?oueny cia/aiiyi iiaa?oiaaiai ?ioaa?aeo

. (1)

I?e neeaaeaii? aeai?eoio ia/eneaiiy oaeeo ?ioaa?ae?a iaiao?aeii
a?aoiaoaaoe oaeo iaeaaeiino? oi/ee N aei ae?eyiee iiaa?oi? A, caa?iaii?
aei A. Oea aeoaea aeooaeueii o aeiaaeeo iaiioeei? iiaa?oi? A (aeae?
aoaea ?icaeyiooi iaeio c oaeeo iiaa?oiiue — aaeioiao ni??aeue). Oiio a
ea?oth/ee aeai?eoi eiiiiiaioii iiaeiai aoiaeeoe aeai?eoi aiae?co
naiicao?iaiiy iiaa?oi?. Aea, ye iieacothoue aeine?aeaeaiiy c
eiii’thoa?ii? a?ao?ee, i?ioean aeyaeaiiy ae?eyiie naiicao?iaiiy iiaa?oi?
iiaaii oi?iae?co?oueny, a ?acoeueoao? /iai ?ioaa?oaaiiy iiaeia
cae?enieoe eeoa ca aeiiiiiaith iaeiaoaeoeaiiai “e?ieoaaiiy” ii iiaa?oi?
A.

(ooo A a ciaiaiieeo icia/a? aaee/eio ieiu? ae?eyiee iiaa?oi? A,
caa?iaii? aei iiaa?oi? A). O oeueiio aeiaaeeo iaiao?aeii i?iaaaeeoe
aiae?c naiicao?iaiiy iiaa?oi? A aeey eiaeii? oi/ee ae?eyiee iiaa?oi? A,
caa?iaii? aei A. Eaaei c?icoi?oe iaia?niaeoeai?noue aeai?eoi?a, ye?
aacothoueny ia no?aoaa?? “e?ieoaaiiy oi/eaie”, ui ?icoaoiaai? ia aca?iii
caa?iaieo ae?eyieao ?aaeueieo iiaa?oiiue A ? A (oiaoi ia no?aoaa??
iiaiiai ia?aai?o oi/ie).

Iniaeeai oea noino?oueny “o?oaiiiae?aieo” eaiaeiaeo iiaa?oiiue, ui
iaeaaeaoue, iai?eeeaae, aei aaeioiaeo ni??aeae (?en.2). Ae?enii, a
oeueiio aeiaaeeo aeaye? aeoee ni??ae? aoaeooue cao?ithaaoe o?aaiaioe
?ioeo aeoe?a. Aeey aecia/aiiy oaeeo ae?eyiie iiaa?oiiue iaiao?aeii
neeaaeaoe aeai?eoi, eiii’thoa?ia ?aae?caoe?y yeiai iio?aao? ia?iaee
aaeeeiai ia’?io a?ao?/ii? ?ioi?iaoe??. I?e oeueiio ia iiaa?oi? aaeioiai?
ni??ae? iaiao?aeii aecia/eoe iiiaeeio oi/ie, ui aoaeooue iaaeaeei?
a?aeiinii aeayei? oi/ee. A ?iaio? cacia/aii, ui oey iiiaeeia oi/ie
neeaaea?oueny c oaeeo iaaeaeeieo oi/ie: i) oeo, ui ?icoaoiaai? “ca
ia???i” iiaa?oi? aaeioiai? ni??ae?, a oaeiae ii) oeo, ui cae?eo? aeoeaie
oe??? ae eaiaeiai? iiaa?oi?.

?en.2. Aoaeo naiicao?iaiiy

?en.3. Iiaoaeiaa ?aae?aeueii-

ia?aeaeueii? i?iaeoe??

Aeey ia/eneaiiy EEA ia i?aeoeoe? aeioe?eueii canoiniaoaaoe
oi?aa?naeueiee iaoiae noa?e iaeeie/iiai ?aae?ona, yeee
cai?iiiioaaa A. Ionaeueo. Nooue iaoiaeo iieyaa? a oaeiio. Nii/aoeo
aeaa?aii ieiueio (, ui ? aeioe/iith aei aeaiaioa dA ? i?ioiaeeoue /a?ac
eiai oeaio?aeueio oi/eo. C oe??? ae oi/ee aoaeo?ii i?anoa?o iaeeie/iiai
?aae?ona, ui nie?a?oueny ia ieiueio (. C oeaio?o i?anoa?e ia ??
iiaa?oith ?aae?aeueii i?iaeoe?th?ii caaeaio iiaa?oith A. Io?eiaio
?aae?aeueio i?iaeoe?th iiaa?oi? A i?oiaiiaeueii ni?iaeoe?th?ii ia
aeioe/io ieiueio (. A ?acoeueoao? iaea?aeeii ?aae?aeueii-ia?aeaeueio
i?iaeoe?th iiaa?oi? A (nei?i/aii RP-i?iaeoe?th A).

Oaa?aeaeaiiy 1. *enaeueia cia/aiiy eieaeueiiai EEA (1) aei??aith?
a?aeiioaiith ieiu? RP-i?iaeoe?? iiaa?oi? A aei ieiu? aaeeeiai e?oaa
noa?e iaeeie/iiai ?aae?ona.

Ioaea, aeey ia/eneaiiy eieaeueiiai EEA iaoiaeii noa?e iaeeie/iiai
?aae?ona iaiao?aeii ioe?ieoe ieiuo RP-i?iaeoe??. A ?iaioao E.I. Eooeaiea
i?iiiio?oueny oeth ioe?ieo cae?enieoe ia aac? iieno RP-i?iaeoe??
iiaa?oi? A o aeaeyae? ia??aiino? f (x, y) ( 0. Aeey oeueiai ne?ae ia?aoe
aeaea?oiao nenoaio eii?aeeiao Oxyz c ii/aoeii a oeaio?aeuei?e oi/oe?
aeaiaioa dA, eii?aeeiaoio ieiueio Oxy noi?noeoe c ieiueiith (, a a?nue
Oz iai?aaeoe o iai?yi? ciai?oiuei? ii?iae? aei aeaiaioa dA (?en. 3).
Oiae? iath/e ia oaac? ciae ooieoe?? f, caniaaie eiii’thoa?ii? a?ao?ee
iiaeia iiaoaeoaaoe ?ano?iaa cia?aaeaiiy RP-i?iaeoe??, yea noi?uo?oueny c
?ano?iaei cia?aaeaiiyi e?oaa iaeeie/iiai ?aae?ona. Iaaeeaeaia cia/aiiy
eieaeueiiai EEA aoaea aei??aithaaoe a?aeiioaiith e?eueeino? ?ano?iaeo
oi/ie, ui neeaaeathoue cia?aaeaiiy RP-i?iaeoe?? oa e?oaa iaeeie/iiai
?aae?ona.

