It is frequently necessary, when using microwave transmission systems, to determine the impedance on one side of a lossless junction in terms of a known impedance on the other side. The paper provides rigorous analysis of the relation between Smith charts for the two sides of the junction by utilizing the theory of a complex variable. Pairs of reference planes are chosen so that an open-circuit at the first plane corresponds to an open-circuit at the second, which is on the other side of the junction. The transforming section is described in terms of the reflection coefficient at the first plane when the line containing the other is matched. The circles of constant reflection coefficient, etc., which constitute a Smith chart corresponding to the first plane, are transformed mathematically into corresponding circles at the second plane. A universal chart is derived which enables circles of constant voltage standing-wave ratio (or reflection coefficient) to be transformed readily from one plane to the other.