Systems and methods are provided for accomplishing fast and accurate 3D registration of curves and surfaces using local differential information, i.e., normals and tangents. In an embodiment, a method solves the curve-vs-surface alignment problem either by using a purely online search scheme, or by taking advantage of the availability of a pre-operative model, which often happens in medical procedures, to further speed-up the computational search by performing offline processing of the pre-operative bone model. The disclosed method is also extended to solve the curve-vs-curve and surface-vs-surface alignment problems, which also have important applications in medical procedures such as arthroscopy and arthroplasty.