Summary: Sufficient conditions are given for \(C^1\) and \(C^2\) (subdivision) curves generated by a particular non-uniform, interpolatory, variational refinement scheme. The ‘energy’ functional being minimized is a discretization of the standard linearized spline functional over piecewise linear curves – a generalization of the minimizing functional used for the uniform scheme of L. Kobbelt [Comput. Aided Geom. Des. 13, No. 8, 743–761 (1996; Zbl 0875.68878)]. The conditions used are uniform bounds on either the energy functional or certain divided differences, along with a condition on the knots, forcing them to be dense and uniform in the limit. To establish \(C^2\), a certain ‘bootstrap’ argument is applied. The argument is based on a generalization of a result of J. A. Gregory and R. Qu [ibid. 13, No. 8, 763–772 (1996; Zbl 0900.68411)] used to show smoothness of curves generated by nonuniform corner cutting.
For the entire collection see [Zbl 1061.41001].