I?iaaaeaiee e?eoe/iee aiae?c e?oa?aoo?ieo aeaea?ae o aaeoc? noai?aiiy
iaoiaeo ia/eneaiiy eooiaeo eiao?oe??io?a aei?ii?ithaaiiy aeey ?ic?aooieo
aia?aiiai?io i?ae aiae?oe/ii caaeaieie iiaa?oiyie ia aac? iiaoaeiae
caniaaie eiii’thoa?ii? a?ao?ee ?aae?aeueii-ia?aeaeueieo i?iaeoe?e
iaei??? c iiaa?oiiue, iieacaa, ui iaae??oaieie ? ieoaiiy:

— aeyaeaiiy aaiiao?e/ieo ?iaa??aio?a aia?aiiai?io i?ae iiaa?oiyie, ui
aeicaieeoue ia/enethaaoe EEA cia/ii i?ino?oa — iaoiaeii “iaa?iaiiai
i?iiaiy”;

— ?ic?iaee i?eaeaoiiai aeey ?iaeaia?ii? i?aeoeee iaoiaeo iiaoaeiae
?aae?aeueii-ia?aeaeueii? i?iaeoe?? caaeaiiai ia’?eoa, eiee:

a) ia’?eo iienaii ??aiyiiyi o iayaiiio aeaeyae?;

a) ia’?eo iienaii ??aiyiiyi o ia?aiao?e/iiio aeaeyae?;

— neeaaeaiiy aeai?eoio ia/eneaiiy cia/aiue eieaeueieo oa ?ioaa?aeueiiai
EEA i?e aia?aiiai?i? i?ae eaiaeiaith iiaa?oiath c a?nnth o aeaeyae?
aaeioiai? e?i?? oa iaeiei aeoeii oe??? ae iiaa?oi? (caaea/a
naiiii?ii?ithaaiiy ni??ae?), aai aeaiaioii aeayei? ?ioi? iiaa?oi?
(caaea/a naiicao?iaiiy ni??ae?);

I?aeaeaii e?eoe/iiio aiae?co a?aeiiee aeai?eoi ia/eneaiiy eieaeueieo
EEA, cai?iiiiiaaiee I.I.?oaoeiaei oa A.I.Eaaaae?aei. Na?aae iaaeie?e?a
iii?/aii, ui aeai?eoi

— ia ia?eoia ?ica’ycaiiy i?iaeaie naiicao?iaiiy iiaa?oi?;

— ia ? aaeeoeaiei a?aeiinii eia?/ieo iia?aoe?e iaae iiaa?oiyie, ye?
i?eeiathoue o/anoue o i?iiaiaaiio oaieiiai?i?;

— iio?aao? nooo?aiai ei?aeooaaoe i?ney aaiiao?e/iiai ?aaeaaoaaiiy
iiaa?oiiue, ye? i?eeiathoue o/anoue o aia?aiiai?i?.

Aeuaiaaaaeaia aeaco? ia aeooaeuei?noue ia?aii? oaie ?iaioe.

A ae?oaiio ?icae?e? aeena?oaoei? ?icaeyaeathoueny oai?aoe/ii iniiae
iaoiaeo noa?e iaeeie/iiai ?aae?ona, yeee iiaeeo?eiaaii ca aeiiiiiaith
R-ooieoe?e.

A ?iaio? canoiniaaii oae? icia/aiiy. Caa?iaieie o?aaiaioaie AI ? AI
iiaa?oiiue A ? A iacaaii ?o ae?eyiee, ye? neeaaeathoueny c? aca?iii
aeaeeieo oi/ie. Caa?iai? o?aaiaioe neeaaeathoueny c /anoeiai ? iiai?noth
caa?iaieo o?aaiaio?a aeo?aeieo iiaa?oiiue. *anoeiai caa?iaieie
o?aaiaioaie A* ? A* iiaa?oiiue A ? A iacaaii ?o ae?eyiee, c oi/ie yeeo
?ioo iiaa?oith aeaeii ia iiai?noth. Iiai?noth caa?iaieie o?aaiaioaie AI
? AI iiaa?oiiue A ? A iacaaii ?o ae?eyiee, c oi/ie yeeo ?ioo iiaa?oith
aeaeii iiai?noth. Iaaieioaii ia oiio, ui a e?oa?aoo?? c i?iiaiaaiai
aia?aiiai?io oaeee aaiiao?e/iee aiae?c ia i?iaaaeeoueny, oiio aeae? i?ae
A oa A aoaeaii ?icoi?oe caa?iai? ae?eyiee iiaa?oiiue.

Cacia/aii, ui i?e ?ic?aooieo i?iiaiaaiai aia?aiiai?io aaeeea cia/aiiy
ia? a?aeiioaiiy iioieo aei?ii?ithaaiiy aei u?eueiino? iioieo, ui
iaaeoiaeeoue c iiaa?oi? A o iaaeieeoi?e i?ino??. Oiaoi IA-B=A(FA-B, aea
A — ieiua ae?eyiee iiaa?oi? A, caa?iaii? aei iiaa?oi? A, a FA-B —
?ioaa?aeueiee EEA. Aaee/eia IA-B ia? ?ici??i?noue ? o?ce/ia cia/aiiy
aca?iii? iiaa?oi? aca?iiae?th/eo iiaa?oiiue A ? A. Ieiua oe??? iiaa?oi?
ia/eneth?oueny ca aeiiiiiaith oi?ioee IA-B = A ( FA-B. Aca?iia iiaa?oiy
aeei?enoiao?oueny a ?ioaa?aeuei?e aaiiao??? (i??a E?iooiia). O ca’yceo c
oeei aca?iio iiaa?oith IA-A iiaeia ?icaeyaeaoe ye i??o /ioe?eaei??ii?
iiiaeeie i?iiai?a, ui ia?aoeiathoue aeia?eueii i???ioiaai? o i?inoi??
iiaa?oi? A ? A. Inoaii? aeoiaeeoue c aecia/aiiy iiaa?oiiue o?e, ui
iai?iththoueny i?iiaiyie, ye i?? aeaiaei??ii? iiiaeeie oi/ie, ye?
yaeythoue niaith aeaea?aea aiiioeaio?e/ieo io/e?a i?iiai?a. A?aoiaoth/e
oea, a ?iaio? i?eeiyoi i?eiouaiiy, ca?aeii yeiai IA-B = IA-A, aai

