Геометричне моделювання перебігу променів в еліптичних та параболічних відбивальних системах: Автореф. дис… канд. техн. наук / Н.І. Середа, Київ. на

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

EE?ANUeEEE IAOe?IIAEUeIEE OI?AA?NEOAO

AOAe?AIEOeOAA ? A?O?OAEOO?E

NA?AAeA IAOAE?ss ?AAI?AIA

Oaee 515.2

AAIIAO?E*IA IIAeAETHAAIIss IA?AA?AO I?IIAI?A

A AE?IOE*IEO OA IA?AAIE?*IEO

A?AeAEAAEUeIEO NENOAIAO

Niaoe?aeuei?noue 05.01.01 —

I?eeeaaeia aaiiao??y, ?iaeaia?ia a?ao?ea

AAOI?AOA?AO

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa oaoi?/ieo iaoe

Ee?a — 1999

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia a Oa?e?anueeiio aea?aeaaiiio iie?oaoi?/iiio
oi?aa?neoao? I?i?noa?noaa ina?oe Oe?a?ie.

Iaoeiaee ea??aiee: — aeieoi? oaoi?/ieo iaoe, i?ioani?

Eooeaiei Eaii?ae Ieeieaeiae/,

i?ioani? eaoaae?e Iiaeaaeii? oaoi?ee,

Oa?e?anueeee ?inoeooo iiaeaaeii?

aaciaee IAN Oe?a?ie

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

Iaeaeeo Aiae??e Aieiaeeie?iae/,

caa?aeoaa/ eaoaae?e I?eeeaaeii? iaoaiaoeee ?

ia/enethaaeueii? oaoi?ee,

Oaa??enueea aea?aeaaia aa?ioaoi?/ia aeaaeai?y;

eaiaeeaeao oaoi?/ieo iaoe, aeioeaio

Aii?eiaiaa A??a Iienei?aia,

aeioeaio eaoaae?e Ia?enii? aaiiao???,

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

Ee?anueeee iaoe?iiaeueiee oi?aa?neoao

aoae?aieoeoaa ? a?o?oaeoo?e

I?ia?aeia i?aai?caoe?y: — Aeiiaoeueeee aea?aeaaiee oaoi?/iee

oi?aa?neoao, eaoaae?a ia?enii? aaiiao???

oa ?iaeaia?ii? a?ao?ee.

Caoeno a?aeaoaeaoueny “______”___________ 1999 ?. i ___ aiaeei? ia
can?aeaii? niaoe?ae?ciaaii? a/aii? ?aaee Ae 26.056.06 o Ee?anueeiio
iaoe?iiaeueiiio oi?aa?neoao? aoae?aieoeoaa i a?o?oaeoo?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 a?ae?ioaoe? Ee?anueeiai iaoe?iiaeueiiai oi?aa?neoaoo
aoae?aieoeoaa i a?o?oaeoo?e ca aae?anith: 252037 Ee?a — 37,
Iiaio?ioeionueeee i?ini., 31.

Aaoi?aoa?ao ?ic?neaiee “_______”______________ 1999
?.

A/aiee nae?aoa?

niaoe?ae?ciaaii? ?aaee

A.I. Ieineee

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. A Iaoe?iiaeuei?e i?ia?ai? ?ioi?iaoecaoe??
iaaieioaii, ui noaiiaeaiiy ae?iaie/iai iioaioe?aeo Oe?a?ie iaiiaeeeaa
aac noai?aiiy ?ioi?iaoe?eiiai caaacia/aiiy nenoai aaoiiaoeciaaiiai
i?iaeooaaiiy. Oea noino?oueny ? i?iaeooaaiiy a?aeaeaaeueieo nenoai, ui
aacothoueny ia o?ce/iiio aoaeo? a?aeaeooy i?iiai?a a?ae e?eai? aai
iiaa?oi? ca caeiiii Aeaea?oa-Niaeiona. A?aeaeaa/? eiioeaio?othoue a
caaeaieo oi/eao i?inoi?o i?iiai?, ye? ia?aeiieai? ?o a?aeaeaaeueieie
iiaa?oiyie. I?eeeaaeaie a?aeaeaaeueieo iiaa?oiiue ? aeca?eaea a
iioe/iiio i?eeaaeiaoaeoaaii?, neeai?iiy noaeue a a?o?oaeoo?i?e
aeonoeoe?, aioaii? eiino?oeoe?? a ?aae?ioaeaneiiao, oieonoth/a
iaeaaeiaiiy a aae?ionoaiiaeao. Aoaeo a?aeaeooy i?iiaiy a?ae e?eai? (aai
iiaa?oi?) a?aoiao?oueny oaeiae a oai??? iaoaiaoe/iiai a?euey?aea oa a
/eneaiieo ?? ai?iaaaeaeaiiyo.

A?aeaeaaeueio nenoaio neeaaeathoue “aeca?eaeueia” e?eaa (iiaa?oiy) oa
aeaea?aei i?iiai?a (ia?aaaaeii oi/eiaa). O yeino? a?aeaeaaeueieo
cae/aeii aeei?enoiaothoueny e?ea? (iiaa?oi?), ui ooai?ai? ia aac?
ae?ioe/ieo oa ia?aaie?/ieo caeaaeiinoae. I?e i?iaeooaaii? i?aoeec?eieo
i?eeaae?a aaiiao?e/io oi?io a?aeaeaaeueii? e?eai? (iiaa?oi?) ne?ae
iae?aoe c o?aooaaiiyi aeaio aaiiao?e/ieo oa?aeoa?enoee: o?iioo a?aeaeoi?
oaee? oa eaoaeaonoeee (ea?noiai? iaeano?). Oea caaacia/o? oaeineiiaeaiiy
i?aoeec?eieo i?eeaae?a uiaei ??aiii??ii? ?ioaineaiino? oa niiaoaciinoi
iioieo a?aeaeoeo i?iiai?a, oa oieonoaaiiy i?iiai?a ia oieaeuei?e i?yi?e
ca?aeii caaeaiiai caeiio ?iciiae?eo ?o u?eueiino?.

