FreeBoundary

% First of all, working algorithm:
function all_binds = getFreeBoundary( T, V )
    edges = [
        T( :, [1 2] );
        T( :, [1 3] );
        T( :, [2 3] );
    ];
    [B,~] = sort( edges, 2 );
    [edges, ia, ic] = unique( B, 'rows', 'stable' );
    [a,~] = hist( ic, unique(ic) );
    x = find( a == 1 );
    all_binds = edges( x, : );
end

% Input to the function is a 3D mesh, output is a line following the path, where the mesh was cut...

% My thought proccess for the whole thing:

% Ok, a mesh is made up of many polygons (triangles in this case) that connect the vertices in a structured way, so an edge is really  only a vector connecting 2 vertices, like [V1 V2] where V1 and V2 are vertex indices.

% Moving on, Cutting a mesh really means duplicating and overlapping some of it's vertices...
% At that point there will be many vertices that are the same, but have different indices, also when a mesh is cut, edges emerge that  are only referenced by a single polygon... And that is what I was supposed to be looking for, in other words: find the unique Vertex pairs.

% The idea was simple, sort all the edges, remove the doubles, and pick out those edges that appear only once.

% So now I needed to get familiar with the unique() and sort() functions...
% I started small, with a 1D array like: [1 1 1 2 3 4 3 2 11 4 5 5 5] and watched what sort and unique do to it, what they take in and % return back... read some docs etc... Once satisfied I upped to a 2D array like:
% arr = [
%   [1 2];
%   [2 3];
%   [1 2];
%   [3 4];
%   [3 4];
%   [4 3]
% ];
% And did the same... And then I came up with something like this:
function all_binds = getFreeBoundary( T, V )
    edges = [
        T( :, [1 2] );
        T( :, [1 3] );
        T( :, [2 3] );
    ];
    [edges, ind_markers, ~] = unique( sort( edges, 2 ), 'rows' );

    occurrences = [];
    for i = 1 : length(ind_markers)-1
        occurrences = [occurrences; ind_markers(i+1)-ind_markers(i) ];
    end
    occurrences = [occurrences; length(edges)-ind_markers(end)+1 ];

    all_binds = edges( occurrences == 1 );
end
% Once tested on an actual mesh it gave completelly wrong results...