ic ▁triangle ▁on ▁the ▁sphere . ▁It ▁is ▁defined ▁by ▁points ▁, ▁, ▁ ▁on ▁the ▁sphere ▁with ▁sides ▁, ▁, ▁ ▁formed ▁from ▁great ▁circle ▁ar cs ▁of ▁length ▁less ▁than ▁. ▁If ▁the ▁lengths ▁of ▁the ▁sides ▁are ▁, ▁, ▁ ▁and ▁the ▁angles ▁between ▁the ▁sides ▁, ▁, ▁, ▁then ▁the ▁spher ical ▁cos ine ▁law ▁states ▁that ▁ ▁The ▁area ▁of ▁the ▁triangle ▁is ▁given ▁by ▁. ▁ ▁Using ▁s tere ographic ▁projection ▁from ▁the ▁North ▁pole , ▁the ▁sphere ▁can ▁be ▁identified ▁with ▁the ▁extended ▁complex ▁plane ▁. ▁The ▁explicit ▁map ▁is ▁given ▁by ▁ ▁Under ▁this ▁correspond ence ▁every ▁rotation ▁of ▁ ▁corresponds ▁to ▁a ▁M ö b ius ▁transformation ▁in ▁, ▁unique ▁up ▁to ▁sign . ▁With ▁respect ▁to ▁the ▁coordinates ▁ ▁in ▁the ▁complex ▁plane , ▁the ▁spher ical ▁metric ▁becomes ▁ ▁The ▁unit ▁sphere ▁is ▁the ▁unique ▁closed ▁orient able ▁surface ▁with ▁constant ▁curv ature ▁+ 1 . ▁The ▁quot ient ▁ ▁can ▁be ▁identified ▁with ▁the ▁real ▁project ive ▁plane . ▁It ▁is ▁non - orient able ▁and ▁can ▁be ▁described ▁as ▁the ▁quot ient ▁of ▁ ▁by ▁the ▁ant ip od al ▁map ▁( multip lication ▁by ▁− 1 ). ▁The ▁sphere ▁is ▁simply ▁connected , ▁while ▁the ▁real ▁project ive ▁plane ▁has ▁fundamental ▁group ▁. ▁The ▁finite ▁sub groups ▁of ▁, ▁corresponding ▁to ▁the ▁finite ▁sub groups ▁of ▁ ▁and ▁the ▁symmetry ▁groups ▁of ▁the ▁plat onic ▁sol ids , ▁do ▁not ▁act ▁freely ▁on ▁, ▁so ▁the ▁corresponding ▁quot ients