Cia/iee aeeaae a ?ica’ycaiiy i?iaeaie ?ic?aooieo a?aeaeaaeueieo nenoai
c?iaeee I.E.Iiaeai?iee oa eiai o/i?: ?.A.Aieioeia, A.I.Aenioeueeee,
I.O.Aeai?aoeueeee, ?.?.Eiaaeaiei, TH.A.Eicae, I.?.Ni?na?aiei,
A.I.*a?iieia oa ?i. Noi?aei? c oeei ieoaiiy ?icaeyaeaee A.I.Aii?eiaiaa,
A.A.Aai?i, E.A.Aaeiooaei, I.Aa?oeaa?aa?, N.I.Eiaaeueia, A.?.Ieoaeeaiei,
A.I.Iaeaeeo, A.N.Iaooiaa, A.A.Iaaeia, I.?.Naaeeaoeueea, ?.A.Neeaeai oa
?i. Iaeiae i?iaaaeai? aeine?aeaeaiiy ia aeicaieythoue aiai?eoe i?i
noai?aiiy iane??ciiai ?ioi?iaoe?eiiai caaacia/aiiy i?iaeooaaiiy
a?aeaeaaeueieo nenoai. Cie?aia oea noino?oueny i?io?ethaaiiy
ae?ini?aeiiiae?aieo a?aeaeaaeueieo iiaa?oiiue, ye? i?ecia/ai? aeey
“??aiii??iiai” ina?oethaaiiy a?ae??ceo oieaeueii? i?yii?, oa ?ic?aooieo
ieineeo ia?aaie?/ieo oieonaoi??a. Iaei??th c i?e/ei oeueiai, ia iao
iiaeyae, aoea a?aenooi?noue iaoaiaoe/ieo i?ioeani??a, ui aeicaieythoue
i?iaaaeeoe aeine?aeaeaiiy ia aiae?oe/iiio ??ai?. ?iaioe E.I.Eooeaiea oa
I.Ae.Iaco?aiei i?iaeiaaeeee aeine?aeaeaiiy a iai?yieo noai?aiiy
?ioi?iaoe?eii? aace aaoiiaoeciaaiiai i?iaeooaaiiy a?aeaeaaeueieo
iiaa?oiiue ia iniia? aaiiao?e/iiai iiaeaethaaiiy ia?aa?ao i?iiai?a o
i?inoi?? eiie?aoii? a?aeaeaaeueii? nenoaie caniaaie iaoaiaoe/ieo
i?ioeani??a (oaeeo ye Maple V R4/R5, Mathematica 3.0, Derive 4.0).

Ioaea, aeey ?ic?aooieo a?aeaeaaeueieo nenoai iaiao?aei? aeai?eoie
aaiiao?e/iiai iiaeaethaaiiy eaoaeaonoee oa o?iio?a oaeeue o
a?aeaeaaeueieo nenoaiao, ui aecia/a? i?aaeiao aeine?aeaeaiue o aaeoc?
i?eeeaaeii? aaiiao??? oa ? iaoith aeine?aeaeaiue aeaii? aeena?oaoe?eii?
?iaioe.

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. ?iaioa aeeiiaia
a ?aieao eiioeaioe?? Iaoe?iiaeueii? i?ia?aie ?ioi?iaoecaoe??
“Iaoeiai-aeine?aei? caeeaaee iiaeii? noai?eoe ? ai?iaaaeeoe ?ioaa?iaai?
?ioi?iaoe?ei? oaoiieia?? iiaoaeiae ?iciiae?eaieo aac ciaiue ? aenia?oieo
nenoai, iniiaaieo ia aaoiiaoe/iiio oi?ioaaii? ciaiue aeey ae??oaiiy
aaaeei oi?iae?co?ieo caaea/ ii ??ciei i?aaeiaoiei iaeanoyi”. ?iaioa
aeeiioaaeanue ca?aeii c ieaiii iaoeiaeo ?ia?o, ye? aaaeooueny a
Oa?e?anueeiio aea?aeaaiiio iie?oaoi?/iiio oi?aa?neoao? ia eaoaae??
ia?enii? aaiiao??? ? ?iaeaia?ii? a?ao?ee.

Iaoa ? caaea/? aeine?aeaeaiiy. Iaoith aeine?aeaeaiiy ? ?ic?iaea
iaoiae?a iieno oa aeai?eoi?a iiaoaeiae cia?aaeaiue eaoaeaonoee ? o?iio?a
a?aeaeoeo oaeeue ia iniia? aaiiao?e/iiai iiaeaethaaiiy ia?aa?ao i?iiai?a
o a?aeaeaaeueieo nenoaiao aeey i?io?ethaaiiy ae?ini?aeiiiae?aieo
a?aeaeaaeueieo iiaa?oiiue, ye? caaacia/othoue “??aiii??ia” ina?oeaiiy
a?ae??ceo oieaeueii? i?yii?, oa aeey ?ic?aooieo ieineeo ia?aaie?/ieo
oieonaoi??a.

Aeey aeinyaiaiiy aieiaii? iaoe aeine?aeaeaiue o aeena?oaoe??
iinoaaeai? oae? iniiai? caaea/?:

1) aeey aeaii? a?aeaeaaeueii? nenoaie ?ic?iaeoe iaoiae iieno: n?i’?

o?iio?a oaeeue, ui a?aeaeoi e?eaith (iiaa?oiath); n?i’? a?aeaeoeo
i?iiai?a oa eaoaeaonoeee e?eai?;

2) neeanoe aeai?eoie iiaoaeiae cia?aaeaiue: n?i’? o?iio?a oaee?, ui
a?aeaeoi e?eaith (iiaa?oiath); eaoaeaonoeee e?eai?; a?aeaeaaeueii?
e?eai? a caeaaeiino? a?ae caaeaii? eaoaeaonoeee ia iniia? iiiyooy
a?aaieueaaioe;

3) cai?iiiioaaoe iaoiae neeaaeaiiy aiae?oe/ieo iien?a o?iio?a oaeeue oa
eaoaeaonoee caniaaie iaoaiaoe/iiai i?ioeani?a Maple V;

4) noai?eoe aeai?eoie o?anoaaiiy i?iiai?a iaoaiaoe/ieo a?euey?ae?a aeey
eiea, ae?ina, eiia?iiaaieo iaeanoae oa aeey ae?ini?aea;

5) ?ica’ycaoe aea? ?aaeuei? caaea/? i?iaeooaaiiy a?aeaeaaeueieo nenoai:
aecia/eoe i?io?eue ae?ini?aeiiiae?aii? a?aeaeaaeueii? iiaa?oi?, yea a
aeicaieeea “??aiii??ii” ina?oethaaoe a?ae??cie oieaeueii? i?yii?, oa
?ic?iaeoe iaoiae ?ic?aooieo ieineiai ia?aaie?/iiai oieonaoi?a.

Iaoeiao iiaecio ?iaioe neeaaeathoue iaoiaee iieno:

— n?i’? o?iio?a oaeeue, ui a?aeaeoi e?eaith (iiaa?oiath);

— eaoaeaonoeee e?eai? (aeey aeaea?aea i?iiai?a ? o iaaeani?e oi/oe?);

— a?aeaeaaeueii? e?eai? a caeaaeiino? a?ae caaeaii? eaoaeaonoeee.

— o?anoaaiiy i?iiai?a iaoaiaoe/ieo a?euey?ae?a aeey eiea, ae?ina,
eiia?iiaaieo iaeanoae oa aeey ae?ini?aea;

— eaoaeaonoeee aeey ieineiai ia?aaie?/iiai oieonaoi?a.

A??ia?aei?noue ?acoeueoao?a i?aeoaa?aeaeo?oueny aeiaaaeaiiyie
oaa?aeaeaiue oa a?ao?/ieie cia?aaeaiiyie o?iio?a oaeeue ? eaoaeaonoee
aeey oanoiaeo i?eeeaae?a, a oaeiae ?ic?aooieaie ?aaeueii?
a?aeaeaaeueii? iiaa?oi? a i?ioean? ai?iaaaeaeaiiy iaoiaeo a i?aeoeeo.

