ature ▁is ▁an ▁intr insic ▁invariant . ▁ ▁In ▁is oth erm al ▁coordinates , ▁first ▁considered ▁by ▁G auss , ▁the ▁metric ▁is ▁required ▁to ▁be ▁of ▁the ▁special ▁form ▁ ▁In ▁this ▁case ▁the ▁Lap lace – Bel tr ami ▁operator ▁is ▁given ▁by ▁ ▁and ▁ ▁satisfies ▁Li ou ville ' s ▁equation ▁ ▁Is oth erm al ▁coordinates ▁are ▁known ▁to ▁exist ▁in ▁a ▁neighbourhood ▁of ▁any ▁point ▁on ▁the ▁surface , ▁although ▁all ▁proofs ▁to ▁date ▁rely ▁on ▁non - t rivial ▁results ▁on ▁partial ▁differential ▁equations . ▁There ▁is ▁an ▁elementary ▁proof ▁for ▁minimal ▁surfaces . ▁ ▁G auss – B on net ▁theorem ▁▁ ▁On ▁a ▁sphere ▁or ▁a ▁hyper bol oid , ▁the ▁area ▁of ▁a ▁ge odes ic ▁triangle , ▁i . e . ▁a ▁triangle ▁all ▁the ▁sides ▁of ▁which ▁are ▁ge odes ics , ▁is ▁proportional ▁to ▁the ▁difference ▁of ▁the ▁sum ▁of ▁the ▁interior ▁angles ▁and ▁. ▁The ▁constant ▁of ▁proportional ity ▁is ▁just ▁the ▁Gaussian ▁curv ature , ▁a ▁constant ▁for ▁these ▁surfaces . ▁For ▁the ▁tor us , ▁the ▁difference ▁is ▁zero , ▁reflect ing ▁the ▁fact ▁that ▁its ▁Gaussian ▁curv ature ▁is ▁zero . ▁These ▁are ▁standard ▁results ▁in ▁spher ical , ▁hyper bol ic ▁and ▁high ▁school ▁trig on ometry ▁( see ▁below ). ▁G auss ▁general ised ▁these ▁results ▁to ▁an ▁arbitrary ▁surface ▁by ▁showing ▁that ▁the ▁integral ▁of ▁the ▁Gaussian ▁curv ature ▁over ▁the ▁interior ▁of ▁a ▁ge odes ic ▁triangle ▁is ▁also ▁equal ▁to