Abstract
This paper addresses geometric problems in manufacturing objects by casting. In casting, molten material is poured into the cavity of the cast and allowed to solidify, after which the cast is removed. The cast has two cast parts to be removed in opposite directions. To manufacture more complicated objects, the cast may also have a side core to be removed in a direction skewed to the removal directions for the cast parts. We address the following problem: Given an object and the removal directions for the cast parts and the side core, can a cast be constructed such that the cast parts and the side core can be removed in the directions specified without colliding with the object or each other? We give necessary and sufficient conditions for the problem, as well as a discrete algorithm to perform the test in O(n 3log n) time for polyhedral objects, where n is the number of vertices, edges, and facets. If the test result is positive, a cast with complexity O(n 3) can be constructed within the same time bound. We also present an example to show that a cast may have Ω(n 3) complexity in the worst case.
Similar content being viewed by others
References
Ahn, H.K., de Berg, M., Bose, P., Cheng, S.W., Halperin, D., Matoušek, J., Schwarzkopf, O.: Separating an object from its cast. Comput. Aided Des. 34, 547–559 (2002)
Ahn, H.K., Cheong, O., van Oostrum, R.: Casting a polyhedron with directional uncertainty. Comput. Geom. Theory Appl. 26(2), 129–141 (2003)
Ahn, H.K., Cheng, S.W., Cheong, O., Snoeyink, J.: The reflex-free hull. Int. J. Comput. Geom. Appl. 14(6), 453–474 (2004)
Ahn, H.K., Cheng, S.W., Cheong, O.: Casting with skewed ejection direction. Algorithmica 44(4), 325–342 (2006)
Bose, P., Bremner, D., van Kreveld, M.: Determining the castability of simple polyhedra. Algorithmica 19, 84–113 (1997)
Chen, L.L., Chou, S.Y., Woo, T.C.: Parting directions for mould and die design. Comput. Aided Des. 25, 762–768 (1993)
Elliot, R.: Cast Iron Technology. Butterworths, London (1988)
Gastrow, H.: Injection Molds. Hanser, Munich (1983)
Halperin, D., Kavraki, L., Latombe, J.-C.: Robot algorithms. In: Attalah, M. (ed.) CRC Algorithms and Theory of Computation Handbook. CRC Press, Boca Raton (1999)
Hui, K.: Geometric aspects of mouldability of parts. Comput. Aided Des. 29(3), 197–208 (1997)
Kwong, K.K.: Computer-aided parting line and parting surface generation in mould design. PhD thesis, The University of Hong Kong, Hong Kong (1992)
Majhi, J., Gupta, P., Janardan, R.: Computing a flattest, undercut-free parting line for a convex polyhedron, with application to mold design. Comput. Geom. Theory Appl. 13, 229–252 (1999)
Pribble, W.I.: Molds for reaction injection, structural foam and expandable styrene molding. In: DuBois, J.H., Pribble, W.I. (eds.) Plastics Mold Engineering Handbook. Van Nostrand Reinhold Company, New York (1987)
Sarnak, N., Tarjan, R.E.: Planar point location using persistent search trees. Commun. ACM 29, 669–679 (1986)
Walton, C.F., Opar, T.J. (eds.): Iron Castings Handbook. Iron Casting Society (1981)
Zuppann, E.C.: Castings made in sand molds. In: Bralla, J.G. (ed.) Handbook of Product Design for Manufacturing, pp. 5.3–5.22. McGraw–Hill, New York (1986)
Author information
Authors and Affiliations
Corresponding author
Additional information
Work by Ahn was supported by the Postech BSRI Research Fund—2007. Work by Cheng was supported by a Direct Allocation Grant of HKUST (HKUST DAG04/05.EG21).
Rights and permissions
About this article
Cite this article
Ahn, HK., Bae, S.W., Cheng, SW. et al. Casting an Object with a Core. Algorithmica 54, 72–88 (2009). https://doi.org/10.1007/s00453-007-9120-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-007-9120-8