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## -*- texinfo -*-
## @deftypefn  {} {[@var{t}, @var{p}] =} __marching_cube__ (@var{xx}, @var{yy}, @var{zz}, @var{val}, @var{iso})
## @deftypefnx {} {[@var{t}, @var{p}, @var{c}] =} __marching_cube__ (@var{xx}, @var{yy}, @var{zz}, @var{v}, @var{iso}, @var{colors})
##
## Return the triangulation information @var{t} at points @var{p} for the
## isosurface values resp. the volume data @var{v} and the iso level
## @var{iso}.  It is considered that the volume data @var{v} is given at
## the points @var{xx}, @var{yy}, and @var{zz} which are of type
## three-dimensional numeric arrays.  The orientation of the triangles is
## chosen such that the normals point from the higher values to the lower
## values.
##
## Optionally the color data @var{colors} can be passed to this function
## whereas computed vertices color data @var{c} is returned as third
## argument.
##
## The marching cube algorithm is well known and described, for example, at
## Wikipedia.  The triangulation lookup table and the edge table used here
## are based on Cory Gene Bloyd's implementation and can be found beyond
## other surface and geometry stuff at Paul Bourke's website
## @uref{http://local.wasp.uwa.edu.au/~pbourke/geometry/polygonise}.
##
## For example:
##
## @example
## @group
## N = 20;
## lin = linspace (0, 2, N);
## [xx, yy, zz] = meshgrid (lin, lin, lin);
##
## v = (xx-.5).^2 + (yy-.5).^2 + (zz-.5).^2;
## [t, p] = __marching_cube__ (xx, yy, zz, v, .5);
##
## figure ();
## trimesh (t, p(:,1), p(:,2), p(:,3));
## @end group
## @end example
##
## Instead of the @code{trimesh} function the @code{patch} function can be
## used to visualize the geometry.  For example:
##
## @example
## @group
## figure ();
## view (-38, 20);
## pa = patch ("Faces", t, "Vertices", p, "FaceVertexCData", p,
##             "FaceColor", "interp", "EdgeColor", "none");
##
## ## Revert normals
## set (pa, "VertexNormals", -get (pa, "VertexNormals"));
##
## ## Set lighting
## set (pa, "FaceLighting", "gouraud");
## light ( "Position", [1 1 5]);
## @end group
## @end example
##
## @end deftypefn

function [T, p, col] = __marching_cube__ (xx, yy, zz, v, iso, colors)

  persistent edge_table = [];
  persistent tri_table = [];

  calc_cols = false;
  lindex = 4;

  if (isempty (tri_table) || isempty (edge_table))
    [edge_table, tri_table] = init_mc ();
  endif

  if (! isnumeric (xx) || ! isnumeric (yy) || ! isnumeric (zz)
      || ! isnumeric (v) || ndims (xx) != 3 || ndims (yy) != 3
      || ndims (zz) != 3 || ndims (v) != 3)
    error ("__marching_cube__: XX, YY, ZZ, v must be 3-D matrices");
  endif

  if (! size_equal (xx, yy, zz, v))
    error ("__marching_cube__: XX, YY, ZZ, v must be of equal size");
  endif

  if (any (size (xx) < [2 2 2]))
    error ("__marching_cube__: grid size must be at least 2x2x2");
  endif

  if (! isscalar (iso))
    error ("__marching_cube__: ISO must be scalar value");
  endif

  if (nargin == 6)
    if (! isnumeric (colors) || ndims (colors) != 3 || ! size_equal (colors, v))
      error ( "COLORS must be a 3-D matrix with the same size as v" );
    endif
    calc_cols = true;
    lindex = 5;
  endif

  n = size (v) - 1;

  ## phase I: assign information to each voxel which edges are intersected by
  ## the isosurface.
  cc = zeros (n(1), n(2), n(3), "uint16");
  cedge = zeros (size (cc), "uint16");

  vertex_idx = {1:n(1), 1:n(2), 1:n(3); ...
                2:n(1)+1, 1:n(2), 1:n(3); ...
                2:n(1)+1, 2:n(2)+1, 1:n(3); ...
                1:n(1), 2:n(2)+1, 1:n(3); ...
                1:n(1), 1:n(2), 2:n(3)+1; ...
                2:n(1)+1, 1:n(2), 2:n(3)+1; ...
                2:n(1)+1, 2:n(2)+1, 2:n(3)+1; ...
                1:n(1), 2:n(2)+1, 2:n(3)+1 };

  ## calculate which vertices have values higher than iso
  for ii = 1:8
    idx = v(vertex_idx{ii, :}) > iso;
    cc(idx) = bitset (cc(idx), ii);
  endfor

  cedge = edge_table(cc+1);  # assign the info about intersected edges
  id = find (cedge);         # select only voxels which are intersected
  if (isempty (id))
    T = p = col = [];
    return;
  endif

