The Distance Geometry Problem (DGP) asks whether a simple weighted undirected graph G can be realized in the Euclidean space so that the distances between embedded vertices correspond to the edge weights. The DGP is a rich and active research field, with many important applications. Several approaches to the DGP are based on the idea of directly placing the vertices of G in space. Our model uses a completely new approach: we focus our attention on the edges, and not on the vertices, and we attempt placing in space the “sticks” that can be associated to each edge of the graph. Sticks have fixed length (hence they always satisfy all distance constraints), and they admit three total degrees of freedom (position of one vertex, plus the stick orientation) in 2D. The automatic satisfaction of all DGP constraints comes at the cost of possibly having several distinct positions associated to the same vertex, potentially a different one for every stick where each vertex is involved. Therefore, we formulate a problem consisting in finding stick configurations where all vertices involved in multiple sticks can find a unique position in space, implying in turn the definition of a valid realization for the original DGP. We initially focus the attention on DGPs where the information on the stick orientations is a priori given, so that to formulate a convex quadratic optimization problem with linear constraints. For the general case, we propose a heuristic which solves, at each iteration, an instance of the quadratic problem.