A ( FA-B = A ( FB-A..
(2)

Ooo A oa A — ieiu? ae?eyiie aca?iicaa?iaieo iiaa?oiiue, a FA-B i FA-A —
?ioaa?aeuei? EEA. Cacia/aii, ui ni?aa?aeiioaiiy IA-B = IA-A a oai???
aia?aiiai?io aei?ii?ithaaiiyi iiaeia aeei?enoiaoaaoe ye aaiiao?e/io
oa?aeoa?enoeeo aei?ii?ithth/eo nenoai, ui caeeoa?oueny iaci?iiith i?e
ci?i? iai?yio ia?aiino aei?ii?ithaaiiy (oiaoi ye aaiiao?e/iee ?iaa??aio
aei?ii?ithaaiiy).

?icaeyiooee ?iaa??aio aeicaiey? a?aoiaoaaoe aaiiao?e/i? iniaeeaino?
aia?aiaei?ii?ithth/eo ? aia?aini?eeiath/eo iiaa?oiiue ? i?iaiaeeoe
e?euee?ni? ?ic?aooiee aia?aiiai?io. Cie?aia, i?eeii ci?ie iai?yio
aei?ii?ithaaiiy c?o/ii aeei?enoiaoaaoe a aeai?eoiao ?ic?aooieo
i?iiaienoiai aia?aiiai?io o aeiaaeeo, yeui iiaa?oiy A ?
“o?oaiiiae?aiith” eaiaeiaith aaeioiaith ni??aeeth. I?aee?aneaii, ui aeey
?iaeaia?ieo ?ic?aooie?a aaee/eio ieiu? aca?iii? iiaa?oi? iaea?aeothoue
ia?aaaaeii ia iniia? aenia?eiaio?a. Oiio aeioe?eueieie aoaeooue
aeine?aeaeaiiy a aaeoc? oai??? /enaeueieo iaoiae?a ia/eneaiiy ieiu?
aca?iii? iiaa?oi? aeey aiae?oe/ii caaeaieo aia?aiaei?ii?ithth/eo ?
aia?aici?eeiath/eo iiaa?oiiue.

?en.4. Iiaoaeiaa RP-i?iaeoe??

noa?e/ii? iiaa?oi?

) = 0.

Oaa?aeaeaiiy 2. ??aiyiiy RP-i?iaeoe?? noa?e c ?aae?onii r ? c oeaio?ii
a oi/oe? (a, b, c) ia? aeaeyae

. (3)

Eii?aeeiaoe RP-i?iaeoe?? oeaio?a noa?e ia/eneththoueny ca oi?ioeaie

Cacia/aii, ui o aeayeeo ie?aieo aeiaaeeao ? iiaeeea?noue ciaeoe oi/iee
iien RP-i?iaeoe??. Iiaa eaea i?i aeaa aeiaaeee: a) iiaa?oiy (A
ae?iaeaeo?oueny o o?ao?o, ui caaeaia ia aeia?euei?e ieiuei? ??aiy; a)
iiaa?oiy (A ? iiaa?oiath ae?oaiai ii?yaeeo.

Oaa?aeaeaiiy 3. Iaoae o?ao?a A caaeaia ia ieiuei? ??aiy x = xp
??aiyiiyi F(y, z) = 0. Oiae? ??aiyiiy RP-i?iaeoe?? A ia? aeaeyae

. (4)

Oaa?aeaeaiiy 4. Iaoae o?ao?o A caaeaii ia ieiuei? ??aiy z = zp
??aiyiiyi F(x, y) = 0. Oiae? ??aiyiiy RP-i?iaeoe?? A iaoeia aeaeyae

(5)

Oaa?aeaeaiiy 5. ??aiyiiy RP-i?iaeoe?? iiaa?oi? ae?oaiai ii?yaeeo

P(x,y,z)=a11x2+a22y2+a33z2+2a12xy+2a13xz+

2a23yz+2a1x+2a2y+2a3z+a=0

ia? aeaeyae

(6)

I?eeeaae. ??aiyiiy eii?/ii? iiaa?oi? c aa?oeiith a oi/oe? (a, 0, b) ?
iienaii? iaaeiei ae?ini?aea iaa?oaiiy

( x2 +(y — y0)2 ) / r f2 + z2/ hf2 = 1

ia? aeaeyae

F (x, y, z)( (a (x-a)/rf2+b (z-b)/hf2) 2 —
(7)

— ((x-a) 2/rf2+(y-y0) 2/rf2 +(z-b) 2/hf2) (a2/rf2+b2/hf2-1) =0.

aoaea ??aiyiiyi RP — i?iaeoe?? ae?ini?aea, aea ooieoe?th F iaaeaii
ae?acii (7).

