gluvi zedapiris nebismier wertilSi SeiZleba gaivlos mxebi sibrtye, romlis orientacia mocemul wertilSi zedapiris orientaciis tolfasia. mxebi sibrtyis orientaciis gansazRvrisTvis gamoiyeneba am wertilSi normalis mimarTuleba. davafiqsiroT sakoordinato RerZTa sistema Sesaswavli zedapiris mimarT: z RerZi mivmarToT gamosaxulebis formirebis sistemis optikuri RerZis gaswvriv, xolo x da y RerZebi ki zedapiris gamosaxulebis sibrtyis paralelurad; maSin zedapiris mocemuli fragmentis orientacia ganisazRvreba marTobuli manZilis saSualebiT z RerZis gaswvriv zedapiridan gamosaxulebis sibrtyemde. es manZili, cxadia, damokidebulia zedapiris daxraze Sesabamisad x da y RerZebis mimarT. es daxrebi SeiZleba warmovidginoT, rogorc z-is kerZo warmoebulebi Sesabamisad x-iT da y-iT..

p=z_x q=z_y. Tu movaxdenT dx mcire wanacvlebas x RerZis gaswvriv, maSin z manZili Seicvleba pdx –iT, Sesabamisad dy wanacvleba y RerZis mimarT Secvlis z manZils qdy-iT. am wanacvlebis Sesabamisi veqtorebi mdebareoben mxeb sibrtyeze, amitom maTi veqtoruli namravlis Sedegad miiReba normali

n=(-p,-q,1)^T, normalis erTeulovani veqtoria n=n/|n| =(-p, -q, 1)/Ц1+p² +q²

ganaTebis wyaros mdebareobis gansasazRvravad ganixilaven zedapiris iseT orientacias, romelic am wyarodan gamosuli sxivebis marTobulia. CavTvaloT, rom aseTi zedapiris normalia n=(-p₀,-q₀,1)^T, xolo erTeulovani veqtori

n₀=(-p₀, -q₀, 1)/Ц1+p₀²+q₀². es veqtori igulisxmeba ganaTebis wyaros mimarTulebis veqtorad /1/.

ganvixiloT, magaliTad, lambertis idealuri zedapiri, romelic masze dacemul sxivs mTlianad aireklavs yvela mimarTulebiT Tanabrad. Tu I aris ganaTebis wyaros intensivoba, maSin arekvlili sxivis intensivoba iqneba I₁=I^.cos j, sadac j aris kuTxe normalsa da wyaros mimarTulebas Soris. Sesabamisi erTeulovani veqtorebis skalaruli gamravlebis Sedegad miviRebT

cos j=1+ p₀^.p + q₀^.q/Ц1+p₀²+q₀^{2 .}Ц1+p₀²+q₀²

es formula asaxavs mocemuli tipis zedapiris sikaSkaSis damokidebulebas mis (p, q) orientaciaze. am funqcias uwodeben arekvlis unarianobis ruqas R(p,q).

es funqcia SeiZleba warmodgindes rogorc SesabamisobaTa cxrilis saxiT, aseve grafikuladac _ (p, q) sakoordinato sibrtyeze erTnairi sikaSkaSis mqone izomeruli mrudeebis saxiT. arekvlis unarianobis ruqa damokidebulia sxeulis zedapiris masalis Tvisebebze da ganaTebis wyaros mdebareobaze. is SeiZleba miRebuli iqnas eqsperimentalurad: magaliTad, aiReba winaswar cnobili formis sxeuli, Semdeg zedapiris mocemul wertilSi Sesabamis orientaciisaTvis gaizomeba Sesabamisi sikaSkaSis mniSvneloba.

im SemTxvevaSi, Tu Cven gvaqvs mxolod erTi arekvlis unarianobis ruqa, maSin gamoTvla unda daviwyoT zedapiris iseTi wertilidan, romlis orientacia (p, q) winaswar mocemuli gvaqvs. Semdeg vakeTebT mcire wanacvlebas (dx, dy) garkveuli mimarTulebiT, gadavdivarT axal wertilSi. Sesabamisad simaRle z Seicvleba mniSvnelobiT dz= p Ч dx + q Ч dy. unda vipovoT orientaciis cvlileba (dp, dq) axal wertilSi. Semogvaqvs damatebiTi pirobis saxiT gamosaxulebis ganaTebulobis gantoleba E(x,y)=R(p,q), sadac E aris gamosaxulebis sikaSkaSis mniSvneloba (x,y) wertilSi. gavakeToT ise, rom wanacvlebis mimarTuleba gamosaxulebis (x,y) sibrtyeze emTxveodes wanacvlebas (p, q) sibrtyeze. sabolood vRebulobT xuT kerZo warmoebulian Cveulebriv diferencialur gantolebas.

xў=R_p ; Yў= R_q ; Zў=p Ч R_p + q Ч R_q

pў=- R_x - p Ч R_z ; qў=- R_y - q Ч R_z

am gantolebaTa amoxsniT sivrceSi miiReba maxasiaTebeli mrudeebi, romlis gaswvriv vgebulobT zedapiris orientacias. TiToeuli mrudis gamoTvla unda iwyebodes wertilidan, romlis orientacia (p, q) winaswar cnobilia. es wertilebi zedapirze qmnian sawyis mruds.

