We consider the problem of embedding the nodes of a hypergraph into Euclidean space under the assumption that the interactions arose through closeness to unknown hyperedge centres. In this way, we tackle the inverse problem associated with the generation of geometric random hypergraphs. We propose two new spectral algorithms; both of these exploit the connection between hypergraphs and bipartite graphs. The assumption of an underlying geometric structure allows us to define a concrete measure of success that can be used to optimize the embedding via gradient descent. Synthetic tests show that this approach accurately reveals geometric structure that is planted in the data, and tests on real hypergraphs show that the approach is also useful for the downstream tasks of detecting spurious or missing data and node clustering.