Oaa?aeaeaiiy 6. Iaoae iiaa?oiy A ia? ni?euei? oi/ee c i?anoa?ith c
?aae?onii, ui aei??aith? iaeeieoe? (?en.5). Oiae? RP-i?iaeoe?y e?i??
ia?aoeio cacia/aieo iiaa?oiiue iiaea iienoaaoeny ??aiyiiyi

) = 0. (8)

?en.5. I?anoa?a

?en.6. Iiaoaeiaa aiiioao??

aaiiao?e/iiai ia’?eoa

Oaa?aeaeaiiy 7. Iaoae iiaa?oiy A: F(x,y,z) = 0 ?icoaoiaaia i?ae
aeaiia eiioeaio?e/ieie i?anoa?aie c ?aae?onaie tmin i tmax ? c
oeaio?aie, ui ni?aiaaeathoue c ii/aoeii eii?aeeiao (?en.6). Iaoae oaeiae
iiaa?oiy A i?aeeyaa? ia?aoai?aiith aiiioao?? a?aeiinii ii/aoeo
eii?aeeiao. C ?yaeo aiiioao?e iaa?aii aea? “oace” — iiaa?oi? C i D,
aeioe/i? aei

?en.7.Aaiiao?e/i? ia?aiao?e

oi?ioee (11)

i?anoa?e c ?aae?onii, ui aei??aith? iaeeieoe?, a?aeiia?aeii ana?aaeei? ?
ciai?. Oiae? C: F(xtmin, ytmin, ztmin) = 0, D: F(xtmax, ytmax, ztmax) =
0.

C oaa?aeaeaiiy 7 aeoiaeeoue ianooiia aaaeeeaa iieiaeaiiy: yeui
ci?ithaaoe ia?aiao? t a ?ioa?aae? [tmin, tmax], oi ??aiyiiyi F(xt, yt,
zt) = 0 , aoaeooue iienai? aiiioao?? iiaa?oi? A, ui “i?ioiaeyoue e??cue”
i?anoa?o c ?aae?onii, ui aei??aith? iaeeieoe?, a?ae “oace” N aei “oace”
D (?en.6).

Oaa?aeaeaiiy 8. Iaoae iiaa?oiy B: F(x,y,z) = 0 ?icoaoiaaia i?ae
eiioeaio?e/ieie noa?aie c ?aae?onaie tmin i tmax (tmin < tmax) a?aeiia?aeii c oeaio?aie a ii/aoeo eii?aeeiao. Oiae? ia ieiuei? 0xy iine?aeiai?noue ??aiyiue , (9) aea ti = [(n - i) tmin + itmax] / n i?e n ( ( ca?aa?oueny aei ??aiyiiy f(x,y) = 0 RP-i?iaeoe?? iiaa?oi? A. Aeae? ?icaeyiooi iien eaiaeiai? iiaa?oi?, o yei? a?nnth ? aaeioiaa e?i?y (ni??aeue). I?e oeueiio aaeioiao e?i?th caaeaii ??aiyiiyie (10) Ooo t - aeiaaeeia aeoae e?i??, h=2(B - e?ie aaeioa, A - ?aae?on “iaiioee” aaeioiai? e?i??. ??aiyiiy aaeioiai? ni??ae? iaoeia aeaeyae (11) . Ooo r > 0 — ?aae?on “ae?ioo” ni??ae?. I?e oeueiio ne?ae a?aoiaoaaoe, ui
eii?aeeiaoai (t, () oa (t, (+2() a?aeiia?aea? iaeia ? oa ae oi/ea ia
ni??ae?, oa ui iiaeiia aeeiioaaoenue oiiaa Ar < A2 + B2. ?io? aaee/eie, ye? aoiaeyoue aei oi?ioee (11), ?ethno?othoueny ia ?en.7. O o?aoueiio ?icae?e? iaaaaeaii aeai?eoi ia/eneaiiy eieaeueieo EEA aeey ni??aeaiiae?aii? iiaa?oi?. ?icaeyiooi aeai?eoi aeey aiae?ca caiicao?iaiiy ni??ae?. Cacia/aii, ui aeia?eueia oi/ea I ia iiaa?oi? aaeioiai? ni??ae? (?en.2) aoaea aeaeeiith c oi/ee niinoa?aaeaiiy E o ?ac? aeeiiaiiy aeaio oiia: i) yeui aeiaeaoia cia/aiiy ia? eineion eooa i?ae aaeoi?ii ii?iae? aei iiaa?oi? a oi/oe? I oa aaeoi?ii IE; ii) yeui i?yea noa?a c oeaio?ii ia aaeioia?e e?i?? ia ia?aoeia? a?ae??cie IE. Iieacaii, ui oiiao ii) aeioe?eueii cai?ieoe ?ioith oiiaith, yea aaco?oueny ia iiiyoo? ii?iaeueii? ooieoe?? a?ae??cea: ii*) yeui aeiaeaoiei aoaea cia/aiiy ae?aco f(xT, yT, zT )-r, aea r>0 —
?aae?on eaiaeo aaeioiai? ni??ae?, f(x,y,z) — ii?iaeueia ooieoe?y
a?ae??cea IE, O(xT, yT, zT ) — oi/ea ia aaeioia?e e?i??.

O ?iaio? aeei?enoaii oaeo ii?iaeueio ooieoe?th a?ae??cea, yeee
niieo/a? oi/ee I(xM,yM,zM) oa E(xK,yK,zK):

, (12)

;

.

Aeey iai/iino? aeaeei? oi/ee cai?iiiiiaaii cia?aaeoaaoe ia
inaaaei?icai?ooe? aaeioiai? ni??ae?. Inaaaei?icai?oeo iaoeiaii oiae?,
eiee “?ic??caoe” iiaa?oith aaeioiai? ni??ae? ca “ia?eae?aiii” ( = 0, a
iio?i oiiaii ?icai?iooe ?ic??caio iiaa?oith ia ieiueio. Ia ?en. 8
iaaaaeaii i?eeeaae ?icai?oee aaeioiai? ni??ae? (e?eueoeyie iicia/aii
iaaeaeei? oi/ee).

?en. 8. Inaaaei?icai?oea 4 — 9-ai aeoe?a iiaa?oi?
ni??ae?

(e?eueoeyie iicia/aii iaaeaeei? oi/ee)

“a”

?en. 9. Iai/i? cia?aaeaiiy aaeioiai? ni??ae? a caeaaeiino?

a?ae a?aenoai? niinoa?aaeaiiy: “a” — 70; “a” — 120; “a” —
210.

sse i?eeeaae, ia ?en. 10 iaaaaeaii cia?aaeaiiy RP-i?iaeoe?? ni??ae?,
a?nue yei? ?icoaoiaaia ia aeaye?e a?aenoai? ia?aeaeueii o?e ieiuei?, ia
yeo nie?a?oueny i?anoa?a iaeeie/iiai ?aae?ona.

