Generalized Lane-Riesenfeld algorithms (Q1950215)

The paper is focused on Lane-Riesenfeld algorithm as an efficient method for subdividing uniform B-splines, using a refine and smooth factorization to get arbitrarily high smoothness through efficient local rules. This algorithm is extended considering that the same operator is used to define the refine and each smoothing stage. Since the operator samples a linear polynomial, the original algorithm preserves only linear polynomial in the functional setting respectively straight lines in the geometric setting. Therefore, the idea of generalization for Lane-Riesenfeld algorithm is to replace the linear sampling rule with an alternative smoothing operator, the invariants being extended by two new families of schemes, one preserving cubic polynomials, the other circles, both requiring only local rules for implementation. A greater number of smoothing stages gives smoother limit curves, hence like the original Lane-Riesenfeld algorithm, the proposed algorithm implies a whole family of subdivision schemes with the same set of invariants. Moreover, as we have schemes reproducing linear polynomials, cubic polynomials or circles, further generalizations might be considered taking into account other functions classes such as rational and exponential functions or ellipses and hyperbolas.