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In this paper we derive novel surface fiducial points that are computed from the differential geometry of the surface. The fiducial intrinsic points are intrinsic, local, and relative invariants, i.e., they are preserved under similarity, affine, and nonlinear transformations that are piecewise affine. As the fiducial points are computed from high order surface shape derivatives, their sensitivity to any noise either due to measurement error or local distortion is high. To reduce these effects, we use a B-Spline curve/surface representation that smoothes out the curve/surface prior to the computation of these intrinsic invariant points. The fiducial points are used in a non-iterative geometric-based method for 3D shape matching and registration of human faces. The matching is achieved by establishing correspondences between fiducial points after a sorting based on a set of absolute local affine invariants derived from them. The performance of the matching based on these fiducial points although shown for face alignment, the overall approach is generic to the alignment a variety of objects for which these intrinsic fiducial points exist and for scenarios where the classes of nonlinear transformations are piecewise affine.