In this article we propose a new 2D triangulation method based on the ball-pivoting algorithm (BPA). The BPA is an interesting advancing front approach for surface reconstruction that uses a ball of fixed radius traversing the 3D sample points by pivoting front edges and attaching triangles to the mesh. Given a set of 2D points, our method applies the BPA on them assuming that they have a constant third coordinate. We show that such geometrical restriction implies in several simplifications on the original BPA implementation. We demonstrate that the resulted triangulation is a solid alpha complex, a special subset of Delaunay Triangulations that is closely related to alpha shapes. The BPA efficency is extremely dependent on the uniformity of the sampling and on the ball radius. We also present an efficient generalization of our method to obtain, in an adaptive way, 2D solid alpha complexes of generic samplings (uniform or non-uniform) free from the influence of ball size.