This paper addresses the problem of fitting flattenable mesh surfaces in onto piecewise linear boundary curves, where a flattenable mesh surface inherits the isometric mapping to a planar region in . The developable surface in differential geometry shows the nice property. However, it is difficult to fit developable surfaces to a boundary with complex shape. The technique presented in this paper can model a piecewise linear flattenable surface that interpolates the given boundary curve and approximates the cross-tangent normal vectors on the boundary. At first, an optimal planar polygonal region is computed from the given boundary curve , triangulated into a planar mesh surface, and warped into a mesh surface in , satisfying the continuities defined on . Then, the fitted mesh surface is further optimized into a flattenable Laplacian (FL) mesh, which preserves the positional continuity and minimizes the variation of cross-tangential normals. Assembled set of such FL mesh patches can be employed to model complex products fabricated from sheets without stretching.