I?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoao?a. Aeeeaaeai? a aeena?oaoe??
?acoeueoaoe aeine?aeaeaiue ? iaoeiaith iniiaith aeey i?aeoe/iiai
aeei?enoaiiy ?ioi?iaoe?eiiai caaacia/aiiy i?iaeooaaiiy a?aeaeaaeueieo
nenoai ia iniia? no/anieo iaoaiaoe/ieo i?ioeani??a. Iaea?aeai?
?acoeueoaoe aeicaieythoue noai?thaaoe oa ai?iaaaeaeoaaoe a ?aaeueio
i?aeoeeo aeai?eoie i?io?ethaaiiy: o?iio?a a?aeaeoeo oaeeue oa
eaoaeaonoee a caeaaeiino? a?ae oi?ie a?aeaeaaeueii? e?eai? (iiaa?oi?);
oa a?aeaeaaeueieo e?eaeo (iiaa?oiiue) a caeaaeiino? a?ae eaoaeaonoeee.

Ai?iaaaeaeaiiy ?acoeueoao?a ?iaioe aeeiiaii a Iaoeiai-aeine?aeiiio
oaoiieia?/iiio ?inoeooo? i?eeaaeiaoaeoaaiiy (IAeO?I, i. Oa?e?a) i?e
i?iaeooaaii? a?aeaeaaeueieo iiaa?oiiue no/anieo ae?ia?a o aaeoc?
eaca?ii? oaoi?ee. ?aae?caoe?y 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 o?iio?a a?aeaeoeo oaeeue
(aea??aoeaeo?a), eaoaeaonoee oa o?anoaaiiy iaoaiaoe/ieo a?euey?ae?a.
Eiie?aoiee aianie aei iaoeiaeo i?aoeue iieyaa? a ?ic?iaoe?
oi?aa?naeueiiai iaoiaeo aaiiao?e/iiai iiaeaethaaiiy ia?aa?ao i?iiai?a o
a?aeaeaaeueieo nenoaiao ?c canoinoaaiiyi iaoaiaoe/iiai i?ioeani?a Maple
V, a aeiaaaeai? oaa?aeaeaiue noiniaii iieno o?iio?a a?aeaeoeo oaeeue oa
eaoaeaonoee.

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??. Iniiai? iieiaeaiiy
aeena?oaoe?eii? ?iaioe aeiiia?aeaeeny oa iaaiai?thaaeenue ia:

— iaoeiai — i?aeoe/ieo eiioa?aioe?yo «No/ani? i?iaeaie aaiiao?e/iiai
iiaeaethaaiiy» (i. Iae?oiiieue, 1996 ?.,1997 ?.,1998 ?.,1999 ?.);

— I?aeia?iaei?e iaoeiai — i?aeoe/i?e eiioa?aioe?? «No/ani? i?iaeaie
aaiiao?e/iiai iiaeaethaaiiy» (i. Oa?e?a, 1998 ?.);

— iaoeiaiio nai?ia?? eaoaae?e ia?enii? aaiiao???, ?iaeaia?ii? ?
eiii’thoa?ii? a?ao?ee EIOAA i?ae ea??ai. A.?.Ieoaeeaiea (Ee?a, 1999?.);

— iaoeiaiio nai?ia?? eaoaae?e ia?enii? aaiiao??? IOOO “EI?” i?ae
ea??ai. A.A.Iaaeiaa (Ee?a, 1998 ?.);

— iaoeiaiio nai?ia?? eaoaae?e i?eeeaaeii? aaiiao??? ? ?iaeaia?ii?
a?ao?ee OOO?A i?ae ea??ai. A.I.Oea/aiea (Oa?e?a, 1999 ?.).

Ioae?eaoe??. Ca ?acoeueoaoaie iaoeiaeo aeine?aeaeaiue iioae?eiaaii 14
?ia?o (c ieo 3 noaoo? o aeaeaiiyo, ye? ?aeiiaiaeiaaii AAE Oe?a?ie).

No?oeoo?a i ianya ?iaioe. Aeena?oaoe?y neeaaea?oueny ?c anooio,
/ioe?ueio ?icae?e?a, aeniiae?a, nieneo e?oa?aoo?e ?c 125 iaeiaioaaiue
oa aeiaeaoe?a. ?iaioa i?noeoue 138 noi??iie iniiaiiai oaenoo oa 46
?enoie?a, iiaoaeiaaieo ca aeiiiiiaith eiii’thoa?a.

CI?NO ?IAIOE

Anooi i?noeoue caaaeueio oa?aeoa?enoeeo ?iaioe. Iaa?oioiaaia
aeooaeuei?noue oaie aeine?aeaeaiue, Noi?ioeueiaai? iaoa oa caaea/?
aeine?aeaeaiue. Iieacaia iaoeiaa iiaecia ? i?aeoe/ia oe?ii?noue
io?eiaieo ?ica’yce?a.

O ia?oiio ?icae?e? ?icaeyiooi aaiiao?e/i? aniaeoe eiino?othaaiiy
a?aeaeaaeueieo nenoai. Iacaaii caaea/?, ui aeieeathoue o i?ioean?
eiino?othaaiiy a?aeaeaaeueieo nenoai. Na?aae caaea/ iacaaii aaiiao?e/ia
iiaeaethaaiiy n?i’? a?aeaeoeo i?iiai?a, o?iio?a a?aeaeoeo oaeeue
(aea??aoeaeo?a) oa eaoaeaonoee a?aeaeoeo i?iiai?a. Iaaaaeaii ?acoeueoaoe
I.Ae.Iaco?aiei, aea iienaii iaoiae aeine?aeaeaiiy oi?ie a?aeaeaa/?a,
yeee ia?aii ye i?ioioei aeaii? ?iaioe. Iaaeaii e?eoeeo uiaei
i?iaeooaaiiy a?aeaeaaeueieo nenoai aac aaiiao?e/iiai iiaeaethaaiiy
ia?aa?ao i?iiai?a. Caaaeyee iai/iiio eiio?ieth aaiiao?e/ia iiaeaethaaiiy
aeicaiey? onoiooe iiaeeea? iiieeee i?e i?iaeooaaii?. Iaaaaeaii eiie?aoi?
i?eeeaaee aeioe?eueiino? aaiiao?e/iiai iiaeaethaaiiy ia?aa?ao i?iiai?a o
a?aeaeaaeueieo nenoaiao. Iaeei ?c ieo noino?oueny iiyniaiiy ia?aaeieno
uiaei “ni?inooaaiiy” ae?oaiai caeiio oa?iiaeeiai?ee, ca?aeii yeiio a
oa?ii?cieueiaai?e nenoai? o?e oaiei ia iiaea ia?aaeaaaoeny o iai?yieo
a?ae oieiaeiiai aei aa?y/iai o?ea. Aeey oi?ioethaaiiy ia?aaeieno
?icaeyiooi eiino?oeoe?th, yea neeaaea?oueny c aeaio ni?aoieonieo
iai?aae?in?a, aeiiiaiaieo aeaiia a?ae??ceaie (?en.1 a). A oieonao F1 ?
F2 ae?in?a ?ici?uaii aeaa aeaea?aea oaiea, ye? iathoue ??ai? ii/aoeia?
oaiia?aoo?e. A ?acoeueoao? aeca?eaeueiiai a?aeaeooy o?ei F2 iaea?aeo?,
e??i i?iiai?a a?ae o?ea F1, ? /anoeio nai?o i?iiai?a, ye? a?aeaeoi
a?ae??ceaie. Oiio F2 iaa??aa?oueny; a oie /an ye o?ei F1
ioieiaeaeo?oueny. Iaaeie?e oaeiai “ni?inooaaiiy” caeiio oa?iiaeeiai?ee
iiynith? aaiiao?e/ia iiaeaethaaiiy ia?aa?ao i?iiai?a (?en.1 a), ui
aeaco? ia iaeiaeiao e?euee?noue i?iiai?a, ye? ia?aoeiathoue oi/ee F1 ?
F2.

