A new contact detection algorithm between three-dimensional non-spherical particles in the discrete element method (DEM) is proposed. Houlsby previously proposed the concept of potential particles where an arbitrarily shaped convex particle can be defined using a 2nd degree polynomial function (Houlsby [1]). The equations in 2-D have been presented and solved using the Newton-Raphson method. Here the necessary mathematics is presented for the 3-D case, which involves non-trivial extensions from 2-D. The polynomial structure of the equations is exploited so that they are second-order cone representable. Second order-cone programmes have been established to be theoretically and practically tractable, and can be solved efficiently using primal-dual interior-point methods (Andersen et al. [13]). Several examples are included in this paper to illustrate the capability of the algorithm to model particles of various shapes. (C) 2012 Elsevier B.V. All rights reserved.