thumb|400px|Two uniformly distributed values, u and v are used to produce the value , which is likewise uniformly distributed. The definitions of the sine and cosine are then applied to the basic form of the Box–Muller transform to avoid using trigonometric functions. The polar form was first proposed by J. Bell and then modified by R. Knop. While several different versions of the polar method have been described, the version of R. Knop will be described here because it is the most widely used, in part due to its inclusion in Numerical Recipes. A slightly different form is described as "Algorithm P" by D. Knuth in The Art of Computer Programming.
ThuMB|400Px|two UNIfOrmly dIstRIBuTed valuEs, U And V aRe USed tO produce tHe vaLue , whiCh is likeWisE uniforMly DiSTriButed. the DEfInitionS of The SINe anD cosInE Are TheN Applied to THe BAsic form oF The box–MulLer TRaNsfOrM to Avoid uSiNg TrigonoMetric FunctioNs. The PolAR forM WAs FiRSt PROposeD By j. bell and THen MoDIfieD By r. kNop. wHIlE SeverAL diFFerent veRsIonS OF tHE poLar meThod hAvE been desCribed, thE VErsIon of r. KNOp will bE described Here beCaUse iT is the MosT Widely usED, In part Due to its InclusiON in nUmeriCal reciPes. A slightly diffeRENT foRM is deScribed As "algorIthm p" by d. knUTh IN the art of Computer proGramMINg.