अनुरूपी निरूपण

anuroopi niroopan ek tal par bani kisi akriti ko doosare ke lie doosari akriti mean ek hi sangat biandoo ho, aur isake atirikt, donoan akritiyoan ke sangatakon barabar ho, anuroopi niroopan (kanph aaurmal riprejeanteshan) kahate hai, kyoanki isamean ek akriti ka doosari akriti mean is prakar niroopan hota hai ki donoan akritiyoan ke chhote chhote bhag anuroop (similar) bane rahate haian.

man lijie, ek tal mean k kh g ek tribhuj hai aur doosare tal mean ki, khi, gi sangat tribhuj hai. yah avashyak nahian hai ki tribhujoan ki bhujaean rriju rekhaean hoan to bhi, jab tribhujoan ke akar bahut chhote ho jaeange, ham unhean rriju rekhaoan ke sadrish hi man sakate haian.

center|

jab biandu kh, g biandu k ki or pravritt hoange, tab sangat biandu khi, gi biandu ki ki or pravritt hoange. yad niroopan anuroopi ho to aant mean tribhuj k kh g aur ki khi gi ke sangat kon saman ho jaeange aur sangat bhujaean anupati ho jaeangi. at: jo do vakr k par milate haian, unaka madhyasth kon un do vakroan ke madhyasth kon ke barabar hoga jo ki par milate haian.

anuroopi niroopan ka sabase prasiddh prayog marketar prakshep kahalata hai. jisake dvara bhoomandal ki akritiyoan ka chitran samatal par kiya jata hai (dr. 'marketar prakshep').

laianbart ne sanh‌ 1772 mean ukt prashn ka adhik vyapak roop se adhyayan kiya. pichhai laiangraanj ne bataya ki is vishay ka sanmishr char ke phalanoan (phankashans aauv e kanpleks veriebul) se kya sanbandh hai. sanh‌ 1822 mean kopinahaigan ki vijnan parishadh ne ek puraskar ke lie yah vishay prastavit kiya ki ek tal ke vibhinn bhag doosare tal par is kaise chitrit kie jaean ki pratibianb ke chhote se chhote bhag maulik tal ke sangat bhagoan ke anuroop hoan? gaus ne sanh‌ 1825 mean is samasya k hal nikala aur vahian se is vishay ke vyapak siddhaant ka aranbh hua. pichhale 50 varshoan mean is kshetr ke any karyakartaoan mean riman, shvarj,aur klain ullekhaniy haian.

man lijie ki s=sh (y, r)+shrash (y, r) sanmishr rashi l=y+shrar ka ek vaishleshik phalan hai, jisamean shr=Ö (-1). yah saralata se siddh kiya ja sakata hai ki phalan ki vaishleshikata ke lie avashyak aur paryapt shartean ye haian:

center|

in samikaranoan ka koshi riman samikaran kahate haian. jab ye samikaran santusht ho jate hai tab, yadi ham y, r samatal ki kisi akriti ka niroopan sh, sh samatal karean, to niroopan anuroopi hoga aur konoan mean koee parivartan nahian hoga. isake lie yah avashyak hai ki donoan phalan sh tatha sh satat hoan aur unake charoan aanshik avakal gunak

center|

bhi satat hoan. akritiyoan ki anuroopata keval un bianduoan par tootegi jahaan uparilikhit charoan avakal gunak shoony ho jaeange.

udaharan ke lie ham koee bhi vaishleshik phalan s=ph (l) lean sakate hai, jaise l2, kojya l athava jya l. yadi ham s=l2 (l shrar)2 lean to sh=y2-r2 aur sh=2 y r.

phir
center|

yadi ham y, r samatal mean rriju rekhaoan ki do sanhatiyaan y=k;r=kh lean, jo paraspar lanb hoan, to sh, sh samatal mean unaki akritiyoan paravalay hoangi : sh2=4k2 (k2-sh) aur sh2=4kh2 (kh2+ sh) jo samanabhi aur samakoniy haian. spasht hai ki y, r samatal ke samakon sh, sh samatal mean bhi samakonoan se hi niroopit hote haian.

isi prakar yadi ham sh, sh samatal mean do rekhapuanj lean : sh=g, sh=gh jo samakoniy haian, to y, r samatal aur ayatakar atiparavalay y2-r2=g aur 2yar=gh unaki sangat akritiyaan hoangi. spasht hai ki is niroopan mean bhi akritiyoan ke konagun akshunn bane rahate haian.

panne ki pragati avastha

adhar

prarambhik

madhyamik

poornata

shodh

tika tippani aur sandarbh

1. san.gran.-e.ar. phorasaiy : thyori aauv phankshans; dablyoo.eph. aausagud : kanapharmal riprejeanteshan aauv sarpheans ap aaun anadar.