Smoothing the antagonism between extraordinary vertex and ordinary neighbourhood on subdivision surfaces

Recent methods for tuning a subdivision scheme create a concentric wave pattern around the extraordinary vertex (EV). We explain it as resulting from the antagonism between the rules which would create a nice limit surface at the EV and the ordinary rules used in the surrounding regular surface. We show that even a scheme which fulfils the most recently proposed conditions for good convergence at the EV may still produce this wave pattern. Then, in order to smooth this antagonism, we define any new vertex as a convex combination of the ideal new vertex from the EV point of view and the one defined with ordinary rules. The weight of the extraordinary rules decreases as the new vertex is topologically farther from the EV. The concentric wave pattern shades off whereas the expected conditions are not too much spoiled. This tuning method remains simple and useful, involving no optimisation process.