Ae?oaee ?icae?e aeena?oaoe?? i?enay/aii ieoaiiyi neeaaeaiiy aeai?eoi?a
aeey ?ic?aooieo a?aeaeaaeueieo nenoai, ye? iaea?aeaii caniaaie Maple V.

Icia/aiiy. Iaoae ia?ii aeaaeeo e?eao (iiaa?oith) L ? oi/eo S ye
aeaea?aei i?iiai?a. Oi/ea Q iaeaaeeoue aea??aoeaeoi?e e?i?? (iiaa?oi?) A
oiae?, eiee aiia ?icoaoiaaia ia a?aeaeoiio a?ae e?eai? (iiaa?oi?) L a
oi/oe? I i?iiai? ? yeui aeeiio?oueny oioiaei?noue (SM (+ (MQ (= t =
const.

Oaa?aeaeaiiy. Aeey e?eai? L:y = f(x) ? oi/ee S(0, y0) ye aeaea?aea
i?iiai?a ??aiyiiy n?i’? aea??aoeaeoieo e?eaeo ia? aeaeyae

;

,

.

? aeaea?aea i?iiai?a a oi/oe? S(0, 0, z0), ??aiyiiy aea??aoeaeoieo
iiaa?oiiue aoaea

;

,

.

Ia ?en.2 iaaaaeaii cia?aaeaiiy aea??aoeaeo?a ia?aaiei?aea iaa?oaiiy a
caeaaeiino? a?ae ?icoaooaaiiy aeaea?aea i?iiai?a.

Oaa?aeaeaiiy. Aeey oeee?iae?a, ??aiyiiy i?io?eth yeiai z=f(x), ? oi/ee
S(0, 0, z0) ye aeaea?aea i?iiai?a, iien aea??aoeaeo?a ia? aeaeyae

;

;

,

.

Aeae? ?icaeyiooi aaiiao?e/ia iiaeaethaaiiy eaoaeaonoeee e?eai? ia
ieiuei?. Iaoae a aeaea?oia?e nenoai? eii?aeeiao Oxy ia?ii e?eao L: y =
f(x). \NEIAIE 108 \f «Symbol» \NEIAIE 108 \f «Symbol» Iicia/eii /a?ac
( eoo i?ae a?nnth Ox oa aeioe/iith aei e?eai? L o oi/oe?

M(z,f(z)) (ooo z \NEIAIE 108 \f «Symbol» — ia?aiao?).

Oaa?aeaeaiiy. sseui aeaea?aei i?iiai?a ?icoaoiaaii a o. S(xS,yS), oi
??aiyiiy n?i’? i?iiai?a, a?aeaeoeo e?eaith L, ia? aeaeyae

.

), oi ia?ii ??aiyiiy

.

c oi/eiaiai aeaea?aea S(xS, yS) iaaea? io/ie i?iiai?a. Oiae? ??aiyiiy
eaoaeaonoeee iaoeia aeaeyae

,

;

.

Ia ?en.3 iaaaaeaii cia?aaeaiiy oa ??aiyiiy eaoaeaonoee ia?aaiee a
caeaaeiino? a?ae ?icoaooaaiiy aeaea?aea i?iiai?a (ooo z1, z2 oa z3 —
cia/aiiy ia?aiao?o z, aeey yeeo ciaiaiiee o iienao eaoaeaonoeee
aei??aith? ioeth).

Oaa?aeaeaiiy. sseui caaeaia e?eaa y=f (x) ? iaaeath/i i?iiai?
ni?yiiaai? acaeiaae aaeoi?a {m, n}, oi ia?ii iien eaoaeaonoeee

;

.

Ia ?en.4 iaaaaeaii eaoaeaonoeee ia?aaiee y = x2/2 — 1/2 (aea tg(=n/m).

Aeae? ?icaeyiooi i?io?ethaaiiy a?aeaeaaeueii? e?eai? a caeaaeiino? a?ae
?? eaoaeaonoeee ia iniia? iiiyooy a?aaieueaaioe. Iaoae iioeea o?ao?a G
iaiaaeaia e?eaith L. I?iaaaeaii c ciai?oiuei? oi/ee A aea? i?yi?, ui
aeioeeathoueny e?eai? L a oi/eao S1 ? S2. ?icaeyiaii o?ao?o G*, ooai?aio
ia’?aeiaiiyi o?ao?e G oa iaeano? o?eeooieea AS1S2.

Icia/aiiy. A?aaieueaaioith L* e?eai? L iaceaa?oueny iiiaeeia oi/ie A,
aeey yeeo ia?eiao? o?ao?e G* iaoeia iino?eia cia/aiiy.

A ?iaio? iieacaii, ui yeui ii?iaeue aei aeayei? e?eai? neeaaea? a
eiaei?e oi/oe? ??ai? eooe c aeioe/ieie aei ?ioi? e?eai? S0S1S2, oi
ia?eiao? o?ao?e AS1S0S2A iino?eiee iacaeaaeii a?ae iieiaeaiiy oi/ee A.

Oaa?aeaeaiiy. sseui a?aaieueaaioa yaey? niaith «aeca?eaeueio» e?eao, oi
?? eaoaeaonoeea ca?aa?oueny c e?eaith L.

Aeuaiaaaaeaia aea? i?aenoaae aeey a?ao?/iiai iaoiaeo iiaoaeiae
a?aaieueaaioe L* e?eai? L: a?aaieueaaioo ooai?eoue ne?ae ie?aoey, ui
?ooa?oueny a iaoe? ca oiiae ?? iaoyao ia eiioo?? L o?ao?e G.

Ia ?en.5 iaaaaeaii cia?aaeaiiy oa caeaaeiino? a?ae eooa iaa?oaiiy ?
ia?eiao?a eaoaeaonoeee (iiiaieooieea) aeiaaeeie: A — eaoaeaonoeee; B —
a?aaieueaaioe, a oaeiae C — ia?eiao?a a?aaieueaaioe.

A o?aoueiio ?icae?e? ye i?eeeaaee a?aeaeaaeueieo nenoai ?icaeyiooi
iaoaiaoe/i? a?euey?aee. Caniaaie iaoaiaoe/iiai i?ioeani?a Maple V ?
iiaith Pascal ceeaaeaii aeai?eoie ? i?ia?aie o?anoaaiiy a?euey?aeieo
i?iiai?a a eie?, a ae?in?, a eiia?iiaaieo o?ao?ao (?en.6) oa a i?inoi??
(a ae?ini?ae?).

