▁are ▁not ▁ 2 - man if olds , ▁just ▁orb if olds . ▁ ▁Hyper bol ic ▁geometry ▁ ▁Non - E uclidean ▁geometry ▁was ▁first ▁discussed ▁in ▁letters ▁of ▁G auss , ▁who ▁made ▁extensive ▁comput ations ▁at ▁the ▁turn ▁of ▁the ▁nin ete enth ▁century ▁which , ▁although ▁priv ately ▁circul ated , ▁he ▁decided ▁not ▁to ▁put ▁into ▁print . ▁In ▁ 1 8 3 0 ▁Lob ache v sky ▁and ▁independently ▁in ▁ 1 8 3 2 ▁Bol y ai , ▁the ▁son ▁of ▁one ▁G auss ' ▁correspond ents , ▁published ▁synth etic ▁versions ▁of ▁this ▁new ▁geometry , ▁for ▁which ▁they ▁were ▁sever ely ▁critic ized . ▁However ▁it ▁was ▁not ▁until ▁ 1 8 6 8 ▁that ▁Bel tr ami , ▁followed ▁by ▁Klein ▁in ▁ 1 8 7 1 ▁and ▁Po inc ar é ▁in ▁ 1 8 8 2 , ▁gave ▁concrete ▁analyt ic ▁models ▁for ▁what ▁Klein ▁dub bed ▁hyper bol ic ▁geometry . ▁The ▁four ▁models ▁of ▁ 2 - dimensional ▁hyper bol ic ▁geometry ▁that ▁emer ged ▁were : ▁the ▁Bel tr ami - K lein ▁model ; ▁the ▁Po inc ar é ▁disk ; ▁the ▁Po inc ar é ▁upper ▁half - plane ; ▁the ▁hyper bol oid ▁model ▁of ▁Wilhelm ▁K illing ▁in ▁ 3 - dimensional ▁M ink owski ▁space . ▁ ▁The ▁first ▁model , ▁based ▁on ▁a ▁disk , ▁has ▁the ▁advantage ▁that ▁ge odes ics ▁are ▁actually ▁line ▁segments ▁( that ▁is , ▁inter