[SnapPea-planning] Dirichlet domain algorithms

[Jeff Weeks]

Dear All,

>to compute the distance functions... recursively...

>It's easy to transport this information 
>one simplex to its neighbors,

  This is the tricky step.  Conceptually it's trivial, of course,
  but computationally if we perform an O(3,1) matrix multiplication
  to tranfer the information, then we're setting off down
  the slippery slope of exponentially accumulating error.
  So the question would be, is there a way to propogate
  the information outward while keeping the numerical error
  under control?

As a test case, it's helpful to visualize, say, a Dehn filling
with coefficients (53, 117).  The Dirichlet domain will look almost
like that of the corresponding cusped manifold, but with some
tiny little faces way out where the cusp used to be.
The question is, what's the robust way to locate the vertices
that live way out on the former cusp?

Of course, one must acknowledge from the get-go that
*any* Dirichlet domain algorithm is sure to fail for 
all (p,q) Dehn fillings with sufficiently large p and q, 
even for such (p,q) Dehn fillings on the figure eight knot complement.
The challenge is to find an algorithm that gets us as far as possible
in this toy test case.  One would then hope that such an algorithm
would fare well with the more realistic Dirichlet domains that
one typically wants to study in practice.

Can anybody see how to propogate the direction and distance information
without undue numerical errors?

Best wishes to all,
Jeff