Restriction and Extension for Planar Splines on Triangulations

Suppose a planar triangulation \(\Delta \) is enlarged to a triangulation \(\Delta ^\sigma \) by adding a single triangle \(\sigma \) which meets \(\Delta \) along one edge (in which case \(\sigma \) is called a flap) or two edges (in which case \(\sigma \) is called a fill). We study the relationship between \(C^r\) splines on \(\Delta \) and \(C^r\) splines on \(\Delta ^\sigma \) . Our focus is mainly on the (Castelnuovo-Mumford) regularity of the graded spline module, since this statistic highly correlates with the maximum degree d for which the dimension of the spline space \(S^r_d(\Delta )\) is not given by Schumaker’s lower bound.