▁B ip olar ▁cyl ind rical ▁coordinates ▁are ▁a ▁three - dimensional ▁orthogonal ▁coordinate ▁system ▁that ▁results ▁from ▁project ing ▁the ▁two - dimensional ▁b ip olar ▁coordinate ▁system ▁in ▁the ▁per pend icular ▁- direction . ▁ ▁The ▁two ▁lines ▁of ▁fo ci ▁▁ ▁and ▁ ▁of ▁the ▁project ed ▁Ap oll onian ▁circles ▁are ▁generally ▁taken ▁to ▁be ▁ ▁defined ▁by ▁ ▁and ▁, ▁respectively , ▁( and ▁by ▁) ▁in ▁the ▁Cart esian ▁coordinate ▁system . ▁ ▁The ▁term ▁" b ip olar " ▁is ▁often ▁used ▁to ▁describe ▁other ▁curves ▁having ▁two ▁singular ▁points ▁( fo ci ), ▁such ▁as ▁elli ps es , ▁hyper bol as , ▁and ▁Cass ini ▁o val s . ▁ ▁However , ▁the ▁term ▁b ip olar ▁coordinates ▁is ▁never ▁used ▁to ▁describe ▁coordinates ▁associated ▁with ▁those ▁curves , ▁e . g ., ▁elli ptic ▁coordinates . ▁ ▁Basic ▁definition ▁ ▁The ▁most ▁common ▁definition ▁of ▁b ip olar ▁cyl ind rical ▁coordinates ▁ ▁is ▁ ▁where ▁the ▁ ▁coordinate ▁of ▁a ▁point ▁ ▁equals ▁the ▁angle ▁ ▁and ▁the ▁▁ ▁coordinate ▁equals ▁the ▁natural ▁log arith m ▁of ▁the ▁ratio ▁of ▁the ▁distances ▁ ▁and ▁ ▁to ▁the ▁foc al ▁lines ▁ ▁( Rec all ▁that ▁the ▁foc al ▁lines ▁ ▁and ▁ ▁are ▁located ▁at ▁ ▁and ▁, ▁respectively .) ▁▁ ▁Sur faces ▁of ▁constant ▁ ▁correspond ▁to ▁cyl ind ers ▁of ▁different ▁rad ii ▁ ▁that ▁all ▁pass ▁through ▁the ▁foc al ▁lines ▁and ▁are ▁not ▁concent ric . ▁