Filling holes under non-linear constraints

Abstract In this paper we handle the problem of filling hole in graphic a surface by means patch that joins original with $$\mathcal {C}^{1}$$ C 1 -smoothness and fulfills an additional non-linear geometrical constraint regarding its area or mean curvature at some points. Furthermore, develop technique to estimate optimum is expected have will allow us determine patches system linear quadratic equations. We present several numerical graphical examples showing effectiveness proposed method.