▁the ▁curv atures ▁of ▁certain ▁plane ▁curves ▁connected ▁with ▁the ▁surface . ▁He ▁later ▁found ▁a ▁series ▁of ▁equivalent ▁definitions . ▁One ▁of ▁the ▁first ▁was ▁in ▁terms ▁of ▁the ▁area - expand ing ▁properties ▁of ▁the ▁G auss ▁map , ▁a ▁map ▁from ▁the ▁surface ▁to ▁a ▁ 2 - dimensional ▁sphere . ▁However , ▁before ▁obtain ing ▁a ▁more ▁intr insic ▁definition ▁in ▁terms ▁of ▁the ▁area ▁and ▁angles ▁of ▁small ▁tri angles , ▁G auss ▁needed ▁to ▁make ▁an ▁in - depth ▁investigation ▁of ▁the ▁properties ▁of ▁ge odes ics ▁on ▁the ▁surface , ▁i . e . ▁paths ▁of ▁short est ▁length ▁between ▁two ▁fixed ▁points ▁on ▁the ▁surface ▁( see ▁below ). ▁ ▁The ▁Gaussian ▁curv ature ▁at ▁a ▁point ▁on ▁an ▁embedded ▁smooth ▁surface ▁given ▁locally ▁by ▁the ▁equation ▁ ▁in ▁E uclidean ▁space ▁( ), ▁is ▁defined ▁to ▁be ▁the ▁product ▁of ▁the ▁principal ▁curv atures ▁at ▁the ▁point ; ▁the ▁mean ▁curv ature ▁is ▁defined ▁to ▁be ▁their ▁average . ▁The ▁principal ▁curv atures ▁are ▁the ▁maximum ▁and ▁minimum ▁curv atures ▁of ▁the ▁plane ▁curves ▁obtained ▁by ▁intersect ing ▁the ▁surface ▁with ▁plan es ▁normal ▁to ▁the ▁tang ent ▁plane ▁at ▁the ▁point . ▁If ▁the ▁point ▁is ▁ ▁with ▁tang ent ▁plane ▁, ▁then , ▁after ▁a ▁rotation ▁about ▁the ▁- axis ▁setting ▁the ▁coefficient ▁on ▁ ▁to ▁zero , ▁ ▁will ▁have ▁the ▁Taylor ▁series ▁expansion ▁ ▁The ▁principal ▁curv atures ▁are ▁ ▁and ▁. ▁In ▁this ▁case , ▁the ▁Gaussian ▁curv ature