A flattenable mesh surface is a polygonal mesh surfaceÂ that can be unfolded into a planar patch withoutÂ stretching any polygon. This paper presents a newÂ method for computing a slightly stretched flattenableÂ mesh surface M from a piecewise-linear surface patchÂ P in 3D, where the shape approximation error betweenÂ M and P is minimized and the strain of stretching onÂ M is controlled. Prior approaches result in either aÂ flattenableÂ surface that could be quite different from theÂ input shape or a (discrete) developable surface has relativeÂ simple shape. The techniques investigated in thisÂ paper overcome these difficulties. First, we introduce aÂ new surface modeling method to conduct a sequenceÂ of nearly isometric deformations to morph aÂ flattenableÂ mesh surface to a new shape which has a betterÂ approximation of the input surface. Second, in orderÂ to get better initial surfaces for fitting and overcomeÂ topological obstacles, a shape perturbation scheme isÂ investigated to obtain the optimal surface fitting result.Â Last, to improve the scalability of our optimal surfaceÂ fitting algorithm, a coarse-to-fine fitting framework isÂ exploited so that very dense flattenable mesh surfacesÂ can be modeled and boundaries of the input surfacescan be interpolated.