A study has been made of vertically incident cosmic-ray trajectories at more than 400 positions on the earth's surface using sixth-degree simulations of the quiescent geomagnetic field. Twenty-six of these positions are near the South African magnetic anomaly, and six other points are in a region of the North Atlantic where Pomerantz and Agarwal have reported anomalous results. The trajectories have been calculated for rigidities separated by intervals as small as 0.01 bv to allow for the effects of the penumbra. When these trajectories are used to determine cosmic-ray cutoff rigidities, differences greater than 15 percent from the Quenby and Wenk threshold values are found in the vicinity of South Africa, the South Atlantic, and the Canary Islands. Various latitude surveys are shown to exhibit consistency when plotted against the cutoff rigidities determined by this computational method. It is concluded that although currently available simulations of the geomagnetic field are not completely adequate over some portions of the earth to describe the cosmic-ray effects in their entirety, there is essentially no difference in the cutoff rigidities determined using two currently accepted field models, even though different penumbral structures are obtained. Asymptotic directions of approach and cutoff rigidities for a number of locations are listed.