▁The ▁surfaces ▁of ▁constant ▁ ▁are ▁non - inter sect ing ▁cyl ind ers ▁of ▁different ▁rad ii ▁ ▁that ▁surr ound ▁the ▁foc al ▁lines ▁but ▁again ▁are ▁not ▁concent ric . ▁ ▁The ▁foc al ▁lines ▁and ▁all ▁these ▁cyl ind ers ▁are ▁parallel ▁to ▁the ▁- axis ▁( the ▁direction ▁of ▁projection ). ▁ ▁In ▁the ▁ ▁plane , ▁the ▁cent ers ▁of ▁the ▁constant - ▁and ▁constant - ▁cyl ind ers ▁lie ▁on ▁the ▁ ▁and ▁ ▁axes , ▁respectively . ▁ ▁Sc ale ▁factors ▁ ▁The ▁scale ▁factors ▁for ▁the ▁b ip olar ▁coordinates ▁ ▁and ▁▁ ▁are ▁equal ▁ ▁whereas ▁the ▁remaining ▁scale ▁factor ▁. ▁▁ ▁Thus , ▁the ▁inf init es imal ▁volume ▁element ▁equals ▁ ▁and ▁the ▁Lap la cian ▁is ▁given ▁by ▁▁ ▁Other ▁differential ▁operators ▁such ▁as ▁▁ ▁and ▁ ▁can ▁be ▁expressed ▁in ▁the ▁coordinates ▁ ▁by ▁subst itut ing ▁ ▁the ▁scale ▁factors ▁into ▁the ▁general ▁formula e ▁ ▁found ▁in ▁orthogonal ▁coordinates . ▁ ▁App lications ▁The ▁classic ▁applications ▁of ▁b ip olar ▁coordinates ▁are ▁in ▁solving ▁partial ▁differential ▁equations , ▁ ▁e . g ., ▁Lap lace ' s ▁equation ▁or ▁the ▁Hel m hol tz ▁equation , ▁for ▁which ▁b ip olar ▁coordinates ▁allow ▁a ▁ ▁separation ▁of ▁variables ▁( in ▁ 2 D ). ▁ ▁A ▁typical ▁example ▁would ▁be ▁the ▁electric ▁field ▁surrounding ▁two ▁ ▁parallel ▁cyl ind rical ▁conduct ors . ▁ ▁Bibli ography ▁ ▁External ▁links ▁Math World ▁description ▁of ▁b ip olar ▁cyl