Least Degree G1-refinable Multi-sided Surfaces Suitable for Inclusion into C1 bi-2 Splines

Abstract Geometrically smooth spline surfaces, generalized to include n -sided facets or configurations of n ≠ 4 quads, can exhibit a curious lack of additional degrees of freedom for modelling or engineering analysis when refined. This paper establishes a minimal polynomial degree for smooth constructions of multi-sided surfaces that guarantees more flexibility in all directions under refinement. Degree bi-4 is both necessary and sufficient for flexibility-increasing  G 1 -refinability within a bi-quadratic C 1 spline complex. Sufficiency is proven by two alternative flexibly G 1 - refinable constructions exhibiting good highlight line distributions.