Starting from a well-known construction of polynomial-based interpolatory 4-point schemes, in this paper we presentudan original affine combination of quadratic polynomial samples that leads to a non-uniform 4-point scheme with edgeudparameters. This blending-type formulation is then further generalized to provide a powerful subdivision algorithmudthat combines the fairing curve of a non-uniform refinement with the advantages of a shape-controlled interpolationudmethod and an arbitrary point insertion rule. The result is a non-uniform interpolatory 4-point scheme that is uniqueudin combining a number of distinctive properties. In fact it generates visually-pleasing limit curves where specialudfeatures ranging from cusps and flat edges to point/edge tension effects may be included without creating undesiredudundulations. Moreover such a scheme is capable of inserting new points at any positions of existing intervals, so thatudthe most convenient parameter values may be chosen as well as the intervals for insertion.udSuch a fully flexible curve scheme is a fundamental step towards the construction of high-quality interpolatory subdivision surfaces with features control.
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