?icaeyiooi iaoiae ioe?iee e?eueeino? ii/aoeiaeo ?oo?a c oi/ee N, i?ney
yeeo i?ii?iue i?ieaea /a?ac oi/eo K ca oiiae k a?aeaeoo?a a?ae eiea
(ae?ina).

Oaa?aeaeaiiy. sseui i?ney k-ai a?aeaeooy a?ae eiea i?ii?iue, yeee
aeoiaeeoue c oi/ee N(a,b), aeinyaa? oi/ee K(n,d), oi e?euee?noue
iiaeeeaeo ii/aoeiaeo iai?yie?a aei??aith? e?eueeino? ei?ai?a ooieoe??

,

aea x — oeaio?aeueiee eoo, A= nis(x+(k-1)((-2()); B= sin
(x+(k-1)((-2()).

I?eeeaae. Iaoae N(0.5, 0.5), K(-0.6, 0) ? k=1. A?ao?eii ooieoe?? aoaea
e?eaa, yea cia?aaeaia ia ?en.7 a (ei?ai? 0,93; 2,27; 2,75 ? 5,03). Oiaoi
?niothoue /ioe?e iai?yiee ?ooo c oi/ee N (?en.7 a).

Aeey aeine?aeaeaiiy ?aciiainieo yaeu a a?aeaeaaeueieo nenoaiao
?icaeyiooi caaea/o “naiiii?ii?iaiiy” oi/ee aeaea?aea i?iiai?a. Oiaoi
aa??aio o?anoaaiiy, eiee oi/ee N ? K ni?aiaaeathoue (K(N). Ia ?en.8 aeey
k=5000 iaaaaeaii a?ao?ee ooieoe?? f(x) a caeaaeiino? a?ae aeaea?aea
i?iiai?a a oi/oe? E.

Oaeiae ?icaeyiooi aeiaaeie iaoaiaoe/iiai a?euey?aea a ae?ini?ae? oa a
ia?aaiei?ae? iaa?oaiiy. Ia ?en.9 iaaaaeaii i?eeeaaee cia?aaeaiiy ia
n?i’? ia?aeaeueieo n?/ieo ieiuei ne?ae?a a?euey?aeii? eoe? a i?inoi??
ae?ini?aea.

A /aoaa?oiio ?icae?e? iaaaaeaii i?eeeaaee ?ic?aooie?a ae?ioe/ieo oa
ia?aaie?/ieo a?aeaeaaeueieo nenoai. Cie?aia, ?ica’ycaii aea? ?aaeuei?
caaea/? i?iaeooaaiiy a?aeaeaaeueieo nenoai o aaeoc? eaca?ii? oaoi?ee:

i) aecia/aii i?io?eue ae?ini?aeiiiae?aii? a?aeaeaaeueii? iiaa?oi?, yea
aeicaiey? “??aiii??ii” ina?oethaaoe a?ae??cie oieaeueii? i?yii?;

ii) ?ic?iaeaii iaoiae ?ic?aooieo ieineiai ia?aaie?/iiai oieonaoi?a.

Iaaieioo?oueny, ui aeena?oaoe?y i?enay/aia aaiiao?e/iei, a ia
o?ceei-oaoiieia?/iei ieoaiiyi uiaei eiino?oeoe?e a?aeiia?aeiiai
onoaoeoaaiiy. A i?e eeoa aeiaiaeeoueny aoaeoeai?noue ?icaeyiooeo
aeai?eoi?a aaiiao?e/iiai iiaeaethaaiiy ia?aa?ao i?iiai?a o cacia/aieo
a?aeaeaaeueieo nenoaiao noiniaii a?aeiia?aeieo ai?iaaaeaeaiue.

Ia?oa caaea/a. Ia ?en.10a iaaaaeaii noaio a?aeaeaa/a aeey iaea/ee
eaca??a, yeee neeaaea?oueny c ae?ini?aeiiiae?aii? a?aeaeaaeueii?
iiaa?oi?, ?c ieoeiiiae?aii? eaiie iaea/ee oa no?eaeiy c aeoeaii?
?a/iaeie, yeee ?icoaoiaaiee a?ey ioai?o iiaa?oi? aeey aeoiaeo eaca?iiai
i?iiaiy. Ia ?en. 10a iaaaaeaii aaiiao?e/ia iiaeaethaaiiy ia?aa?ao
i?iiai?a ca oiiae, ui aeaea?aeii i?iiai?a ? o?e oi/ee, ye? ?icoaoiaai?
ia eaii? iaea/ee, ?, ui a?aeaeooy i?iiai?a a?ae “aeca?eaeueii?” iiaa?oi?
? iaeii?aciaei. Aeine?aeaeaiiy iieacaee, ui ia??aiii??iith ?
caeaaei?noue a?ae iieiaeaiiy ia oieaeuei?e i?yi?e u?eueiino? i?iiai?a,
ye? iaaeathoue ia no?eaeaiue. Oiio aeieeea caaea/a iienaoe
a?aeaeaaeueio e?eao (iiaa?oith), yea a aeicaieeea aia?a?th aeaea?aea o
aeaeyae? a?ae??cea oieaeueii? i?yii? “??aiii??ii” ?iciiae?eeoe ia
a?ae??ceo no?eaeiy.

Aeey a?aooaaiiy aaaaoi?aciaiai a?aeaeooy a?ae iiaa?oi? ?icaeyiaii
ia?a??c ae?ini?aea n?/iith ieiueiith, yea i?ioiaeeoue /a?ac oieaeueio
i?yio (?en. 11a). Aeey oi/ee A(xa, ya) ??aiyiiy eaiee AM o?a?eoi???
oi/ee aoaea

.

Oaa?aeaeaiiy. Aa?oeia A ianooiii? eaiee eaiaii? o?a?eoi??? ia?
eii?aeeiaoe A(acos(, bsin(), aea eoo ( aecia/a?oueny ye ei??iue ??aiyiiy

.

.

I?ney ia/eneaiiy cia/aiiy eooa ( ia?ii ??aiyiiy eaiee IA

.

Aeeeaaeaia aeicaiey? cae?enithaaoe o?anoaaiiy i?iiai?a a ae?in? (?en.
11a).

Ia ?en.12 iaaaaeaii ?acoeueoaoe i?io?ethaaiiy a?aeaeaaeueii? iiaa?oi?,
yea caaacia/o? ??aiii??iee ?iciiae?e a?aeaeoeo i?iiai?a, ui iaaeathoue
ia a?ae??cie oieaeueii? i?yii?. I?e oeiio ci?ithaaeeny ianooii?
ia?aiao?e:

iieiaeaiiy aeoeaii? ?a/iaeie (no?eaeiy) — ia?aiao? bn (120(200);

u?euei?noue a?aeaeoeo i?iiai?a ia no?eaei? — ia?aiao? bi (1(5);

iieiaeaiiy o. aeca?eaea a?aeiinii no?eaeiy — ia?aiao? A (150(230).

