Notice of RetractionThe continuity of Cubic extension Bézier curves and surfaces

Notice of Retraction

After careful and considered review of the content of this paper by a duly constituted expert committee, this paper has been found to be in violation of IEEE´s Publication Principles.

We hereby retract the content of this paper. Reasonable effort should be made to remove all past references to this paper.

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Cubic extension Bézier curves (CE-Bézier curves) are constructed by a natural extension of the traditional cubic Bézier curves. While retaining noticeable properties of cubic Bézier curves, CE-Bézier curves have the advantages such as adjustable shape parameters and better approximation property. In this paper, terminal behaviors of CE-Bézier curves, which are closely related to shape parameters, are carefully examinated and geometric interpretations of shape parameters are explicitly presented. Subsequently, the necessary and sufficient condition ensuring G¹ and G² continuity requirements is respectively established for the first category CE-Bézier curves. Parallel to CE-Bézier curves, Geometric condition ensuring G¹ continuity requirement between CE-Bézier surfaces are rigorously investigated, and relevant examples of geometric modeling are provided. Theoretical results and concrete examples have validated that, with appropriately selected control parameters, mosaic technique of CE-Bézier curves and surfaces can be widely applied to the representation and modeling of engineering curves and surfaces.