  ## phase II: calculate the list of intersection points
  xyz_off = [1, 1, 1; 2, 1, 1; 2, 2, 1; 1, 2, 1; 1, 1, 2; 2, 1, 2; 2, 2, 2; 1, 2, 2];
  edges = [1 2; 2 3; 3 4; 4 1; 5 6; 6 7; 7 8; 8 5; 1 5; 2 6; 3 7; 4 8];
  offset = sub2ind (size (v), xyz_off(:, 1), xyz_off(:, 2), xyz_off(:, 3)) - 1;
  pp = zeros (length (id), lindex, 12);
  ccedge = [vec(cedge(id)), id];
  ix_offset=0;
  for jj = 1:12
    id__ = bitget (ccedge(:, 1), jj);
    id_ = ccedge(id__, 2);
    [ix iy iz] = ind2sub (size (cc), id_);
    id_c = sub2ind (size (v), ix, iy, iz);
    id1 = id_c + offset(edges(jj, 1));
    id2 = id_c + offset(edges(jj, 2));
    if (calc_cols)
      pp(id__, 1:5, jj) = [vertex_interp(iso, xx(id1), yy(id1), zz(id1), ...
        xx(id2), yy(id2), zz(id2), v(id1), v(id2), colors(id1), colors(id2)), ...
        (1:rows (id_))' + ix_offset ];
    else
      pp(id__, 1:4, jj) = [vertex_interp(iso, xx(id1), yy(id1), zz(id1), ...
        xx(id2), yy(id2), zz(id2), v(id1), v(id2)), ...
        (1:rows (id_))' + ix_offset ];
    endif
    ix_offset += rows (id_);
  endfor

  ## phase III: calculate the triangulation from the point list
  T = [];
  tri = tri_table(cc(id)+1, :);
  for jj = 1:3:15
    id_ = find (tri(:, jj) > 0);
    p = [id_, lindex*ones(rows (id_), 1), tri(id_, jj:jj+2)];
    if (! isempty (p))
      p1 = sub2ind (size (pp), p(:,1), p(:,2), p(:,3));
      p2 = sub2ind (size (pp), p(:,1), p(:,2), p(:,4));
      p3 = sub2ind (size (pp), p(:,1), p(:,2), p(:,5));
      T = [T; pp(p1), pp(p2), pp(p3)];
    endif
  endfor

  p = [];
  col = [];
  for jj = 1:12
    idp = pp(:, lindex, jj) > 0;
    if (any (idp))
      p(pp(idp, lindex, jj), 1:3) = pp(idp, 1:3, jj);
      if (calc_cols)
        col(pp(idp, lindex, jj),1) = pp(idp, 4, jj);
      endif
    endif
  endfor

endfunction

function p = vertex_interp (isolevel, p1x, p1y, p1z,
                            p2x, p2y, p2z,valp1,valp2, col1, col2)

  if (nargin == 9)
    p = zeros (length (p1x), 3);
  elseif (nargin == 11)
    p = zeros (length (p1x), 4);
  else
    ## FIXME: Do we need input validation on internal functions?
    error ("__marching_cube__: wrong number of arguments");
  endif

  id = abs (valp1-valp2) < (10*eps) .* (abs (valp1) + abs (valp2));
  if (any (id))
    p(id, 1:3) = [ p1x(id), p1y(id), p1z(id) ];
    if (nargin == 11)
      p(id, 4) = col1(id);
    endif
  endif

  mu = zeros (length (p1x), 1);
  nid = ! id;
  if (any (nid))
    mu(nid) = (isolevel - valp1(nid)) ./ (valp2(nid) - valp1(nid));
    p(nid, 1:3) = [p1x(nid) + mu(nid) .* (p2x(nid) - p1x(nid)), ...
      p1y(nid) + mu(nid) .* (p2y(nid) - p1y(nid)), ...
      p1z(nid) + mu(nid) .* (p2z(nid) - p1z(nid))];
    if (nargin == 11)
      p(nid, 4) = col1(nid) + mu(nid) .* (col2(nid) - col1(nid));
    endif
  endif

endfunction

function [edge_table, tri_table] = init_mc ()