Ae?oaa caaea/a. ?icaeyiaii n?i’th ni?aoieonieo ia?aaie (?en. 13a).
Ia?a??c ieineiai oieonaoi?a neeaaea?oueny c o?aaiaio?a cacia/aieo
ia?aaie (?en. 13a). Eiino?oeoeaii oieonaoi?o iaaeathoue oi?io
oeee?iae?e/ii? iiaa?oi? (?en. 13a) aai iiaa?oi? iaa?oaiiy (?en.13a).
Aeey aaiiao?e/iiai ?ic?aooieo oieonaoi?a iaiao?aeii aecia/eoe ia?aiao?e
iioe/ii aea?aaeaioii? ia?aaiee. Inoaii? icia/a?, ui o ieineiai
oieonaoi?a oa ia?aaiee iiaeii? aooe aeecuee? eaoaeaonoeee aeey
aeiionoeieo iai?yie?a ?o ina?oeaiiy.

Aeey ieineiai ia?aaie?/iiai oieonaoi?a iioe/ii aea?aaeaioio ia?aaieo
i?iiiio?oueny ciaoiaeeoe oeyoii aaiiao?e/iiai iiaeaethaaiiy ??
eaoaeaonoeee (?en.14) oa iiaeaeueoei ii??aiyiiyi ?? ai?ieneiaioa c
a?aeiieie eaoaeaonoeeaie ia?aaiee (?en 4).

Ai?iaaaeaeaiiy ?acoeueoao?a ?iaioe aeeiiaii a IAeO?I (i. Oa?e?a) i?e
i?iaeooaaii? a?aeaeaaeueieo iiaa?oiiue no/anieo ae?ia?a eaca?ii?
oaoi?ee.

AENIIAEE

Aeena?oaoe?y i?enay/aia ?ic?iaoe? iaoiae?a ? aeai?eoi?a iieno oa
iiaoaeiae cia?aaeaiue eaoaeaonoee ? o?iio?a a?aeaeoeo oaeeue
(aea??aoeaeo?a) ia iniia? aaiiao?e/iiai iiaeaethaaiiy ia?aa?ao i?iiai?a
o a?aeaeaaeueieo nenoaiao c iaoith i?io?ethaaiiy ae?ini?aeiiiae?aii?
a?aeaeaaeueii? iiaa?oi?, yea caaacia/eoue “??aiii??ia” ina?oeaiiy
a?ae??ceo oieaeueii? i?yii?, oa aeey ?ic?aooieo ieineeo ia?aaie?/ieo
oieonaoi??a.

I?e oeueiio iaea?aeai? oae? ?acoeueoaoe, ui iathoue iaoeiao oa
i?aeoe/io oe?ii?noue.

1. Iaaaaeaii e?eoe/iee aiae?c uiaei i?iaeooaaiiy a?aeaeaaeueieo nenoai
aac aaiiao?e/iiai iiaeaethaaiiy ia?aa?ao i?iiai?a. Cacia/aii, ui
aaiiao?e/ia iiaeaethaaiiy caaaeyee iai/iiio eiio?ieth aeicaiey? onoiooe
iiaeeea? iiieeee i?e i?iaeooaaii?.

2. Aeey aeaii? a?aeaeaaeueii? nenoaie ?ic?iaeaii iaoiae iieno n?i’?
aea??aoeaeo?a oa iaa?aeii? n?i’? a?aeaeoeo i?iiai?a (eaoaeaonoeee
e?eai?).

3. Neeaaeaii aeai?eoie iiaoaeiae cia?aaeaiue n?i’? aea??aoeaeo?a ?
eaoaeaonoeee e?eai?, a oaeiae a?aeaeaaeueii? e?eai? a caeaaeiino? a?ae
caaeaii? eaoaeaonoeee ia iniia? iiiyooy a?aaieueaaioe.

4. Cai?iiiiiaaii iaoiae neeaaeaiiy aiae?oe/ieo iien?a aea??aoeaeo?a oa
eaoaeaonoee caniaaie iaoaiaoe/iiai i?ioeani?a Maple V.

5. Noai?aii aeai?eoie oa neeaaeaii i?ia?aie (caniaaie Maple V ? iiaith
PASCAL) o?anoaaiiy i?iiai?a iaoaiaoe/ieo a?euey?ae?a aeey eiea, ae?ina,
eiia?iiaaieo iaeanoae oa aeey ae?ini?aea.

6. ?ica’ycaii aea? ?aaeuei? caaea/? i?iaeooaaiiy a?aeaeaaeueieo nenoai
o aaeoc? eaca?ii? oaoi?ee:

i) aecia/aii i?io?eue ae?ini?aeiiiae?aii? a?aeaeaaeueii? iiaa?oi?, yea
aeicaiey? “??aiii??ii” ina?oethaaoe a?ae??cie oieaeueii? i?yii?;

ii) ?ic?iaeaii iaoiae ?ic?aooieo ia?aaie?/iiai ieineiai oieonaoi?a.

7. A??ia?aei?noue iaea?aeaieo ?acoeueoao?a i?aeoaa?aeaeaii oeyoii
aeiaaaeaiiy oaa?aeaeaiue oa iiaoaeiaith ca aeiiiiiaith AII a?ao?/ieo
cia?aaeaiiyie aea??aoeaeo?a ? eaoaeaonoee aeey oanoiaeo i?eeeaae?a.

8. Ai?iaaaeaeaiiy ?acoeueoao?a ?iaioe aeeiiaii a Iaoeiai-aeine?aeiiio
oaoiieia?/iiio ?inoeooo? i?eeaaeiaoaeoaaiiy (IAeO?I, i. Oa?e?a) i?e
i?iaeooaaii? a?aeaeaaeueieo iiaa?oiiue no/anieo ae?ia?a. ?aae?caoe?y
i?aeoaa?aeaeo?oueny aeia?aeeith i?i aeei?enoaiiy cai?iiiiiaaii? o ?iaio?
iaoiaeeee.

Iniiai? iieiaeaiiy aeena?oaoe?? iioae?eiaaii o oaeeo ?iaioao:

Iniiai? ioae?eaoe??:

1. Na?aaea I.?. Aeine?aeaeaiiy ?ioaineaiino? io/ea i?iiai?a o
ae?ioe/iiio a?aeaeaa/?. // I?eeeaaeia aaiiao??y oa ?iaeaia?ia a?ao?ea:
I?aea?aeii/a iaoeiai-oaoi?/ia ca??ea. Aei. 63. A?aei. ?aae A.?.
Ieoaeeaiei. Ee?a: EAeOOAA, 1998. — n.206-209

2. ?aaa A.A., Na?aaea I.?. Cia?aaeaiiy o?iioo oaee?, yea a?aeaeoa a?ae
oeee?iae?e/ii? neioni?aeaeueii? iiaa?oi?. // I?iaeaiu iiaea?iie
aaciianiinoe. Na. iao/iuo o?oaeia. THaeeaeiue aui. *anoue 2. Oa?ueeia:
OEIA IAAe Oe?aeiu, 1998. — n.35 — 38

3. Na?aaea I.E. O?aaeoi?ee iaoaiaoe/aneiai aeeuey?aea aioo?e
yeeeinieaea. // O?oaeu Oaa?e/aneie ainoaea?noaaiiie aa?ioaoie/aneie
aeaaeaiee. O.3. Aui. 4. “I?eeeaaeiay aaiiao?ey e eiaeaia?iay a?aoeea” —
Iaeeoiiieue: OAAOA. 1998. — n.111-114

