Dimensionality reduction (DR) is widely used to condense data for subsequent application of machine learning algorithms and to learn about the high-level structure of the data by providing low-dimensional embeddings used for visualization. Existing DR techniques such as t-SNE or UMAP are not guaranteed to represent the high-dimensional topological structure of the data faithfully. While topological embedding methods aim at modeling the underlying structure as closely as possible, they remain a black box for the user and do not allow for interactive exploration. In this project we aim at developing dimensionality reduction methods to fit the data on a low-dimensional shape template in an interactive fashion. To this end, we formalize the notion of shape templates and investigate how to automatically extract them from the data. We then integrate these templates into existing dimensionality reduction methods and design feedback mechanisms to evaluate the fit of the data on the template. This interactive setting can then be used to gain insights into the high-dimensional shape of the data that would remain hidden when using a static embedding from existing methods.