rogorc yovelTvis, sawyisi mrudi sxeulze winaswar mocemuli ara gvaqvs. amisaTvis saWiroa sxeulis gamosaxulebis segmentacia - zedapirze unda gamovyoT iseTi wertilebi, romelzec sikaSkaSis gazomva erTmniSvnelovnad mogvcemda an wertilSi zedapiris orientaciis mniSnelobas. aseTi wertilebs tonalur gamosaxulebaze Seesabameba yvelaze naTeli an yvelaze bneli wertilebi. aq gvaqvs zedapiris iseTi orientacia, romlis drosac arekvlis unarianobis funqcia Rebulobs Sesabamisad maqsimum an minimum mniSvnelobas. am wertilebs ewodeba gansakuTrebuli wertilebi. yvela danarCeni wertilisTvis sruldeba piroba R(p,q) < R(p₀,q₀), an R(p,q) > R(p₀,q₀), sadac (p₀,q₀) aris gansakuTrebuli wertilis orientacia. samwuxarod am wertilebSi maxasiaTebeli mrudeebi ar gaivleba, radganac aq arekvlis unarianobis funqciis kerZo warmoebuli nulis tolia. amitom sawyis mrudi unda gatardes mis gverdze avliT, risTvisac unda Seiswavlos gansakuTrebuli wertilis midamo. midamos napiri, romelzedac gvecodineba zedapiris orientacia SeiZleba gamodges sawyis mrudad. midamo SeiZleba iyos Cazneqili an amozneqili zedapiris centri, amitom gansakuTrebul wertils ixilaven, rogorc sferos centrs, Tumca rogorc vnaxavT, es daSveba ara srulia.

ganvixiloT zedapiris asaxva sibrtyeze f: R³ ® R² , romelsac zedapiris nebismieri a wertili gadahyavs sibrtyis f(a) wertilSi. uitnis gansakuTrebulobis Teoriis Tanaxmad, nebismieri wertilisTvis moiZebneba zedapirze x₁ x₂ da sibrtyeze y₁ y₂ koordinatTa sistema, romlisTvisac asaxva gamoisaxeba erT-erTi qvemoT mocemuli sami formiT /2/:

a) y₁ = x₁ ; y₂ =x₂ (1)

b) y₁ = x₁² ; y₂ = x₂ (2)

g) y₁ = x₁³ + x₁ Ч x₂ ; y₂ = x₂ (3)

morsis Teoremis Tanaxmad /3/, asaxvis f: Rⁿ ® R funqcia kritikuli wertilis axlos daiyvaneba formamde:

± x² ... ± x² _n + f(x_m)

sadac n - asaxvis meoradi kerZo warmoebulebis kvadratuli formis, anu heses matricis rangia, xolo m matricis korangi. iseTi kritikul wertilebs, romlebisaTvisac matrica gadagvarebulia, anu matricis rangi ar udris mis ganzomilebas, da Sesabamisad korangi nulze metia, gansakuTrebuli kritikuli wertilebi hqvia. arsebobs mxolod ori gansakuTrebuli wertili, romlebsac Seesabameba (2) da (3) asaxva /4/.

ganvixiloT am zedapiris x1 x3 sibrtyis paraleluri kveTebi: davafiqsiroT mniSvneloba x2=a. TiToeul a mniSvnelobas x1 x3 sibrtyeze Seesabameba mrude, kuburi parabola: x3= x³- a Ч x1 . mrudis forma damokidebulia a-s niSanze:

Tu a<0, gvaqvs ori kritikuli wertili: maqsimumi da minimumi. Tu a=0 gvaqvs erTi, gadaRunvis wertili, xolo Tu a>0, kritikuli wertili ara aris. Tu davadebT erTmaneTs mrudeebs da vimoZravebT x2 RerZis gaswvriv, miviRebT sivrceSi zedapirs, romelsac gaaCnia uitnis nakeci wertilebi, am wertilebSi zedapiris mxebi sibrtyis paraleluria. im wertils, romelSic x2 RerZze moZraobisas gavivliT, da sadac mrudis kritikuli wertilebis raodenoba icvleba, ewodeba uitnis naoWi wertili.

x1x2 sibrtyis paralelur sibrtyeze proeqtirebisas (sur 2) uitnis nakeci wertilebs Seesabameba parabola, romlis erT Stos Seesabameba zedapiris maqsimumi, meore Stos zedapiris minimumi. Stoebi erTiandebian P uitnis naoW wertilSi.

y1x2 sibrtyis paralelur sibrtyeze proeqtirebisas (sur 3.) nakec wertilebs Seesabameba naxevarkuburi parabola, naoW wertils ki parabolis wvero, gansakuTrebuli wertili P.

aigo naoWi zedapiris tonaluri gamosaxuleba. 1-suraTze miRebulia gamosaxulebebi sxvadasxva xedvis a kuTxiT

a - kuTxe x3 RerZsa da xedvis veqtors Soris, zedapiri moZraobs saaTis isris mimarTulebiT

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2. Математическая энциклопедия. М. ''Советская Энциклопедия'' 1984, т.4 с.126
3. Постон Т., Стюарт И. Теория катастроф и её приложения М. ''Мир'' 1980, 608с.
4. Арнольд Б.И. Теория катастроф М. ''Наука'' 1990, 128 с.