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In this work, we give an algorithm for constructing a normalized B-spline basis over a Worsey-Piper split of a bounded domain of R 3 . These B-splines are all positive, have local support and form a partition of unity. Therefore, they can be used for constructing local approximants and for many other applications in CAGD. We also introduce the Worsey-Piper B-spline representation of C 1 quadratic polynomials or splines in terms of their polar forms. Then, we use this B-representation for constructing several quasi-interpolants which have an optimal approximation order.

In this paper, we describe a recursive method for computing interpolants defined in a space spanned by a finite number of continuous functions in R d . We apply this method to construct several interpolants such as spline interpolants, tensor product interpolants and multivariate polynomial interpolants. We also give a simple algorithm for solving a multivariate polynomial interpolation problem and constructing the minimal interpolation space for a given finite set of interpolation points.

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