
saraina bhramanam ledhu

jyaamitiloo, bhramanam-pratibimbam, bhramana-pratibimbam, rotary pratibimbam ledhaa rotoinversion anie kooda piluvabade anuchita bhramanam anedi euclidean pradaesamloe oka isometry, idhi oka aksham chuttoo bhramanam mariyu aaa akshaniki lambamgaa unna samatalamlo pratibimbam kalayika. pratibimbam mariyu vilomam anevi prati okkati anuchita bhramanaaniki pratyeka sandarbhaalu. edainaa anuchitamaina bhramanam anedi oka anubandha parivartana mariyu, samanvaya muulaanni sthiramgaa unche sandarbhaallo, oka sarala parivartana.
idhi rekhaganita samaruupata, paramaanu samaruupata mariyu sphatika shaastram sandarbhamlo samaruupata operationga upayoginchabadutundi, ikkada bhramanam mariyu pratibimbam kalayika dwara maarani oka vastuvu "anuchita bhramana samaruupata" kaligi untundani chebutaaru.
moodu koonaalu.
3 kolatalalo, sarikani bhramanam anedi aksham chuttoo bhramanam mariyu aksham meeda oka binduvu loe vilomam kalayikaga samanamga nirvachinchabadutundi. eee kaaranamgaa dheenini rotainversion ledhaa rotary inversion anie kooda pilustaaru. rendu nirvachanalu samanamga untaayi endukante θ konam dwara bhramanam taruvaata pratibimbam anedi θ + 180° dwara bhramanam taruvaata vilomam (vilomam yokka binduvunu pratibimbam yokka samatalamlo unchadam) vale adhe parivartana. rendu nirvachanaalalo, karyakalapalu prayaanistaayi.
oke oka sthira binduvunu kaligi unna trimiteeya samaruupata tappanisarigaa anuchita bhramanam. 
oka vastuvu yokka anuchita bhramanam daani addam chitram yokka bhramanaanni utpatti chestundi. eee akshanni bhramana-pratibimbam aksham antaaru. prativartanaku mundu ledhaa taruvaata bhramana konam 360°/"n" (ikkada "n" sarri undaali) ayithe dheenini "n"-madata anuchita bhramanam antaaru. vyaktigata anuchita bhramanaalaku paeru pettadaniki aneka vibhina vyavasthalu unnaih
paroksha isometriga.
vistruta konamlo, anuchita bhramanaanni edainaa paroksha isometriga nirvachinchavacchu; i.ii., e (3) \e + (3) yokka moolakamh anduvalla idhi oka samatalamlo swachchamaina pratibimbam kooda kaavacchu ledhaa glide samatalaanni kaligi undavacchu. paroksha isometry anedi-1 yokka nirnayakaanni kaligi unna orthogonal maatrikatoe kuudina anubandha parivartana.
saraina bhramanam anedi oka saadhaarana bhramanam. vistruta konamlo, saraina bhramanam anedi pratyaksha samamitigaa nirvachinchabadindi; i.ii., e + (3) yokka moolakamh idhi gurtimpu, aksham venta anuvaadamtoe kuudina bhramanam ledhaa swachchamaina anuvaadam kooda kaavacchu. pratyaksha isometry anedi 1 yokka nirnayakaanni kaligi unna orthogonal maatrukato kuudina anubandha parivartana.
irukaina ledhaa vistruta indriyaalalo, rendu anuchita bhramanaala kuurpu saraina bhramanam, mariyu sarikani mariyu saraina bhramanam yokka kuurpu sarikani bhramanam.
bhouthika vyavasthalu.
oka bhouthika vyavastha yokka samaruupatanu anuchita bhramanamlo adhyayanam chesetappudu (udhaa.zii.oka vyavasthaku addam samaruupata sthaayi unte), vahakalu mariyu sudovectorla (alaage scalarlu mariyu sudoscalorlu, mariyu saadhaaranamgaa tensorlu mariyu sudotensorla madhya) madhya thedaanu gurtinchadam chaalaa mukhyam, endukante rendodi saraina mariyu anuchita bhramanaala krinda bhinnamgaa maarutundi (3 kolatalalo, sudovectorlu vilomam krinda asthiramgaa untaayi).

