Transforming curves into curves with the same shape

Curves are considered to have the same shape when they are related by a similarity transformation of a certain kind. This paper extends earlier work on parallel curves to curves with the same shape. Some examples are given more or less explicitly. A generalization is used to show that the theory is ordinal and to show how the theory may be applied to measure sensation. The problem of actually transforming curves into curves with the same shape is reduced to the problem of rendering another set of curves parallel. Connections with groups and rings are developed to place the work in a familiar context. These connections and the earlier work on parallel curves are used to obtain necessary and sufficient conditions for the existence of transformations, to study the uniqueness of transformations and to show how transformations can be calculated.