Computing injective maps with low distortions is a long-standing problem in computer graphics. Such maps are particularly challenging to obtain in the presence of positional constraints, because an injective initial map is often not available. Recently, several energies were proposed and shown to be highly successful in optimizing injectivity from non-injective initial maps while satisfying positional constraints. However, minimizing these energies tends to produce elements with significant isometric distortions. This paper presents simple variants of these energies that retain their desirable traits while promoting isometry. While our method is not guaranteed to provide an injective map, we observe that, on large-scale 2D and 3D data sets, minimizing the proposed isometric variants results in a similar level of success in recovering injectivity as the original energies but a significantly lower isometric distortion.