Multiresolution Analysis on a spherical domain based on a flexible C 2 subdivision scheme over a valence 3 extraordinary vertex

It is known from a result of Prautzsch and Reif [19] that it is impossible to construct a flexible C2 scheme over extraordinary vertices unless the regular subdivision scheme (assumed in [19] to be based on polynomial splines) is capable of producing polynomial patches of total degree 8 in the triangle case and bi-degree 6 in the quadrilateral case. Prautzsch-Reif’s degree estimate, however, comes with a caveat, namely, that in the triangle mesh setting it is not applicable to the valence 3 case. In fact, the result in [19] is inconclusive about this valence 3 case. We show in this paper that, based on the 3-directional box-spline with directions [1, 0], [0, 1], [1, 1] repeated thrice, which produces degree 7 patches, it is actually possible to construct a flexible C2 scheme over a valence 3 extraordinary vertex. Moreover, the characteristic map of this scheme coincides with a so-called valence 3 Bers’ chart that shows up in the study of Riemann surfaces. As an application, this C2 scheme gives rise to smooth hierarchical approximations of functions defined on a spherical domain. In the quadrilateral case, Prautzsch and Reif’s degree estimate applies to all valences greater than or equal to 3, suggesting that there is no chance of extending our result to quadrilateral meshes. Nonetheless, we discuss an unexpected extension to quadrilateral meshes. Acknowledgments. We are grateful to Tom Duchamp and Hartmut Prautzsch for very helpful information directly related to the materials in this paper. We also acknowledge discussions with Pencho Petrushev and Qingtang Jiang on spherical multiresolution analysis. Yu is also grateful to a software donation by Alias. Finally, we thank Michael Overton for suggesting and supporting the collaboration.