Minimizing the total projection of a set of vectors, with applications to layered manufacturing

Man Chung Hon, Ravi Janardan, Jörg Schwerdt, Michiel Smid

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


In layered manufacturing (LM), a three-dimensional polyhedral solid is built as a stack of two-dimensional slices. Each slice (a polygon) is built by filling its interior with a sequence of parallel line segments (of some small non-zero width), in a process called hatching. A critical step in hatching is choosing a direction which minimizes the number of segments. In this paper, this problem is approximated as the problem of finding a direction which minimizes the total projected length of a certain set of vectors. Efficient algorithms are proposed for the latter problem, using techniques from computational geometry. Experimental and theoretical analyses show that this approach yields results that approximate closely the optimal solution to the hatching problem. Extensions of these results to several related problems are also discussed.

Original languageEnglish (US)
Pages (from-to)57-68
Number of pages12
JournalCAD Computer Aided Design
Issue number1
StatePublished - Jan 2003

Bibliographical note

Funding Information:
Portion of this work was done while Ravi Janardan was visiting the University of Magdeburg,under a joint grant from NSF and DAAD for international research.

Funding Information:
This work was done while Man Chung Hon was at the Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455, USA. Research supported, in part, by NSF grant CCR-9712226.


  • Hatching
  • Layered manufacturing
  • Projection

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