[SnapPea-planning] snappea request (fwd)

[Saul Schleimer]

---------- Forwarded message ----------
Date: Sun, 24 May 2009 20:52:42 -0500
From: Nathan Dunfield <nmd at illinois.edu>
To: Saul Schleimer <saulsch at math.rutgers.edu>
Cc: nathaniel at thething.is
Subject: Re: snappea request

Saul,

Back when I was in graduate school, I wrote a program to probe the
boundary of hyperbolic Dehn surgery space in the manner that you
describe.  It worked by the following method:

- For each lattice point x in a large square, do a change of
peripheral basis so that x is a meridian.   Then move in along the
corresponding ray in the following manner:

/*

This function finds the point on the boundry of hyperbolic Dehn
surgery space along the ray in the meridian direction.   The algorithm
works like this: At each stage we consider two points along the ray,
lower_pt and upper_pt, where lower_pt < upper_pt, lower_pt is some
kind of degenerate solution and upper_pt is a geometric solution
(allowing neg. oriented tetra.).  We then consider the midpoint of
upper_pt and lower_pt and test it to see if it lies in hyperbolic Dehn
surgery space. If the midpoint does then it becomes the new upper_pt,
else it becomes the new lower_pt. The algorithm terminates once
upper_pt and lower_pt are closer than the constant sensitivity or if
max_iter is exceeded.  The function puts the final lower_pt in the
varible final_lower_point and returns the solution type of the
final_lower_pt.

The function does not change manifold.  Setting the intial_lower_pt to
0 is OK as in this case, the function does not check to see if the
intial_lower_pt represents a non-geometric solution.

The function find_bdry_of_hyperbolic_dehn_surgery_space calls
sub_find... several times with different starting points in order to
deal with the problem of sub_find reporting much to small of numbers.

*/

I've attached the source code.  My recollection is that it worked fine
but was slow.  But that was more than 10 years ago, so it runs pretty
quickly now...

--------

Also, Saul, I think it would be a good idea if you forwarded your
messages, and this one, to the snappea-planning list for posterity.

	Best,

	Nathan




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