Using a classical non-linear theory, we analytically investigate possible ways for transforming the shape of a curved elastic membrane while keeping it tensioned and moderately strained. This is a critical issue because, as a rule, membranes must be considerably stretched in order to avoid wrinkling and slackening. If the final configuration is fixed, the membrane can be cut and formed according to the final shape, but this cannot be done if more configurations, considerably distant from one another, have to be achieved. Nevertheless, we propose large transformation movements that can be obtained starting from flat membranes while maintaining their strain as limited. We discuss in detail the paradigmatic example of the hyperbolic-paraboloid-shaped membrane. These opportunities are suitable for applications of transformable architecture because they do not require excessive tensioning, compatible with the strength of materials used for this kind of structures.