?en.10. ?ano?iaa cia?aaeaiiy RP-i?iaeoe?? aaeioiai? ni??ae?

Iicia/eii /a?ac W anth iiaa?oith aaeioiai? ni??ae?, a /a?ac V
-iiaa?oith iaeiiai c ?? aeoe?a. Aeaiaioa?i? ae?eyiee oeeo iiaa?oiiue
iicia/eii ye dW oa dV. Iaoae r — a?aenoaiue, yea aei??th?oueny acaeiaae
ieoo?aiai iai?yio aei?ii?ithaaiiy, ui a?aeiia?aea? ae?eyieai dW oa dV.
*a?ac ( oa ( iicia/eii aino?? eooe, ye? ooai?th? iai?yi c
ia?iaiaeeeoey?aie aei iiaa?oi? ni??ae?, i?iaaaeaieie /a?ac oeaio?aeuei?
oi/ee dW oa dV. I??o naiiii?ii?ithaaiiy ni??ae? aoaeaii aecia/aoe oeyoii
ia/eneaiiy ?ioaa?aeueiiai eooiaiai eiao?oe??ioa aei?ii?ithaaiiy (EEA)

, (13)

aea aioo??oi?e ?ioaa?ae ( iicia/eii eiai FdV-W ) aecia/a? eieaeueiee EEA
i?ae aeaiaioa?iith ae?eyieith dV iiaa?oi? iaeiiai aeoea ni??ae? ? an??th
?? iiaa?oiath W.

Eieaeueiee EEA ia/enethaaany oeyoii iiaoaeiae caniaaie eiii’thoa?ii?
a?ao?ee ?aae?aeueii — ia?aeaeueii? i?iaeoe?? ni??ae? W ia ieiuei?, ui ?
aeioe/iith aei iiaa?oi? aeoea V a oeaio?aeuei?e oi/oe? aeaiaioa?ii?
ae?eyiee dV.

?ioaa?ae aeey ciaoiaeaeaiiy ieiu? aeaiaioa dV iiaa?oi? ni??ae? ?
ae?ioe/iei ? oi/ii ia ia/eneth?oueny. Oiio aeaiaio iiaa?oi? ni??ae? aoa
cai?iaiee ia aeaiaio iiaa?oi? oi?a (?en.11). Ciaeaeaii oi?ioee aeey
ia/eneaiiy ieiu? iiaa?oi? oi?a i?ae aeaiia eiai ia?aeaeyie.

I?e ia/eneaii? ?ioaa?aeueieo EEA i?eionea?oueny, ui RP-i?iaeoe??
iiaa?oi? ni??ae? ? iaci?iieie aeey on?o dV, aeey yeeo ia?aiao? t
ci?ith?oueny o iaaeao “a?ae i?ion i?aaeoea aei iethn i?aaeoea”.

Cia/aiiy ?ioaa?aeueiiai EEA ciaoiaeeeinue ca oi?ioeith

?en.11. Ai?ieneiaoe?y

aeoea ni??ae?

iiaa?oiath oi?a

(14)

— cia/aiiy eieaeueiiai EEA aeey aeaiaioa dSi c oeaio?aeueiith
oi/eith, ?icoaoiaaiith ia iiaa?oi? ii?iaeueii? ieiueie i?e t=0.

Iaaaaeaii iien aeai?eoio ia/eneaiiy ?ioaa?aeueieo EEA.

1. Iae?a?oueny cia/aiiy ia?aiao?a aaeioiai? e?i?? (iai?eeeaae, t=0) .
Oeei caaea?oueny ii?iaeueia n?/ia ieiueia eaiaeo ni??ae?.

2. Ia n?/i?e ieiuei? aecia/a?oueny eoo (i (ii/aoeiaee eoo (1=0).
Ia/eneththoueny a?aeiia?aei? aei oeueiai eooa eii?aeeiaoe oi/ee (x,y,z)
? eii?aeeiaoe {xA, yA, zA}aaeoi?a ii?iae? aei iiaa?oi? ni??ae?.

3. O nenoai? eii?aeeiao Oxyz ni??aeue ia?aiineoueny oae, uia oi/ea (x,
y, z) ia ?? iiaa?oi? ca?aeany a c ii/aoeii eii?aeeiao.

?en. 12. RP-i?iaeoe?? ni??ae? a?aeiinii oi/ie ?? aeoea

4. Oeyoii iaa?oaiiy ni??ae? o i?inoi?? aeinyaa?oueny ca?a iai?yio
aaeoi?a {xA, yA, zA} c a?nnth Oz.

caiai’yoiao?oueny.

6. Eooo (1 iaaea?oueny i?e??no (2 = (1 + (, ?, ii/eiath/e c ioieoo 2
oeueiai aeai?eoio, an? iia?aoe?? iiaoi?ththoueny. sseui aaee/eia eooa (E
aeyaeoueny a?eueoith, i?ae 3600, oi ne?ae ia?aeoe aei aeeiiaiiy i.7.

7. Ia/eneththoueny cia/aiiy dSi, ui a?aeiia?aeathoue eooai (i.

8. Ca oi?ioeith (14) ia/eneth?oueny cia/aiiy ?ioaa?aeueiiai EEA.

Ia ?en.12 cia?aaeaia iecea RP-i?iaeoe?e iiaa?oi? ni??ae? aeey eooa( o
iaaeao 0 ( ( ( 3600 c e?ieii 300. Ia?aiao?e ni??ae?: e?euee?noue aeoe?a
— 21, ?aae?on eaiaeo — r = 1, ?aae?on iaiioee in? ni??ae? — A = 10,
eiao?oe??io naiicao?iaiiy ni??ae? (oiaoi a?aeiioaiiy aaee/eie e?ieo
ni??ae? aei ?aae?ono ?? eaiaeo) — 5. Ia ?en.13 iaaaaeaii a?ao?ee
caeaaeiino? ?ioaa?aeueiiai EEA a?ae eiao?oe??ioa naiicao?iaiiy ni??ae?.
O iaaaaeaiiio i?eeeaae? aeine?aeaeaii naiiii?ii?ithaaiiy iaeeiaaeoeyoiai
aeoea.

