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

###
Willaim Floyd, Brian Weber, and Jeffrey Weeks

December 21, 2001

### Abstract

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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