The two-dimensional (2D) Delaunay triangulation-based structure, i.e., Delaunay triangle, has been widely used in fingerprint authentication. However, we also notice the existence of three-dimensional (3D) Delaunay triangulation, which has not been extensively explored. Inspired by this, in this paper, the features of both 2D and 3D Delaunay triangulation-based structures are investigated and the findings show that a 3D Delaunay structure, e.g., Delaunay tetrahedron, can provide more feature types and a larger number of elements than a 2D Delaunay structure, which was expected to provide a higher discriminative capability. However, higher discrimination does not necessarily lead to better performance, especially in biometric applications, when biometric uncertainty is unavoidable. Experimental results show that the biometric uncertainty such as missing or spurious minutiae causes more negative influence on the 3D Delaunay triangulation than that on the 2D Delaunay triangulation in three out of four experimental data sets.