A geometrical technique is developed by which different methods of sampling the state of a one-dimensional distributed system may be related. Using this approach, transformations are derived relating scanned sampling to continuous time, discrete spatial sampling and to space distributed time sampling. Utilization of the technique is demonstrated in the derivation of stability relations for the case of a statically unstable string that is to be made stable by a scanned detector and controller. Of possible practical significance is the observation that scanned sampling can provide many of the characteristics of a spatially distributed sensor or controller, especially if the scan period is short. A limitation, which arises from the imposition of causality on real systems, is observed in the example.