Abstract
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 admits a PTAS and can be solved by a (1 + ε)-approximation algorithm in O(n logn) time for any constant ε> 0, where n is the input size of the problem.
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Aono, M., Breen, D.E., Wozny, M.J.: Modeling methods for the design of 3D broadcloth composite parts. Computer-Aided Design 33(13), 989–1007 (2001)
Aona, M., Denti, P., Breen, D.E., Wozny, M.J.: Fitting a woven cloth model to a curved surface: Dart insertion. IEEE Computer Graphics and Applications 16(5), 60–70 (1996)
Azariadis, P.N., Sapidis, N.S.: Planar development of free-form surfaces: Quality evaluation and visual inspection. Computing 72(1-2), 13–27 (2004)
Borradaile, G., Kenyon-Mathieu, C., Klein, P.N.: A polynomial-time approximation scheme for Steiner tree in planar graphs. In: Proc. of 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1285–1294 (2007)
Borradaile, G., Klein, P.N., Mathieu, C.: Steiner tree in planar graphs: An O(n logn) approximation scheme with singly exponential dependence on epsilon. In: Proc. of 10th International Workshop on Algorithms and Data Structures, pp. 276–287 (2007)
Garey, M.R., Johnson, D.S.: The rectilinear Steiner tree problem is NP-complete. SIAM Journal on Applied Mathematics 32(4), 826–834 (1977)
Karp, R.: On the computational complexity of combinatorial problems. Networks 5, 45–68 (1975)
Kim, S.M., Kang, T.J.: Garment pattern generation from body scan data. Computer-Aided Design 35(7), 611–618 (2003)
Kobbelt, L.P., Bischoff, S., Botsch, M., Kähler, K., Rössl, C., Schneider, R., Vorsatz, J.: Geometric modeling based on polygonal meshes. In: Eurographics 2000 Tutorial (2000)
McCartney, J., Hinds, B.K., Seow, B.L.: The flattening of triangulated surfaces incorporating darts and gussets. Computer-Aided Design 31(4), 249–260 (1999)
Parida, L., Mudur, S.P.: Constraint-satisfying planar development of complex surfaces. Computer-Aided Design 25(4), 225–232 (1993)
Sheffer, A.: Spanning tree seams for reducing parameterization distortion of triangulated surface. In: Proc. of International Conference on Shape Modeling and Applications, pp. 61–68 (2002)
Wang, C.L., Smith, S.F., Yuen, M.F.: Surface flattening based on energy model. Computer-Aided Design 34(11), 823–833 (2002)
Wang, C.L., Wang, Y., Tang, K., Yuen, M.F.: Reduce the stretch in surface flattening by finding cutting paths to the surface boundary. Computer-Aided Design 36(8), 665–677 (2004)
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Chen, D.Z., Misiołek, E. (2008). Optimal Surface Flattening. In: Preparata, F.P., Wu, X., Yin, J. (eds) Frontiers in Algorithmics. FAW 2008. Lecture Notes in Computer Science, vol 5059. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69311-6_25
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DOI: https://doi.org/10.1007/978-3-540-69311-6_25
Publisher Name: Springer, Berlin, Heidelberg
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