The Achilles' Heel of O(3,1)?

Willaim Floyd, Brian Weber, and Jeffrey Weeks
December 21, 2001


What's the best way to represent an isometry of hyperbolic 3-space $\bH^3$ ? Geometers traditionally worked in $\text{SL}(2,\bC)$, but for software development many now prefer the Minkowski space model of $\bH^3$ 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 $\bH^3$ 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 $\text{SL}(2,\bC)$?" In other words, is numerical instability the Achilles' heel of $\text{O}(3,1)$?

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