Accurate local surface geometry estimation in discrete surfaces is an important problem with numerous applications. Principal curvatures and principal directions can be used in applications such as shape analysis and recognition, object segmentation, adaptive smoothing, anisotropic fairing of irregular meshes, and anisotropic texture mapping. In this paper, a novel approach for accurate principal direction estimation in discrete surfaces is described. The proposed approach is based on local directional curve sampling of the surface where the sampling frequency can be controlled. This local model has a large number of degrees of freedoms compared with known techniques and so can better represent the local geometry. The proposed approach is quantitatively evaluated and compared with known techniques for principal direction estimation. In order to perform an unbiased evaluation in which smoothing effects are factored out, we use a set of randomly generated Bezier surface patches for which the principal directions can be computed analytically.