sections ▁of ▁E uclidean ▁lines ▁with ▁the ▁open ▁unit ▁disk ). ▁The ▁last ▁model ▁has ▁the ▁advantage ▁that ▁it ▁gives ▁a ▁construction ▁which ▁is ▁completely ▁parallel ▁to ▁that ▁of ▁the ▁unit ▁sphere ▁in ▁ 3 - dimensional ▁E uclidean ▁space . ▁Because ▁of ▁their ▁application ▁in ▁complex ▁analysis ▁and ▁geometry , ▁however , ▁the ▁models ▁of ▁Po inc ar é ▁are ▁the ▁most ▁widely ▁used : ▁they ▁are ▁inter change able ▁thanks ▁to ▁the ▁M ö b ius ▁transformations ▁between ▁the ▁disk ▁and ▁the ▁upper ▁half - plane . ▁ ▁Let ▁ ▁be ▁the ▁Po inc ar é ▁disk ▁in ▁the ▁complex ▁plane ▁with ▁Po inc ar é ▁metric ▁ ▁In ▁polar ▁coordinates ▁ ▁the ▁metric ▁is ▁given ▁by ▁ ▁The ▁length ▁of ▁a ▁curve ▁ ▁is ▁given ▁by ▁the ▁formula ▁ ▁The ▁group ▁ ▁given ▁by ▁ ▁acts ▁trans it ively ▁by ▁M ö b ius ▁transformations ▁on ▁ ▁and ▁the ▁stabil izer ▁subgroup ▁of ▁ 0 ▁is ▁the ▁rotation ▁group ▁ ▁The ▁quot ient ▁group ▁ ▁is ▁the ▁group ▁of ▁orientation - pres erving ▁is omet ries ▁of ▁. ▁Any ▁two ▁points ▁, ▁ ▁in ▁ ▁are ▁joined ▁by ▁a ▁unique ▁ge odes ic , ▁given ▁by ▁the ▁portion ▁of ▁the ▁circle ▁or ▁straight ▁line ▁passing ▁through ▁ ▁and ▁ ▁and ▁orthogonal ▁to ▁the ▁boundary ▁circle . ▁The ▁distance ▁between ▁ ▁and ▁ ▁is ▁given ▁by ▁ ▁In ▁particular ▁ ▁and ▁ ▁is ▁the ▁ge odes ic ▁through ▁ 0 ▁along ▁the ▁real ▁axis , ▁paramet riz ed ▁by ▁ar cl