Approximate pyramidal shape decomposition

Ruizhen Hu, Honghua Li, Hao Zhang, Daniel Cohen-Or

Research output: Contribution to journalArticlepeer-review


A shape is pyramidal if it has a flat base with the remaining boundary forming a height function over the base. Pyramidal shapes are optimal for molding, casting, and layered 3D printing. However, many common objects are not pyramidal. We introduce an algorithm for approximate pyramidal shape decomposition. The general exact pyramidal decomposition problem is NP-hard. We turn this problem into an NP-complete problem which admits a practical solution. Specifically, we link pyramidal decomposition to the Exact Cover Problem (ECP). Given an input shape S, we develop clustering schemes to derive a set of building blocks for approximate pyramidal parts of S. The building blocks are then combined to yield a set of candidate pyramidal parts. Finally, we employ Knuth's Algorithm X over the candidate parts to obtain solutions to ECP as pyramidal shape decompositions. Our solution is equally applicable to 2D or 3D shapes, and to shapes with polygonal or smooth boundaries, with or without holes. We demonstrate our algorithm on numerous shapes and evaluate its performance.

Original languageEnglish
JournalACM Transactions on Graphics
Issue number6
StatePublished - 19 Nov 2014


  • 3D printing
  • Pyramidality
  • Shape decomposition

All Science Journal Classification (ASJC) codes

  • Computer Graphics and Computer-Aided Design


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