Abstract: Popular subdivision algorithms like Catmull-Clark and Loop are C² almost everywhere, but suffer from shape artifacts and reduced smoothness exactly near the so-called "extraordinary vertices" that motivate their use. Subdivision theory explains that inherently, for standard stationary subdivision algorithms, curvature-continuity and the ability to model all quadratic shapes requires a degree of at least bi-6. The existence of a simple-to-implement C² subdivision algorithm generating surfaces of good shape and piecewise degree bi-3 in the polar setting is therefore a welcome surprise. This paper presents such an algorithm, the underlying insights, and a detailed analysis. In bi-3 C² polar subdivision the weights depend, as in standard schemes, only on the valence, but the valence at one central polar vertex increases to match Catmull-Clark-refinement.