×

The Achilles’ heel of \(O(3,1)\)? (English) Zbl 1029.65017

Summary: What’s the best way to represent an isometry of hyperbolic 3-space \(\mathbb{H}^3\)? Geometers traditionally worked in \(\text{SL}(2,\mathbb{C})\), but for software development many now prefer the Minkowski space model of \(\mathbb{H}^3\) and the orthogonal group \(\text{O}(3,1)\). One powerful advantage is that ideas and computations in \(S^3\) using matrices in \(\text{O}(4)\) carry over directly to \(\mathbb{H}^3\) and \(\text{O}(3,1)\). Furthermore, \(\text{O}(3,1)\) handles orientation reversing isometries exactly as it handles orientation preserving ones. Unfortunately in computations one encounters a nagging dissimilarity between \(\text{O}(4)\) and \(\text{O}(3,1)\): while numerical errors in \(\text{O}(4)\) are negligible, numerical errors in \(\text{O}(3,1)\) tend to spiral out of control. The question we ask (and answer) in this article is, “Are exponentially compounded errors simply a fact of life in hyperbolic space, no matter what model we use? Or would they be less severe in \(\text{SL}(2,\mathbb{C})\)?” In other words, is numerical instability the Achilles’ heel of \(\text{O}(3,1)\)?

MSC:

65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
57M50 General geometric structures on low-dimensional manifolds
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Beardon A. F., The Geometry of Discrete Groups. (1983) · Zbl 0528.30001 · doi:10.1007/978-1-4612-1146-4
[2] Shafarevich I. R., Algebra I. Encyclopaedia of Mathematical Sciences 11 (1990)
[3] Thurston W. P., Three-Dimensional Geometry and Topology 1 (1997) · Zbl 0873.57001
[4] Weeks J. R., SnapPea: A computer program for creating and studying hyperbolic 3-manifolds (2001)
[5] Weeks J. R., ”Computer graphics in curved spaces,” (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.