Three methods are proposed to construct a piecewise Hermite interpolation surface (PHIS), which is a piecewise algebraic surface interpolating a set of given points with associated normal directions. The surface is obtained by blending together some low-degree surface patches. Both the first and the second methods are completely local and give a surface with Gⁿ -continuity. In the third construction, we reduce the number of surface patches by joining as many cubic patches as possible.This method gives a global solution with G¹ -continuity. These three different methods can be used to meet different requirements of the designers.