A crossover method for determining zonal and meridional ocean current components is examined using data at three crossovers of TOPEX/Poseidon and JASON-1 ground tracks over 2002 - 2006. To implement this method, a geoid model around Taiwan is constructed using surface and airborne gravity data. The modeled and observed geoidal heights at coastal benchmarks are consistent to 5 cm RMS with the means removed. The error and limitation of this method are discussed, concluding that, in order to obtain current velocities at a 10 cm s^-1 accuracy and a 6-km resolution, the dynamic ocean topography (DOT) at a mm-level accuracy is needed, which is not possible to achieve today. By filtering DOT to a spatial scale of 100 km or coarser, a 10 cm s^-1 accuracy of velocity may be obtained. One crossover (A) is situated south of Taiwan and near the Kuroshio, the second (B) is at the axis of the Kuroshio and the third is located in the northern Taiwan Strait. These three crossovers feature different ocean current patterns. At a spatial scale of 120 km, the agreement among the altimeter, the Princeton Ocean Model (POM), and the drifter-derived velocities is the best at B, followed by that at A, and then C. In fact, at C the altimeter-derived velocities contradict the POM-derived values, and the tide model error is to be blamed. Further improvement on geoid modeling is suggested.