Fast Algorithms for Generating Parametric Cubic Spline Interpolation Curves

For any two or three dimensional dataset given as an open or closed piecewise linear function, two algorithms are established for constructing G1 and G2 parametric cubic splines that interpolate the given dataset. These algorithms do not require any additional tangent direction information at each datapoint or any boundary conditions. One algorithm is specif-ically formulated for the unique local least square parametric cubic spline interpolation, so that a given dataset can be best approximated through appropriately defined local least squares approximation. Simple examples will be used to illustrate the algorithms and various datasets will also be demonstrated to show the efficiency of the algorithms.