▁the ▁t ors ion ▁of ▁a ▁space ▁curve ▁are ▁analog ous ▁to ▁the ▁curv ature ▁of ▁a ▁plane ▁curve . ▁For ▁example , ▁they ▁are ▁coefficients ▁in ▁the ▁system ▁of ▁differential ▁equations ▁for ▁the ▁F ren et ▁frame ▁given ▁by ▁the ▁F ren et – Ser ret ▁formulas . ▁ ▁Definition ▁▁ ▁Let ▁C ▁be ▁a ▁space ▁curve ▁paramet riz ed ▁by ▁arc ▁length ▁ ▁and ▁with ▁the ▁unit ▁tang ent ▁vector ▁t . ▁If ▁the ▁curv ature ▁ ▁of ▁C ▁at ▁a ▁certain ▁point ▁is ▁not ▁zero ▁then ▁the ▁principal ▁normal ▁vector ▁and ▁the ▁bin ormal ▁vector ▁at ▁that ▁point ▁are ▁the ▁unit ▁vectors ▁▁▁▁▁▁ ▁where ▁the ▁prime ▁denotes ▁the ▁derivative ▁of ▁the ▁vector ▁with ▁respect ▁to ▁the ▁parameter ▁. ▁The ▁t ors ion ▁ ▁measures ▁the ▁speed ▁of ▁rotation ▁of ▁the ▁bin ormal ▁vector ▁at ▁the ▁given ▁point . ▁It ▁is ▁found ▁from ▁the ▁equation ▁▁▁▁▁ ▁which ▁means ▁▁▁▁ ▁Rem ark : ▁The ▁derivative ▁of ▁the ▁bin ormal ▁vector ▁is ▁per pend icular ▁to ▁both ▁the ▁bin ormal ▁and ▁the ▁tang ent , ▁hence ▁it ▁has ▁to ▁be ▁proportional ▁to ▁the ▁principal ▁normal ▁vector . ▁The ▁negative ▁sign ▁is ▁simply ▁a ▁matter ▁of ▁convention : ▁it ▁is ▁a ▁by - product ▁of ▁the ▁historical ▁development ▁of ▁the ▁subject . ▁ ▁Ge ometric ▁relev ance : ▁The ▁t ors ion ▁ ▁measures ▁the ▁turn around ▁of ▁the ▁bin ormal ▁vector . ▁The ▁larger ▁the ▁t ors ion ▁is , ▁the ▁faster ▁the ▁bin ormal ▁vector ▁rot ates ▁around ▁the ▁axis ▁given ▁by