The crosspolarisation of antenna systems involving 4-arm-structures depends on two factors: the quality of the feeding network and the shape of the antenna arms. Using a mathematical description for the separate treatment of these factors, planar archimedean and logarithmic spirals are investigated to optimise the arm shape. It is shown that a proper choice of the antenna shape can improve the crosspolarisation discrimination of the antenna system by more than 15 dB. It is also demonstrated that an optimised antenna shape yields significantly improved polarisation performance for the entire main beam region, even in the case of a poor excitation network.