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In this paper we present a non-stationary 4-point ternary interpolatory subdivision scheme which provides the user with a tension parameter that, when increased within its range of definition, can generate C 2 -continuous limit curves showing considerable variations of shape.As a generalization we additionally propose a locally-controlled C 2 -continuous subdivision scheme, which allows a different tension value to be assigned to every edge of the original control polygon.

Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multidegree splines that can be derived by existing approaches. We then propose a new alternative method for constructing and evaluating the B-spline basis, based on the use of so-called transition functions. Using the transition functions we develop general algorithms for knot-insertion, degree elevation and conversion to Bézier form, essential tools for applications in geometric modeling. We present numerical examples and briefly discuss how the same idea can be used in order to construct geometrically continuous multi-degree splines.

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In this paper we consider the problem of designing piecewise polynomial local interpolants of non-uniformly spaced data. We provide a constructive approach that, for any assigned degree of polynomial reproduction, continuity order, and support width, allows for generating the fundamental spline functions of minimum degree having the desired properties. Finally, the proposed construction is extended to handle open sets of data and to the case of multiple knots