4. Na?aaea I.?. Aaiiao?e/ia iiaeaethaaiiy o?a?eoi??? a?euey?aeii? oi/ee
a ae?in?. // O?oaeu Oaa?e/aneie ainoaea?noaaiiie aa?ioaoie/aneie
aeaaeaiee. O.5. Aui. 4. “I?eeeaaeiay aaiiao?ey e eiaeaia?iay a?aoeea” —
Iaeeoiiieue: OAAOA. 1999. — n.75-79

5. Na?aaea I.E. I?ioeee?iaaiea naiaea aeey ?aaiiia?iiai inaauaiey
ieiuaaeee io?aaeaiiui naaoii. // O?oaeu Oaa?e/aneie ainoaea?noaaiiie
aa?ioaoie/aneie aeaaeaiee. O.5. Aui. 4. “I?eeeaaeiay aaiiao?ey e
eiaeaia?iay a?aoeea”-Iaeeoiiieue: OAAOA. 1999. — n.90-93

6. Na?aaea I.?. Aaiiao?e/ia iiaeaethaaiiy ia?aa?ao i?iiai?a o nenoai?
aeaio ni?aoieonieo iai?aae?in?a. // O?oaeu Oaa?e/aneie ainoaea?noaaiiie
aa?ioaoie/aneie aeaaeaiee. O.5. Aui. 4. “I?eeeaaeiay aaiiao?ey e
eiaeaia?iay a?aoeea”-Iaeeoiiieue: OAAOA. 1999. — n.132-135

Aeiaeaoeia? ioae?eaoe??:

7. Eooeaiei E.I., Iaco?aiei A.Ae., Na?aaea I.E. Iienaiea eaoaeaonoeee e
ea?noiaie iaeanoe. A na. o?oaeia 3 Iaaeaeoia?iaeiie iao/ii-i?aeoe/aneie
eiioa?aioeee «Nia?aiaiiua i?iaeaiu aaiiao?e/aneiai iiaeaee?iaaiey.
*anoue 1. — Iaeeoiiieue: OAAOA. 1996.- n.46-47

8. Iaco?aiei A.Ae., Na?aaea I.E. Iienaiea eaoaeaonoeee e ea?noiaie
iaeanoe i?e iiiiue R-ooieoeee. A na. iao/iuo o?oaeia e 70-eaoeth
A.E.?aa/aaa “R-ooieoeee a caaea/ao iaoaiaoe/aneie oeceee e i?eeeaaeiie
aaiiao?ee”. — Xa?ueeia: OEIA IAN Oe?aeiu, 1996. — n.77-82

9. Na?aaea I.E. I?eiaiaiea eiiiuethoa?iie a?aoeee a oai?ee
iaoaiaoe/aneeo aeeuey?aeia. // O?oaeu Oaa?e/aneie ainoaea?noaaiiie
aa?ioaoie/aneie aeaaeaiee. O.1. Aui. 4. “I?eeeaaeiay aaiiao?ey e
eiaeaia?iay a?aoeea”-Iaeeoiiieue: OAAOA. 1997. — n.95-98

10. Na?aaea I.E. Ii?aaeaeaiea aaiiao?e/aneie oi?iu io?aaeaoaey
inaaoeoaey aeey iaea/ee eaca?ia. A na. o?oaeia 4 Iaaeaeoia?iaeiie
iao/ii-i?aeoe/aneie eiioa?aioeee «Nia?aiaiiua i?iaeaiu aaiiao?e/aneiai
iiaeaee?iaaiey. *anoue 3. — Iaeeoiiieue: OAAOA. 1997. — n. 53-56

11. Eooeaiei E.I., Na?aaea I.E. Eiee/anoai aiciiaeiuo o?aaeoi?ee
iaoaiaoe/aneiai aeeuey?aea a e?oaa. — A ca. i?aoeue I?aeia?iaeii?
iaoeiai-i?aeoe/ii? eiioa?aioe?? “No/ani? i?iaeaie aaiiao?e/iiai
iiaeaethaaiiy”. *anoeia 2. — Oa?e?a: O?IA IAN Oe?a?ie, 1998. — n.55-59

12. Na?aaea I.?. Ioe?iea ?ioaineaiino? io/ea i?iiai?a o a?aeaeaa/? c
a?aooaaiiyi eoo?a iaae?iiy. — A ca. i?aoeue I?aeia?iaeii?
iaoeiai-i?aeoe/ii? eiioa?aioe?? “No/ani? i?iaeaie aaiiao?e/iiai
iiaeaethaaiiy”. *anoeia 2. — Oa?e?a: O?IA IAN Oe?a?ie, 1998. — n.60-64

13. Na?aaea I.E. I?ioeee?iaaiia io?aaeaoaey n caaeaiiuie
oa?aeoa?enoeeaie. — A ca. i?aoeue I?aeia?iaeii? iaoeiai-i?aeoe/ii?
eiioa?aioe?? “No/ani? i?iaeaie aaiiao?e/iiai iiaeaethaaiiy”. *anoeia 2.
— Oa?e?a: O?IA IAN Oe?a?ie, 1998. — n.65-68

14. Iaco?aiei A.Ae., Na?aaea I.E. Aaiiao?e/aneia iiaeaee?iaaiea oiaea
eo/ae i?e ?an/aoa io?aaeathuae iiaa?oiinoe NA*-aioaiiu.- A ca. i?aoeue
I?aeia?iaeii? iaoeiai-i?aeoe/ii? eiioa?aioe?? “No/ani? i?iaeaie
aaiiao?e/iiai iiaeaethaaiiy”. *anoeia 4. — Oa?e?a: O?IA IAN Oe?a?ie,
1998. — n.38-46

Aiioaoe?y

Na?aaea Iaoaey ?aai?aia. Aaiiao?e/ia iiaeaethaaiiy ia?aa?ao i?iiai?a a
ae?ioe/ieo oa ia?aaie?/ieo a?aeaeaaeueieo nenoaiao. -?oeiien.

Aeena?oaoeiy ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa oaoii/ieo iaoe
ca niaoeiaeueiinoth 05.01.01 — I?eeeaaeia aaiiao?iy, ?iaeaia?ia
a?aoiea.- Ee?anueeee iaoe?iiaeueiee oiiaa?neoao aoaeiaieoeoaa i
a?oioaeoo?e. Oe?a?ia, Ee?a, 1999.

