▁in ▁. ▁ ▁Sur faces ▁of ▁constant ▁Gaussian ▁curv ature ▁ ▁If ▁a ▁surface ▁has ▁constant ▁Gaussian ▁curv ature , ▁it ▁is ▁called ▁a ▁surface ▁of ▁constant ▁curv ature . ▁ ▁The ▁unit ▁sphere ▁in ▁ ▁has ▁constant ▁Gaussian ▁curv ature ▁+ 1 . ▁The ▁E uclidean ▁plane ▁and ▁the ▁cyl inder ▁both ▁have ▁constant ▁Gaussian ▁curv ature ▁ 0 . ▁The ▁surfaces ▁of ▁revolution ▁with ▁ ▁have ▁constant ▁Gaussian ▁curv ature ▁– 1 . ▁Part icular ▁cases ▁are ▁obtained ▁by ▁taking ▁, ▁ ▁and ▁. ▁The ▁latter ▁case ▁is ▁the ▁classical ▁pseud os phere ▁generated ▁by ▁rot ating ▁a ▁tract rix ▁around ▁a ▁central ▁axis . ▁In ▁ 1 8 6 8 ▁Eugen io ▁Bel tr ami ▁showed ▁that ▁the ▁geometry ▁of ▁the ▁pseud os phere ▁was ▁directly ▁related ▁to ▁that ▁of ▁the ▁hyper bol ic ▁plane , ▁discovered ▁independently ▁by ▁Lob ache v sky ▁( 1 8 3 0 ) ▁and ▁Bol y ai ▁( 1 8 3 2 ). ▁Al ready ▁in ▁ 1 8 4 0 , ▁F . ▁M inding , ▁a ▁student ▁of ▁G auss , ▁had ▁obtained ▁trig on ometric ▁formulas ▁for ▁the ▁pseud os phere ▁identical ▁to ▁those ▁for ▁the ▁hyper bol ic ▁plane . ▁This ▁surface ▁of ▁constant ▁curv ature ▁is ▁now ▁better ▁understood ▁in ▁terms ▁of ▁the ▁Po inc ar é ▁metric ▁on ▁the ▁upper ▁half ▁plane ▁or ▁the ▁unit ▁disc , ▁and ▁has ▁been ▁described ▁by ▁other ▁models ▁such ▁as ▁the ▁Klein ▁model ▁or ▁the ▁hyper bol oid ▁model , ▁obtained ▁by