Bug when changing projection line in color_braiding

[jeanluct]

Bug discovered while writing taffy_xrods.m (talk). (This was originally filed with issue #2, but has been split into a separae issue.)

This is a very 'symmetric' case where noise is needed for the trajectories. By changing the projection angle it should have fixed it? But we still get an error.

#!matlab

npts = 200; r = .75; rodr = .05; n = 5;

th = linspace(0,2*pi,npts); th = th(end:-1:1);

z = zeros(npts,n);
z(:,1) = 0 + r*exp(1i*(th-pi));
z(:,2) = 0;
z(:,3) = 1 + r*exp(1i*th);
z(:,4) = 1;
z(:,5) = .5*ones(size(z(:,1)));

XY = zeros(npts,2,n); XY(:,1,:) = real(z); XY(:,2,:) = imag(z);

% Bug? If we specify a projection angle we sometimes get an error.
b = braid(XY,.1)

This returns the error

#!matlab

Error using color_braiding (line 181)
crossdat inconsistency at crossing 6, time 124.110878, index 2, with permutation [2
5  3  1  4].

Error in braidlab.braid (line 126)
        br = color_braiding(b,1:size(b,1),secnd);

The longer version attached also plots the trajectories and the braid.

[jeanluct]

From Jean-Luc Thiffeault on 2013-01-20 22:41:18+00:00

The bug is not in the projection angle calculation, I'm pretty sure.

Chaging npts makes the bug appear and disappear. The simplest choice is npts=3 (!)

Actually, I think I know what the bug is. Even though we check for coincident particles at first, it's possible that coincidence will arise because of interpolation! Grrr!

[jeanluct]

From Jean-Luc Thiffeault on 2013-01-21 04:25:44+00:00

Here's the simplest example I can come up with: 3 strings of 4 points.

#!matlab

npts = 4; r = .75; n = 3; th = linspace(0,2*pi,npts); th = th(end:-1:1);

z = zeros(npts,n);
z(:,1) = 0 + r*exp(1i*(th-pi));
z(:,2) = 1 + r*exp(1i*th);
z(:,3) = .5*ones(size(z(:,1)));

XY = zeros(npts,2,n); XY(:,1,:) = real(z); XY(:,2,:) = imag(z);

% Bug?  If we specify a projection angle we sometimes get an error.
b = braid(XY,1)

with output

#!matlab

Error using color_braiding (line 204)
crossdat inconsistency at crossing 1, time 1.585051, index 1, with permutation [1  2  3].

Error in braidlab.braid (line 126)
        br = color_braiding(b,1:size(b,1),secnd);

Here's a plot of the braid from the two axes:

It's pretty clear the triple-crossings are creating the problem. The braid is perfectly well-defined (identity braid in this case), but the algorithm gets confused. Not sure yet if the X or Y triple crossing is the problem.

[jeanluct]

From Jean-Luc Thiffeault on 2013-01-21 15:33:04+00:00

Resolve issue #3: If the crossing string is not the one on the right,
start looking for a simultaneous crossing. If we find one, then swap
and use that crossing instead.

→ <<cset 7dfcea223346>>