An E-beam lithography machine with a high precision X-Y stage is used to measure a grid plate with grid points at approximately known coordinates. Two interferometer beams, one parallel to the X-axis and one parallel to the Y-axis, measure stage displacements. A calibration procedure is described in which no geometric assumptions other than repeatability are made concerning the stage, e.g. it is not assumed that interferometer mirrors are spherical or parabolic, nor are specific geometric features, such as axis misalignment, modelled. Although the assumptions are very general, it is possible to observe the grid in just three orientations to determine both an inverse distortion function (calibration) and absolute rectangular coordinates for the grid points, providing that only one additional item of information is available, namely the absolute distance between a pair of points on the grid. The mathematical theory is based upon the group of symmetries generated by the three orientations of the grid and an associated lattice of rotational fixpoints. A cali-bration program has been written applying the theory to the practical problem of calibrating an E-beam lithography system. Simulations based on actual E-beam measurements confirm the theory for the practical problem. The calibration program is a powerful tool for observing and measuring the behavior of a working E-beam system. An E-beam machine can be calibrated using the program to an accuracy limited only by the repeatability of the machine, currently on the order of hundredths of a micrometer.