We accurately determine the fundamental system parameters of the neutron–star X-ray transient Cen X–4 solely using phase-resolved high–resolution UVES spectroscopy . We first determine the radial-velocity curve of the secondary star and then model the shape of the phase-resolved absorption line profiles using an X-ray binary model . The model computes the exact rotationally broadened phase-resolved spectrum and does not depend on assumptions about the rotation profile , limb-darkening coefficients and the effects of contamination from an accretion disk . We determine the secondary star-to-neutron star binary mass ratio to be 0.1755 \pm 0.0025 , which is an order of magnitude more accurate than previous estimates . We also constrain the inclination angle to be 32 ^ { \circ } ^ { +8 ^ { \circ } } _ { -2 ^ { \circ } } . Combining these values with the results of the radial velocity study gives a neutron star mass of 1.94 ^ { +0.37 } _ { -0.85 } M _ { \odot } consistent with previous estimates . Finally , we perform the first Roche tomography reconstruction of the secondary star in an X-ray binary . The tomogram reveals surface inhomogeneities that are due to the presence of cool starspots . A large cool polar spot , similar to that seen in Doppler images of rapidly-rotating isolated stars is present on the Northern hemisphere of the K7 secondary star and we estimate that \sim 4 per cent of the total surface area of the donor star is covered with spots . This evidence for starspots supports the idea that magnetic braking plays an important role in the evolution of low-mass X-ray binaries .