Caoeua?oueny aeena?oaoe?y oa 14 iaoeiaeo ?ia?o, a yeeo aeeeaaeathoueny
?ic?iaeai? iaoiaee ? aeai?eoie iieno oa iiaoaeiae cia?aaeaiue
eaoaeaonoee ? o?iio?a a?aeaeoeo oaeeue (aea??aoeaeo?a) ia iniia?
aaiiao?e/iiai iiaeaethaaiiy ia?aa?ao i?iiai?a o a?aeaeaaeueieo nenoaiao
c iaoith i?io?ethaaiiy ae?ini?aeiiiae?aii? a?aeaeaaeueii? iiaa?oii?, yea
caaacia/o? “??aiii??ia” ina?oeaiiy a?ae??ceo oieaeueii? i?yii?, oa aeey
?ic?aooieo ieineeo ia?aaie?/ieo oieonaoi??a. Aeey aeaii? a?aeaeaaeueii?
nenoaie ?ic?iaeaii iaoiae iieno n?i’? aea??aoeaeo?a. Oaeiae ?ic?iaeaii
iaoiae iieno iaa?aeii? n?i’? a?aeaeoeo i?iiai?a (eaoaeaonoeee e?eai?).
Neeaaeaii aeai?eoie iiaoaeiae cia?aaeaiue n?i’? aea??aoeaieo?a ?
eaoaeaonoeee e?eai?. Iaea?aeaii cia?aaeaiiy a?aeaeaaeueii? e?eai? a
caeaaeiino? a?ae caaeaii? eaoaeaonoeee ia iniia? iiiyooy a?aaieueaaioe.
?icaeyiooi iaoiae neeaaeaiiy aiae?oe/ieo iien?a aea??aoeaeo?a oa
eaoaeaonoee caniaaie iaoaiaoe/iiai i?ioeani?a Maple V. Neeaaeaii
aeai?eoie o?anoaaiiy i?iiai?a iaoaiaoe/ieo a?euey?ae?a aeey eiea,
ae?ina, eiia?iiaaieo iaeanoae oa aeey ae?ini?aea. ?ica”ycaii aea?
?aaeuei? caaea/? i?iaeooaaiiy a?aeaeaaeueieo nenoai:

1) aecia/aii i?io?eue ae?ini?aeiiiae?aii? a?aeaeaaeueii? iiaa?oi?, yea
aeicaiey? “??aiii??ii” ina?oethaaoe a?ae??cie oieaeueii? i?yii?;

2) ?ic?iaeaii iaoiae ?ic?aooieo ieineiai ia?aaie?/iiai oieonaoi?a.

Eeth/iai neiaa: aiaeaeaaeueia iiaa?oiy, o?iio a?aeaeoi? oaee?,
eaoaeaonoeea, ea?noiaa iaeanoue.

Aiiioaoeey

Na?aaea Iaoaeey Eaaiiaia. Aaiiao?e/aneia iiaeaee?iaaiea oiaea eo/ae a
yeeeioe/aneeo e ia?aaiee/aneeo io?aaeaoaeueiuo nenoaiao. -?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 iaoeeiiaeueiue oieaa?neoao no?ieoaeuenoaa e a?oeoaeoo?u.
Oe?aeia, Eeaa, 1999.

Caueuaaony aeenna?oaoeey e 14 iao/iuo ?aaio, a eioi?uo eceiaeaiu
?ac?aaioaiiua iaoiaeu e aeai?eoiu iienaiey e iino?iaiey ecia?aaeaiee
eaoaeaonoee e o?iioia io?aaeaiiuo aiei (yeae?aoeaeoia) ia iniiaa
aaiiao?e/aneiai iiaeaee?iaaiey oiaea eo/ae a io?aaeaoaeueiuo nenoaiao n
oeaeueth i?ioeee?iaaiey yeeeinieaeiiiaeiaiie io?aaeaoaeueiie
iiaa?oiinoe, eioi?ay iaania/eaaao “?aaiiia?iia” inaauaiea io?acea
oieaeueiie i?yiie, e aeey ?an/aoa ieineeo ia?aaiee/aneeo oieonaoi?ia.
Aeey aeaiiie io?aaeaoaeueiie nenoaiu ?ac?aaioai iaoiae iienaiey
naiaenoaa yeae?aoeaeoia. Oaeaea ?ac?aaioai iaoiae iienaiey iaeaathuae
naiaenoaa io?aaeaiiuo eo/ae (eaoaeaonoeee e?eaie). Ninoaaeaiu aeai?eoiu
iino?iaiey ecia?aaeaiee naiaenoaa yeae?aoeaeoia e eaoaeaonoeee e?eaie.
Iieo/aiu ecia?aaeaiey io?aaeaoaeueiie e?eaie a caaeneiinoe io caaeaiiie
eaoaeaonoeee ia iniiaa iiiyoey aeyaieueaaiou. ?anniio?ai iaoiae
ninoaaeaiey aiaeeoe/aneeo iienaiee yeae?aoeaeoia e eaoaeaonoee
n?aaenoaaie iaoaiaoe/aneiai i?ioeanni?a Maple V. Ninoaaeaiu aeai?eoiu
o?anne?iaaiey eo/ae iaoaiaoe/aneeo aeeuey?aeia aeey ie?oaeiinoe,
yeeeina, eiiaeie?iaaiiuo iaeanoae e aeey yeeeinieaea. ?aoaiu aeaa
?aaeueiua caaea/e i?iaeoe?iaaiey io?aaeaoaeueiuo nenoai:

1) ii?aaeaeai i?ioeeue yeeeinieaeiiiaeiaiie io?aaeaoaeueiie
iiaa?oiinoe, iicaieythuae “?aaiiia?ii” inaauaoue io?acie oieaeueiie
i?yiie;

2) ?ac?aaioai iaoiae ?an/aoa ieineiai ia?aaiee/aneiai oieonaoi?a.

Eeth/aaua neiaa: io?aaeaoaeueiay iiaa?oiinoue, o?iio io?aaeaiiie aieiu,
eaoaeaonoeea, ea?noiaay iaeanoue.

SUMMARY

Sereda Nataliy Ivanovna. Geometric modelling a course of rays in
elliptic and parabolic reflective systems. — Manuscript.

The competition thesis for Scientific Degree of Candidate of Technical
Sciences’ in speciality 05.01.01 — Applied geometry, engineering
graphics.- the Kyiv National University of Building and Architecture.
Ukraine, Kyiv, 1999.

The thesis and 14 scientific works is protected, in which the
developed methods both algorithms of exposition and construction of
images catacaustiks and fronts of reflected waves (ekvireflekts) because
of geometric modelling of a course of rays in reflective systems are
explained with the purpose of profiling kvaziellipsoid of a reflective
surface, which ensures «uniform» illumination of a segment focal direct,
and for calculation flat parabolic focusators. For the given reflective
system the method of exposition of a set ekvireflect is developed. A
method of exposition enveloping sets of reflected rays (catacaustik by a
curve) also is developed. The algorithms of a construction of images of
a set ekvireflect and catacaustik by a curve are composed. The images
reflective curve in an association from specific catacaustiks because of
concepts bievolvents are obtained. The method of compiling of analytical
expositions ecvireflects and catacaustiks by tools of the mathematical
processor Maple V is considered. The algorithms tracing of rays of
mathematical billiards for a circle, ellipse, combined areas and for an
ellipsoid are composed. Two actual tasks of projection of reflective
systems are decided:

1) is defined a profile kvaziellipsoid of a reflective surface, which
allows «is uniform» to illuminate a segment focal direct;

2) the method of calculation flat parabolic focusator is developed.

The key words: a reflector surface, drive-round of family rays, the
catacaustics, the cars field.

I?aeienaii aei ae?oeo 25.08.99 ?. Oi?iao 60×80 1\16

Ae?oe. ?ecia?ao. Oi. ae?oe.a?e. 1,25

Oe?aae 100 Aeae. ? 7 Cai.? 117

O?IA IAN Oe?a?ie, 310023, i. Oa?e?a, aoe. *a?ieoaanueeiai, 94.

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