mallus-dupin pramey

mallus-dupin pramey jyaamitiya prakashiki mein ack pramey haye jisaki khoj 1808 mein attien-luis mallus dwara kii gaee thee aur 1822 mein charles dupin dwara spasht kiya gayaa thaa. hamilton ney ise apani hamiltonian prakashiki vidhi key ack saral anuprayog key roop mein saabit kiya.
ack samroop madhyam mein prakaash kiranon kii ack pencil parr vichaar karein joe kissy satah key lambavat haye. kiranon kii pencil koo manmaane dhang seey paraavartan aur apavartan key madhyam seey paarit karein, phir ise kissy anya sajaatiya madhyam mein ubharane dein. pramey mein kahaa gayaa haye kii prakaash kiranon kii parinami pencil abhee bhee kissy anya satah key lambavat haye.
prabhaav.
kuchh kiran pencil kissy bhee satah key lambavat naheen hoti hain. yeh sampark jyaamiti seey sambandhit haye. 
udaaharan key liye, liye parr maanak sampark sanrachana parr vichaar karein, jise aseem talon key kshetra key roop mein samajha jaa sakata haye. abb, pratyek anant tala key lambavat, ack prakaash kiran banaaen. yeh hamein prakaash kiranon kii ack "samtal ghumaavdaar" pencil detaa haye. pencil key liye koee satah lambavat naheen haye, kyonki aisee koee satah naheen haye joe pratyek anant tala key liye sparshrekha hoe sakati haye. 
maan leejiye kii vahaan haye, tou x-y tala mein ack anant varg banaaen, aur satah key saath maarg kaa paalan karein. ack paripath key baad maarg usee jaid-samanvaya parr vaapas naheen aayega. virodhabhas.
isi tarah, koee bhee golakar pencil kaa ack crum banaa sakata haye, jisamen seey pratyek ack-sheet kaa hyperboloid banaati haye. parinami "mudey hue cylinder" pencil kissy bhee satah key lambavat naheen hogi.
mallus-dupin pramey kaa tatparya haye kii paraavartan aur apavartan kii koee bhee matra aisee kiranon kii pencil koo samanantar kiranon kii pencil, yaa ack bindu parr abhisaran karane vaali kiranon kii pencil, yaa kissy satah key lambavat kiranon kii pencil mein parivartit naheen curr sakati haye.
nirmaan aur pramaan.
kiranon kii ack pencil diye jaane parr, maan leejiye kii kiranon kii pencil shuru mein ack satah m key lambavat haye.
kiranon kii pencil koo paraavartit karane aur apavartit karane vaali samagri kii ack manmaane pranali key madhyam seey paarit karein. udaaharan key liye, dou kiranon parr vichaar karein joe a aur a 'parr apavartit hoti hain; joe b aur b' parr paraavartit hoti hain; aur apavartit bindu c aur c 'parr. 
abb pencil seey dou kiranen lenh [mabcp] aur [m 'a' b 'c' p]. maan leejiye kii donon kiranen kramshah m aur m 'binduon parr m key lambavat houn.
bindu pee 'chunein taaki [aim. aeye. bee. see. pee.] aur [aim. aeye. bee. see. pee.'] kaa ack hii optical path hoe; phir sabhi pee, pee 'kaa samaan optical path key saath sett ack ghumaavdaar satah pee banaata haye. 

