Two algorithms for establishing PACSIC (parametric cubic spline interpolation curve) for any given dataset as an open or closed PLF (piecewise linear function) were established. Both algorithms do not require any prescribed tangent directions or any boundary conditions, though these directions can also be pre-defined if necessary. The direction at each vertex was simply determined by the two neighboring vertices. The LS (least square) G1 PACSIC, in particular, can be applied to approximate any 2d or 3d dataset. Simple examples were used to illustrate the flexibility of the algorithms, and various illustrative examples were also shown to demonstrate the efficiency of both algorithms. For completeness, the G2 PACSIC was briefly investigated, where the G2 conditions were given in terms of elastic parameters. Certainly, similar algorithms for generic PACSIC and LS PACSIC can also be developed if the tangent directions at vertices are evaluated by using three or more neighboring vertices; or even by raising the degree from cubic to quartic.

An innovative and efficient algorithm was established for generating a G1 parametric bicubic spline interpolation surface (PACSIS) for any polygonal mesh. The resulting G1 surface interpolates all vertices of the quads; and the G1 smoothness was guaranteed by the special creation of new control points for each quad so that all tangent planes at any vertex match. The algorithm was demonstrated by a variety of PLY files representing different polygonal meshes. There will be more applications of PACSIS to areas such as signal and image processing, animation, data science, feature representation, mesh rendering, visualization, as well as computer graphics.