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 {O}(3,1).
One powerful advantage is that ideas and computations in $S^3$
using matrices in {O}(4) carry over directly to $\bH^3$ and
{O}(3, 1).
Furthermore, {O}(3,1) handles orientation reversing isometries
exactly as it handles orientation preserving ones.
Unfortunately in computations one encounters a nagging dissimilarity
between {O}(4) and {O}(3,1):
while numerical errors in {O}(4) are negligible,
numerical errors in {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 {O}(3,1)?