?en. 13. Caeaaei?noue

?ioaa?aeueiiai EEA a?ae w

Aeai?eoie ia/eneaiiy eieaeueiiai EEA aacothoueny ia ciaoiaeaeaii?
a?aeiioaiiy cia/aiue ieiu RP-i?iaeoe?? iiaa?oi? ? e?oaa c ?aae?onii
iaeeieoey. I?e oeueiio iaeiny ia oaac?, ui a?aeiiee iien ?? RP-i?iaeoe??
??aiyiiyi o aeaeyae? f(x,y) = 0. Ioaea, an? oi/ee ?ano?o o e?oc? c
?aae?onii, ui aei??aith? iaeeieoe?, ca aeiiiiiaith ooieoe?? f(x,y)
iiaeia ?icae?eeoe ia aeaa eeane — o?, ui iaeaaeaoue, ? o?, ui ia
iaeaaeaoue aei RP-i?iaeoe??.

Oea eaaei c?iaeoe, yeui a?aoiaoaaoe ciae ooieoe?? f(x,y). Iaoae aeey
?ano?o c e?ieii ( cia?aaeaiiy RP-i?iaeoe?? iiaa?oi? ni??ae? ? e?oaa
iaeeie/iiai ?aae?ona neeaaeathoueny, c M( ? N( i?enae?a. Oiae? cia/aiiy
eieaeueiiai EEA aei??aith? F(dA-A=M(/N(. Aeae? ia/eneaiiy
iiaoi?ththoueny aeey ?ano?o c e?ieii (/2. ?ic?aooiie eieaeueiiai EEA
i?eieiy?oueny, yeui aoaea aeeiioaaoeny ia??ai?noue

, (15)

aea (-caaeaia aaee/eia.

A /aoaa?oiio ?icae?e? iaaaaeaii i?eeeaaee ai?iaaaeaeaiiy iaoiaeo
ia/eneaiiy EEA. Cie?aia, iaaaaeaii aeai?eoie ?ic?aooieo aia?aiiai?io
i?ae eiie?aoieie aaiiao?e/ieie ia’(eoaie oeio aaeioiaa ni??aeue (
iiaa?oiy oeee?iae?a..

?en. 14. Nenoaia iiaa?oiiue

aaeioiaa ni??aeue ( oeee?iae?

Ni?eueiith iciaeith aeey ?icaeyiooeo caaea/ ? oa, ui aia?a?th (oaieiao,
na?oeiao, oiui) a ieo aei?ii?ith? aaeioiaa ni??aeue (oiaoi iiaa?oiy, ui
ia? oi?io eaiaeiai? iiaa?oi?, o yei? a?nnth ? aaeioiaa e?i?y), a
ni?eeiath/ith ? oeee?iae?e/ia iiaa?oiy (?en. 14). Ia?oueny ia oaac?, ui
aia?a?th o iaaeieeoi?e i?ino?? aei?ii?ith? any iiaa?oiy ni??ae?.
Iaiao?aeii ioe?ieoe /anoeo aia?a??, yea iio?aiey? ia iaaio ae?eyieo
iiaa?oi? oeee?iae?a.

E?ei oiai, i?e ?ic?aooieao iaiao?aeii a?aoiaoaaoe, ui aeaye? aeoee
ni??ae? aoaeooue ae?aioaaoe aei?ii?ithaaiiy ?ioeo aeoe?a. Oea
ni?e/eiyoeia ia?i? aeo?aoe aia?a?? aei?ii?ithaaiiy. Ioaea, ia i?aeoeoe?
iaiao?aeii ioe?ieoe /anoeo aia?a??, yea iio?aiey? ia iiaa?oith iaeiiai c
iaia?aae ia?aieo aeoe?a oe??? ae ni??ae?.

Inoaiiy caaea/a ia? aaaeeeaa i?aeoe/ia cia/aiiy. Oiio i?e ?ic?aooieao
iaiao?aeii i?iaaaeeoe aaiiao?e/iee aiae?c naiicao?iaiiy aeoe?a aaeioiai?
ni??ae?. ?ica’ycaiiy oe??? caaea/? aeicaiey? caanoe aei i?i?ioio aeo?aoe
aia?a?? oeyoii ?aoe?iiaeueiiai aeai?o aaiiao?e/ieo ia?aiao??a
eiiiiiaio?a aia?aiiai?io.

A ?iaio? iaea?aeai? cia/aiiy caeaaeiinoae ?ioaa?aeueiiai eooiaiai
eiao?oe??ioa aei?ii?ithaaiiy a?ae eiao?oe??ioa naiicao?iaiiy ni??ae?.

Na?aae eiie?aoieo caaea/, ye? ?icaeyiooi o ?iaio?, iacaaii

— ?ic?aooiie i?iiaiaaiai aia?aiiai?io o iioiaieieiiieo e?i?yo ca’yceo;
iieacaii, yeee eieaeueiee eooiaee eiao?oe??io aei?ii?ithaaiiy iaiao?aeii
?ic?aooaaoe, aeey oiai uia caaacia/eoe iioiaieieio oe?ieo nioao
i?iioneaiiy neaiaeo;

— ?ic?aooiie ?aaeeio iie?ia?ecaoe?? ?cieyoe?eiiai iie?eooy aeaeo?e/ieo
ae?io?a; iieacaii, ui ooo iaiao?aeii i?iaanoe aaiiao?e/iee aiae?c
naiicao?iaiiyi aeoe?a i?o?iiiai? ni??ae?, ui ? aeaea?aeii oaiea.

— ?ic?aooiie i?iiaiaaiai oaieiiai?io aeey oaieiianiiai?iieo aa?aaao?a
o?i?/iiai ae?iaieoeoaa.

