ON THE G1 CONTINUITY OF PIECEWISE BEZIER SURFACES - A REVIEW WITH NEW RESULTS

The tensor product Bézier patch is currently one of the most widely used models in CAGD for free-form surface modelling. In a piecewise representation the patches are distributed on a mesh. If any piecewise surface has to be modelled using non-degenerate Bézier patches, it is necessary to use a mesh of unrestricted topology, i.e. with any number of patches meeting at a node. In order to obtain a smooth surface the geometric continuities between adjacent patches must be controlled. A lot of research has been devoted to this problem and various solutions have been proposed. This paper reviews the various studies dealing with the G1 smooth connection between adjacent Bézier patches and those dealing with the techniques of free-form surface modelling using Bézier patches. First, the constraints guaranteeing G1 continuity between two adjacent Bézier patches are analysed. This analysis reveals several important geometric properties hidden in these constraints, usually expressed analytically. From these results the G1 smooth connection between N (N > 2) patches meeting at a common corner is studied. The resulting G1 constraints are deduced, and it is shown how to satisfy them in the definition of the control points of the Bézier patches. Degeneration problems around a four-patch corner adjacent to a non-four-patch corner are then analysed, and the supplementary conditions to be satisfied are developed in order to guarantee the G1 continuity around a degenerate four-patch corner. After that, the various methods proposed to model complex surfaces using Bézier patches are reviewed. Based on this analysis, new alternative approaches for modelling free-form G1 continuous surfaces are presented. © 1990.