ant ▁derivative ▁with ▁respect ▁to ▁the ▁velocity ▁vector ▁of ▁the ▁curve . ▁Par allel ▁transport ▁along ▁ge odes ics , ▁the ▁" stra ight ▁lines " ▁of ▁the ▁surface , ▁can ▁also ▁easily ▁be ▁described ▁directly . ▁A ▁vector ▁in ▁the ▁tang ent ▁plane ▁is ▁transport ed ▁along ▁a ▁ge odes ic ▁as ▁the ▁unique ▁vector ▁field ▁with ▁constant ▁length ▁and ▁making ▁a ▁constant ▁angle ▁with ▁the ▁velocity ▁vector ▁of ▁the ▁ge odes ic . ▁For ▁a ▁general ▁curve , ▁this ▁process ▁has ▁to ▁be ▁modified ▁using ▁the ▁ge odes ic ▁curv ature , ▁which ▁measures ▁how ▁far ▁the ▁curve ▁dep arts ▁from ▁being ▁a ▁ge odes ic . ▁ ▁A ▁vector ▁field ▁ ▁along ▁a ▁unit ▁speed ▁curve ▁, ▁with ▁ge odes ic ▁curv ature ▁, ▁is ▁said ▁to ▁be ▁parallel ▁along ▁the ▁curve ▁if ▁ ▁it ▁has ▁constant ▁length ▁ ▁the ▁angle ▁ ▁that ▁it ▁makes ▁with ▁the ▁velocity ▁vector ▁ ▁satisfies ▁ ▁This ▁re capt ures ▁the ▁rule ▁for ▁parallel ▁transport ▁along ▁a ▁ge odes ic ▁or ▁piece wise ▁ge odes ic ▁curve , ▁because ▁in ▁that ▁case ▁, ▁so ▁that ▁the ▁angle ▁ ▁should ▁remain ▁constant ▁on ▁any ▁ge odes ic ▁segment . ▁The ▁existence ▁of ▁parallel ▁transport ▁follows ▁because ▁ ▁can ▁be ▁computed ▁as ▁the ▁integral ▁of ▁the ▁ge odes ic ▁curv ature . ▁Since ▁it ▁therefore ▁depends ▁continu ously ▁on ▁the ▁ ▁norm ▁of ▁, ▁it ▁follows ▁that ▁parallel ▁transport ▁for ▁an ▁arbitrary ▁curve ▁can ▁be ▁obtained ▁as ▁the ▁limit ▁of ▁the ▁parallel ▁transport ▁on ▁approxim ating