The paper deals with a procedure for constructing a composite surface interpolating a polygonal curve mesh defined from 3D scattered points. The procedure consists of a poly-angulation, construction of a curve net, and interpolation of the curve net. The poly-angulation contains a stage that changes a triangular edge net obtained from a triangulation into a poly-angular edge net. A curve net is constructed by replacing edges on the edge net with cubic Bezier curves. Finally, inside of an n-sided polygon is interpolated by n subdivided triangular subpatches. The method interpolates given point data with relatively few triangular subpatches. For an n-sided polygon, our method constructs an interpolant with n subdivided triangular subpatches while the existing triangular surface modeling needs 3(n-2) subpatches. The obtained surface is composed of quartic triangular patches which are $G^1$-continuous to adjacent patches.