▁is ▁a ▁unique ▁ge odes ic ▁connecting ▁the ▁base ▁point ▁to ▁each ▁sufficiently ▁nearby ▁point . ▁The ▁direction ▁of ▁the ▁ge odes ic ▁at ▁the ▁base ▁point ▁and ▁the ▁distance ▁uniqu ely ▁determine ▁the ▁other ▁endpoint . ▁These ▁two ▁bits ▁of ▁data , ▁a ▁direction ▁and ▁a ▁magnitude , ▁thus ▁determine ▁a ▁tang ent ▁vector ▁at ▁the ▁base ▁point . ▁The ▁map ▁from ▁tang ent ▁vectors ▁to ▁end points ▁smooth ly ▁swe eps ▁out ▁a ▁neighbourhood ▁of ▁the ▁base ▁point ▁and ▁defines ▁what ▁is ▁called ▁the ▁" ex pon ential ▁map ", ▁defining ▁a ▁local ▁coordinate ▁chart ▁at ▁that ▁base ▁point . ▁The ▁neighbourhood ▁swe pt ▁out ▁has ▁similar ▁properties ▁to ▁balls ▁in ▁E uclidean ▁space , ▁namely ▁any ▁two ▁points ▁in ▁it ▁are ▁joined ▁by ▁a ▁unique ▁ge odes ic . ▁This ▁property ▁is ▁called ▁" ge odes ic ▁convex ity " ▁and ▁the ▁coordinates ▁are ▁called ▁" normal ▁coordinates ". ▁The ▁explicit ▁calculation ▁of ▁normal ▁coordinates ▁can ▁be ▁accomplished ▁by ▁considering ▁the ▁differential ▁equation ▁satisfied ▁by ▁ge odes ics . ▁The ▁convex ity ▁properties ▁are ▁consequences ▁of ▁G auss ' s ▁lemma ▁and ▁its ▁general is ations . ▁R ough ly ▁speaking ▁this ▁lemma ▁states ▁that ▁ge odes ics ▁starting ▁at ▁the ▁base ▁point ▁must ▁cut ▁the ▁spher es ▁of ▁fixed ▁radius ▁cent red ▁on ▁the ▁base ▁point ▁at ▁right ▁angles . ▁Ge odes ic ▁polar ▁coordinates ▁are ▁obtained ▁by ▁combining ▁the ▁exponential ▁map ▁with ▁polar ▁coordinates ▁on ▁tang ent ▁vectors ▁at ▁the ▁base ▁point . ▁The ▁Gaussian ▁curv ature ▁of