Iaaieioo?oueny, ui aeena?oaoe?y i?enay/aia aaiiao?e/iei, a ia
o?ceei-oaoiieia?/iei ieoaiiyi uiaei eiino?oeoe?e onoaoeoaaiiy. A i?e
eeoa aeiaiaeeoueny aoaeoeai?noue ?icaeyiooeo aeai?eoi?a ia/eneaiiy
eooiaeo eiao?oe??io?a aei?ii?ithaaiiy o a?aeiia?aeieo ai?iaaaeaeaiiyo

AENIIAEE

A ?iaioi aeeiiaii aeineiaeaeaiiy o aaeoci aaiiao?e/iiai iiaeaethaaiiy
i?iiaiaaiai aia?aiiai?io i?ae iiaa?oiyie neeaaeii? i?inoi?iai? oi?ie.
Cie?aia, noai?aii oai?aoe/io iniiao eiii’thoa?ieo i?ia?ai ia/eneaiiy
eooiaeo eiao?oe??io?a aei?ii?ithaaiiy (EEA) aeey ?ic?aooieo aia?aiiai?io
i?ae iiaa?oiyie, ye? caaeaii aiae?oe/ii o aeaeyae? ??aiyiue. I?e oeueiio
iaea?aeaii oaei ?acoeueoaoe, ui iathoue iaoeiao oa i?aeoe/io oeiiiinoue.

1. Aeyaeaii aaiiao?e/i? ?iaa??aioe i?ioeano aia?aiiai?io i?ae
iiaa?oiyie, ui aeicaiey? ia/enethaaoe EEA iaoiaeii “iaa?iaiiai i?iiaiy”.

2. ?ic?iaeaii i?eeiyoiee aeey ?iaeaia?ii? i?aeoeee iaoiae iiaoaeiae
?aae?aeueii-ia?aeaeueii? i?iaeoe?? aeaiiai ia’?eoa, eiee:

a) ia’?eo iienaii ??aiyiiyi o iayaiiio aeaeyae?;

a) ia’?eo iienaii ??aiyiiyi o ia?aiao?e/iiio aeaeyae?.

3. Cai?iiiiiaaii iaoiae ia/eneaiiy ?ioaa?ae?a niaoe?aeueiiai aeaeo
(i??e E?iooiia).

4. ?ic?iaeaii iaoiae ?ic?aooieo eieaeueieo EEA.

5. Neeaaeaii aeai?eoie ia/eneaiiy cia/aiue eieaeueieo oa ?ioaa?aeueiiai
EEA i?e aia?aiiai?i? i?ae oaeeie nenoaiaie iiaa?oiiue:

a) eaiaeiaa iiaa?oiy c a?nnth o aeaeyae? aaeioiai? e?i?? ( aeaiaio
aeayei? iiaa?oi?;

a) eaiaeiaa iiaa?oiy c a?nnth o aeaeyae? aaeioiai? e?i?? ( iaeei aeoie
oe??? ae iiaa?oi? (caaea/a naiiii?ii?ithaaiiy ni??ae?);

6. ?ic?iaeaii iaoiae ?ic?aooieo i?iiaiaai? aia?a??, ui aeinyaa? oi/ee
ciai? ni??ae? c o?aooaaiiy aoaeoo ?? naiicao?iaiiy.

7. ?ica’ycaii ?yae oanoiaeo i?eeeaaeia ia/eneaiiy EEA.

8. ?acoeueoaoe ?iaioe ai?iaaaeaeaii: a) a iaoeiai — aeine?aeiiio
oaoiieia?/iiio ?inoeooo? i?eeaaeiaoaeoaaiiy I?iiaoi?iio Oe?a?ie -aeey
?ic?aooieo aaiiao?e/ieo ia?aiao??a i?eeaae?a iioe/ieo nenoai ia?aaea/?
?ioi?iaoe??; a) a Oe?a?inueeiio iaoeiai-aeine?aeiiio oa
eiino?oeoi?nueeiio ?inoeooo? o?i?/iiai iaoeiiaoaeoaaiiy — aeey
?ic?aooieo aaiiao?e/ieo ia?aiao??a o?i?/ieo oaieiiai?iiee?a; a) ia
i?aei?e?inoa? “ssNII LTD” — aeey ?ic?aooieo aaiiao?e/ieo ia?aiao??a
onoaoeoaaiiy aeey aeaioiaeaiiy ?cieyoe?? aeaeo?e/ieo ae?io?a.

Iniiaii iieiaeaiiy aeena?oaoei? iioaeieiaaii o oaeeo ?iaioao:

1. Eooeaiei E.I., Oiiai I.A. I?eaeeaeaiiue iaoiae au/eneaiey eieaeueiuo
oaeiauo eiyooeoeeaioia eceo/aiey // I?eeeaaeia aaiiao??y oa ?iaeaia?ia
a?ao?ea. — Ee?a: EAeOOAA, 1996. — Aei. 60. — N. 46-49.

2. Oiiai I.A. Au/eneaiea oaeiauo eiyooeoeeaioia i?e yia?aiiaiaia
iaaeaeo o?oaiia?aciuie iiaa?oiinoyie // O?. 4-e Iaaeae. iao/ii-i?aeoe/.
eiio. “Nia?aiaiiua i?iaeaiu aaiiao?e/aneiai iiaeaee?iaaiey”. —
Iaeeoiiieue: OAAOA, 1997. — *. 3. — N. 43-48.

3. Oiiai I.A. Aaiiao?e/aneea eiaa?eaiou oai?ee eo/aaiai yia?aiiaiaia
iaaeaeo iiaa?oiinoyie // O?. Oaa?e/aneie ainoaea?noaaiiie
aa?ioaoie/aneie aeaaeaiee. — Iaeeoiiieue: OAAOA, 1997. — Aui. 4.
I?eeeaaeiay aaiiao?ey e eiaeaia?iay a?aoeea. — O. 1. — N. 99-102.

4. Oiiai I.A. Aiae?c aeaeeiino? oi/ie, ye? ?icoaoiaai? ia eaiaeia?e
o?oaiiiae?ai?e iiaa?oi? // Ca. i?. I?aeia?. iaoeiai-i?aeoe/ii? eiio.
“No/ani? i?iaeaie aaiiao?e/iiai iiaeaethaaiiy”. — Oa?e?a: O?IA IAN
Oe?a?ie, 1998. — *. 2. — N. 69-73.

5. Oiiai I.A. Aeine?aeaeaiiy naiiii?ii?ithaaiiy eaiaeiai? iiaa?oi?, o
yei? a?nnth ? aaeioiaa e?i?y. // I?eeeaaeia aaiiao??y oa ?iaeaia?ia
a?ao?ea. — Ee?a: EAeOOAA, 1998. — Aei. 63. — N. 213-216.

