Presented in this thesis are two $G^1$-surface interpolation schemes for 3D point data : a triangular surface interpolation for 3D scattered data and a rectangular surface interpolation for unevenly spaced 3D point data array.
Triangular B$\acute{e}$zier interpolant is widely used in widely sued in constructing smooth surfaces from scattered data in 3D. Before applying triangular interpolant, the input 3D points have to be triangulated. To obtain smooth triangular grid, a smoothness criterion is proposed. Each triangle is filled with TBP (triangular B$\acute{e}$zier patch) with $G^1$-continuity (tangent plane continuity). A new $G^1$-continuity condition is introduced, which makes it possible for a TBP to have $G^1$-continuous joins with adjacent TBPs while preserving its boundary curves.
Errorneous point data would result in surface irregularities. Reflection line method is adopted to detect unpleasant regions, and these regions are cured by slightly moving the near-by-points in the direction of decreasing the unfairness measure.
Ferguson surface and non-uniform B-Spline surface are widely used in automatic surface fitting from an array of 3D points. But they suffer from local flatness or bulges when the physical spacing of data in uneven. This thesis describes a method of constructing $G^1$-continuous composite bisextic B$\acute{e}$zier surface which is free from any local flatness and bulges even with a very unevenly spaced point array. This scheme has some nice features, for example: a) it is a completely local scheme, and b) iso parametric curves of the entire surface are smooth across patch boundaries.
