1. An automatic system for determining a position and direction-of-motion of a vehicle, which system comprises set of position markers fixed in known positions, a density sensor mounted on the vehicle and positioned so it determines density along a line as the vehicle moves, a computer receiving density values determined by the density sensor, and maintaining an estimate of the position and direction-of-motion of the vehicle, and a computer control system that directs the computer to perform computations for estimating the position and direction-of-motion of the vehicle from the density values as the sensor passes across any one of the position markers; each of the position markers comprising one or more density patterns in a finite two-dimensional region; each of the density patterns filling a two-dimensional region of a position marker; each of the density patterns having a known orientation vector; in the two-dimensional region, a density of each density pattern being a known periodic function of position in a direction of its orientation vector, and independent of position in a direction orthogonal to its orientation vector; at each point of the two-dimensional region the position marker having a density which is a combination of its density-pattern densities at the same point; the computer storing the density values it receives as a path density function until it determines that it has a quantity of them that is sufficient for performing the computations needed to estimate a current position and direction-of-motion of the vehicle; the computer then performing mathematical operations necessary to determine frequency, phase, and amplitude values of various sinusoidal components of the path density function; the computer then using so-determined amplitude value of each sinusoidal component of the path density function to determine which density pattern corresponds to that sinusoidal component; the computer then performing mathematical operations necessary to use so-determined frequency and phase values of sinusoidal components of the path density function and spatial frequency and phase values of corresponding known density patterns of position markers to determine the vehicle's direction-of-motion and an offset of a current estimate of the vehicle's position from a vehicle's actual position, and then using that offset to correct the current estimate of the vehicle's position.