Smooth Bi-3 spline surfaces with fewest knots

Converting a quadrilateral input mesh into a C1C1 surface with one bi-3 tensor-product spline patch per facet is a classical challenge. We give explicit local averaging formulas for the spline control points. Where the quadrilateral mesh is not regular, the patches have two internal double knots, the least number and multiplicity to always allow for an unbiased G1G1 construction.

Research highlights
► We give explicit local averaging formulas for converting a quadrilateral input mesh into a smooth surface with one bi-3 tensor-product spline patch per facet.
► In regular regions, the surface is simply a collection of C2C2-connected splines.
► Near vertices of valence other than four, the spline patches have two internal double knots.
► This is least number and multiplicity of knots for a surface with one bi-3 tensor-product spline patch per quadrilateral facet, to guarantee that the surface is everywhere automatically tangent continuous unless less smoothness is explicitly prescribed.