Rational orthogonal approximations to orthogonal matrices

Victor J. Milenkovic, Veljko Milenkovic

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


Several algorithms are presented for approximating an orthogonal rotation matrix M in three dimensions by an orthogonal matrix with rational entries. The first algorithm generates an approximation M2(M, ε) with accuracy ε and (2b + 2)-bit numerators and a common (2b + 2)-bit denominator (bit-size 2b + 2), where b = ⌋-1g ε⌉ (ε ≈ 2-b). The second algorithm uses basis reduction to generate an approximation Mv(M, ε) with accuracy εv/1.5 and bit-size vb for some 1.5 ≤ v ≥ 6 (but v cannot be controlled except by trial and error). A third algorithm, based on integer programming, generates optimal Mopt(M, ε) with accuracy ε and bit-size proven to be no more than 1.5b. In practice, the second algorithm generates an approximation with v ≈ 1.5 and is much faster than the third algorithm. The best bit-sizes which one could obtain using previously known results in two dimensions (Canny et al., 1992) are more than 3b bits for numerator and denominator. Applications are described for the approximation functions in the area of solid modeling.

Original languageEnglish (US)
Pages (from-to)25-35
Number of pages11
JournalComputational Geometry: Theory and Applications
Issue number1-2
StatePublished - Jan 1997
Externally publishedYes


  • Computational geometry
  • Polyhedral modeling
  • Robust geometry
  • Solids modeling

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics


Dive into the research topics of 'Rational orthogonal approximations to orthogonal matrices'. Together they form a unique fingerprint.

Cite this