Multidimensional projection is emerging as an important visualization tool in applications involving the visual analysis of high-dimensional data. However, existing projection methods are either computationally expensive or not flexible enough to enable fully interactive data manipulation. That is, they do not support the feedback of user interaction into the projection process. A mechanism that dynamically adapts the projection based on direct user interaction would go a long way towards making the technique more useful with a large range of applications and data sets. In this paper we propose the Piecewise Laplacian-based Projection (PLP), a novel multidimensional projection technique, that, due to the local nature of its formulation, enables a versatile mechanism to interact with projected data and to allow interactive changes to dynamically alter the projection map, a unique capability of the technique. We exploit the flexibility provided by PLP in two interactive projection-based applications, one designed to organize pictures visually and another to build music playlists. These applications illustrate the usefulness of PLP in handling high-dimensional data in a flexible and highly visual way. We also compare PLP with the currently most promising projections in terms of precision and speed. The results show that PLP perform very well also according to these quality criteria.