Discrete & Computational Geometry

, Volume 7, Issue 3, pp 227–253

On the difficulty of triangulating three-dimensional Nonconvex Polyhedra


  • Jim Ruppert
    • Computer Science DivisionUniversity of California at Berkeley
  • Raimund Seidel
    • Computer Science DivisionUniversity of California at Berkeley

DOI: 10.1007/BF02187840

Cite this article as:
Ruppert, J. & Seidel, R. Discrete Comput Geom (1992) 7: 227. doi:10.1007/BF02187840


A number of different polyhedraldecomposition problems have previously been studied, most notably the problem of triangulating a simple polygon. We are concerned with thepolyhedron triangulation problem: decomposing a three-dimensional polyhedron into a set of nonoverlapping tetrahedra whose vertices must be vertices of the polyhedron. It has previously been shown that some polyhedra cannot be triangulated in this fashion. We show that the problem of deciding whether a given polyhedron can be triangulated is NP-complete, and hence likely to be computationally intractable. The problem remains NP-complete when restricted to the case of star-shaped polyhedra. Various versions of the question of how many Steiner points are needed to triangulate a polyhedron also turn out to be NP-hard.

Copyright information

© Springer-Verlag New York Inc. 1992