Flattening topologically spherical surface

The problem of optimal surface flattening in 3‐D finds many applications in engineering and manufacturing. However, previous algorithms for this problem are all heuristics without any quality guarantee and the computational complexity of the problem was not well understood. In this paper, we prove that the optimal surface flattening problem is NP‐hard. Further, we show that the problem of flattening a topologically spherical surface admits a PTAS and can be solved by a (1+ϵ)‐approximation algorithm in O(nlog n) time for any constant ϵ>0, where n is the input size of the problem.