Oiiai I.A. Aaiiao?e/ia iiaeaethaaiiy i?iiaiaaiai aia?aiiai?io i?ae
aiae?oe/ii caaeaieie iiaa?oiyie. — ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa oaoi?/ieo iaoe
ca niaoe?aeuei?noth 05.01.01 — i?eeeaaeia aaiiao??y, ?iaeaia?ia
a?ao?ea. — Ee?anueeee aea?aeaaiee oaoi?/iee oi?aa?neoao aoae?aieoeoaa ?
a?o?oaeoo?e, Oe?a?ia, Ee?a, 1998.

Aeena?oaoe?th i?enay/aii ?ic?iaoe? aaiiao?e/iiai iaoiaeo ia/eneaiiy
eooiaeo eiao?oe??io?a aei?ii?ithaaiiy (EEA) uiaei i?iiaiaaiai
aia?aiiai?io i?ae aiae?oe/ii caaeaieie iiaa?oiyie. A iaoiae? cae?eniaii
iiaoaeiao ?aae?aeueii-ia?aeaeueieo i?iaeoe?e (RP-i?iaeoe?e) iiaa?oiiue
caniaaie eiii’thoa?ii? a?ao?ee. Eooia? eiao?oe??ioe aei?ii?ithaaiiy
io?eiaii aeey aeiaaee?a aia?aiiai?io i?ae eaiaeiaeie iiaa?oiyie.
Aeai?eoie ?aae?ciaaii o aeaeyae? PASCAL-i?ia?ai. A iaoiae? a?aoiaaii
iayai?noue oa?aeoa?iiai aeey eaiaeiaeo iiaa?oiiue aoaeoo naiicao?iaiiy.
?ica’ycaii caaea/o naiiii?ii?ithaaiiy aaeioiai? eaiaeiai? iiaa?oi? aeey
aeoea. ?acoeueoaoe aeena?oaoe?? ciaeoee ai?iaaaeaeaiiy a iaoeiaeo
aeine?aeaeaiiyo iioe/ieo nenoai ia?aaea/? ?ioi?iaoe??, a ?ic?iaeao
no/anieo oaieiiai?iiee?a, o i?iaeooaaii? onoaiiaie iie?ia?ecaoe??
?cieyoe?eiiai iie?eooy ae?io?a.

Eeth/ia? neiaa: aaiiao?e/ia iiaeaethaaiiy, eooiaee eiao?oe??io
aei?ii?ithaaiiy (EEA), ?aae?aeueii-ia?aeaeueia i?iaeoe?y (RP-i?iaeoe?y),
aaeioiaa eaiaeiaa iiaa?oiy.

Oiiai I.A. Aaiiao?e/aneia iiaeaee?iaaiea eo/aaiai yia?aiiaiaia iaaeaeo
aiaeeoe/anee caaeaiiuie iiaa?oiinoyie. — ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa oaoie/aneeo iaoe
ii niaoeeaeueiinoe 05.01.01 — i?eeeaaeiay aaiiao?ey, eiaeaia?iay
a?aoeea. — Eeaaneee ainoaea?noaaiiue oaoie/aneee oieaa?neoao
no?ieoaeuenoaa e a?oeoaeoo?u, Oe?aeia, Eeaa, 1998.

Aeenna?oaoeey iinayuaia nicaeaieth aaiiao?e/aneiai iaoiaea ii?aaeaeaiey
oaeiauo eiyooeoeeaioia eceo/aiey (OEE) i?e eo/aaii yia?aiiaiaia iaaeaeo
aiaeeoe/anee caaeaiiuie iiaa?oiinoyie. A iaoiaea inouanoaeaii iino?iaiea
?aaeeaeueii-ia?aeeaeueiuo i?iaeoeee (RP-i?iaeoeee) iiaa?oiinoae
n?aaenoaaie eiiiuethoa?iie a?aoeee. Oaeiaua eiyooeoeeaiou eceo/aiey
iieo/aiu aeey neo/aaa yia?aiiaiaia iaaeaeo eaiaeiauie iiaa?oiinoyie.
Aeai?eoiu ?aaeeciaaiu a aeaea PASCAL-i?ia?aii. A iaoiaea o/oai
oa?aeoa?iue aeey eaiaeiauo iiaa?oiinoae yooaeo naiicaneiiaiey. ?aoaia
caaea/a naiiiaeo/aiey aeioiaie eaiaeiaie iiaa?oiinoe aeey aeoea.
?acoeueoaou aeenna?oaoeee iaoee i?eiaiaiea a iao/iuo enneaaeiaaieyo
iioe/aneeo nenoai ia?aaea/e eioi?iaoeee, a ?ac?aaioeao nia?aiaiiuo
oaieiiaiaiieeia, a i?iaeoe?iaaiee onoaiiaie iieeia?ecaoeee
ecieyoeeiiiiai iie?uoey i?iaiaeia.

Eeth/aaua neiaa: aaiiao?e/aneia iiaeaee?iaaiea, oaeiaie eiyooeoeeaio
eceo/aiey (OEE), ?aaeeaeueii-ia?aeeaeueiay i?iaeoeey (RP-i?iaeoeey),
aeioiaay eaiaeiaay iiaa?oiinoue.

Shoman O.V. Geometrical modelling of energy exchange by radiation
between surfaces presented by analytical form. — Manuscript.

Thesis for a candidate’s degree by speciality 05.01.01 — applied
geometry, engineering graphics. — Kyiv State Technical University of
Building and Architecture, Ukraine, Kyiv, 1998.

The dissertation is devoted to the creation of geometrical method for
view factor definition during radiation energy exchange between surfaces
presented by analytical form. In this method construction of radial
parallel projections (RP-projections) of surfaces has been carried out
by means of computer graphics. View factors have been obtained for cases
of energy exchange between channel surfaces. Algorithms have been
realized as PASCAL-programs. Typical channel surfaces effect of hiding
parts of surface by itself has been taken into account in the method.
The self-irradiation problem of screw channel surface has been solved
for the whole turn. The results of the work have found an utility in the
scientific investigations of optical systems of transmitting
information, in elaborating modern thermal energy exchange devices, in
the design of wire isolation polymerisation units.

Key words: geometrical modelling, view factor, radial parallel
projection (RP-projection), screw channel surface.

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