On the G1 continuity of piecewise Bézier surfaces: a review with new results

Abstract The tensor product Bezier 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 Bezier 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 G 1 smooth connection between adjacent Bezier patches and those dealing with the techniques of free-form surface modelling using Bezier patches. First, the constraints guaranteeing G 1 continuity between two adjacent Bezier patches are analysed. This analysis reveals several important geometric properties hidden in these constraints, usually expressed analytically. From these results the G 1 smooth connection between N ( N > 2) patches meeting at a common corner is studied. The resulting G 1 constraints are deduced, and it is shown how to satisfy them in the definition of the control points of the Bezier 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 G 1 continuity around a degenerate four-patch corner. After that, the various methods proposed to model complex surfaces using Bezier patches are reviewed. Based on this analysis, new alternative approaches for modelling free-form G 1 continuous surfaces are presented.

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