  edge_table = double ([
    0x0000, 0x0109, 0x0203, 0x030a, 0x0406, 0x050f, 0x0605, 0x070c, ...
    0x080c, 0x0905, 0x0a0f, 0x0b06, 0x0c0a, 0x0d03, 0x0e09, 0x0f00, ...
    0x0190, 0x0099, 0x0393, 0x029a, 0x0596, 0x049f, 0x0795, 0x069c, ...
    0x099c, 0x0895, 0x0b9f, 0x0a96, 0x0d9a, 0x0c93, 0x0f99, 0x0e90, ...
    0x0230, 0x0339, 0x0033, 0x013a, 0x0636, 0x073f, 0x0435, 0x053c, ...
    0x0a3c, 0x0b35, 0x083f, 0x0936, 0x0e3a, 0x0f33, 0x0c39, 0x0d30, ...
    0x03a0, 0x02a9, 0x01a3, 0x00aa, 0x07a6, 0x06af, 0x05a5, 0x04ac, ...
    0x0bac, 0x0aa5, 0x09af, 0x08a6, 0x0faa, 0x0ea3, 0x0da9, 0x0ca0, ...
    0x0460, 0x0569, 0x0663, 0x076a, 0x0066, 0x016f, 0x0265, 0x036c, ...
    0x0c6c, 0x0d65, 0x0e6f, 0x0f66, 0x086a, 0x0963, 0x0a69, 0x0b60, ...
    0x05f0, 0x04f9, 0x07f3, 0x06fa, 0x01f6, 0x00ff, 0x03f5, 0x02fc, ...
    0x0dfc, 0x0cf5, 0x0fff, 0x0ef6, 0x09fa, 0x08f3, 0x0bf9, 0x0af0, ...
    0x0650, 0x0759, 0x0453, 0x055a, 0x0256, 0x035f, 0x0055, 0x015c, ...
    0x0e5c, 0x0f55, 0x0c5f, 0x0d56, 0x0a5a, 0x0b53, 0x0859, 0x0950, ...
    0x07c0, 0x06c9, 0x05c3, 0x04ca, 0x03c6, 0x02cf, 0x01c5, 0x00cc, ...
    0x0fcc, 0x0ec5, 0x0dcf, 0x0cc6, 0x0bca, 0x0ac3, 0x09c9, 0x08c0, ...
    0x08c0, 0x09c9, 0x0ac3, 0x0bca, 0x0cc6, 0x0dcf, 0x0ec5, 0x0fcc, ...
    0x00cc, 0x01c5, 0x02cf, 0x03c6, 0x04ca, 0x05c3, 0x06c9, 0x07c0, ...
    0x0950, 0x0859, 0x0b53, 0x0a5a, 0x0d56, 0x0c5f, 0x0f55, 0x0e5c, ...
    0x015c, 0x0055, 0x035f, 0x0256, 0x055a, 0x0453, 0x0759, 0x0650, ...
    0x0af0, 0x0bf9, 0x08f3, 0x09fa, 0x0ef6, 0x0fff, 0x0cf5, 0x0dfc, ...
    0x02fc, 0x03f5, 0x00ff, 0x01f6, 0x06fa, 0x07f3, 0x04f9, 0x05f0, ...
    0x0b60, 0x0a69, 0x0963, 0x086a, 0x0f66, 0x0e6f, 0x0d65, 0x0c6c, ...
    0x036c, 0x0265, 0x016f, 0x0066, 0x076a, 0x0663, 0x0569, 0x0460, ...
    0x0ca0, 0x0da9, 0x0ea3, 0x0faa, 0x08a6, 0x09af, 0x0aa5, 0x0bac, ...
    0x04ac, 0x05a5, 0x06af, 0x07a6, 0x00aa, 0x01a3, 0x02a9, 0x03a0, ...
    0x0d30, 0x0c39, 0x0f33, 0x0e3a, 0x0936, 0x083f, 0x0b35, 0x0a3c, ...
    0x053c, 0x0435, 0x073f, 0x0636, 0x013a, 0x0033, 0x0339, 0x0230, ...
    0x0e90, 0x0f99, 0x0c93, 0x0d9a, 0x0a96, 0x0b9f, 0x0895, 0x099c, ...
    0x069c, 0x0795, 0x049f, 0x0596, 0x029a, 0x0393, 0x0099, 0x0190, ...
    0x0f00, 0x0e09, 0x0d03, 0x0c0a, 0x0b06, 0x0a0f, 0x0905, 0x080c, ...
    0x070c, 0x0605, 0x050f, 0x0406, 0x030a, 0x0203, 0x0109, 0x0000 ]);

  tri_table = [
  -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 8, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 1, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  1, 8, 3, 9, 8, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  1, 2, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 8, 3, 1, 2, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  9, 2, 10, 0, 2, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  2, 8, 3, 2, 10, 8, 10, 9, 8, -1, -1, -1, -1, -1, -1, -1;
  3, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 11, 2, 8, 11, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  1, 9, 0, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  1, 11, 2, 1, 9, 11, 9, 8, 11, -1, -1, -1, -1, -1, -1, -1;
  3, 10, 1, 11, 10, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 10, 1, 0, 8, 10, 8, 11, 10, -1, -1, -1, -1, -1, -1, -1;
  3, 9, 0, 3, 11, 9, 11, 10, 9, -1, -1, -1, -1, -1, -1, -1;
  9, 8, 10, 10, 8, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  4, 7, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  4, 3, 0, 7, 3, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 1, 9, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  4, 1, 9, 4, 7, 1, 7, 3, 1, -1, -1, -1, -1, -1, -1, -1;
  1, 2, 10, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  3, 4, 7, 3, 0, 4, 1, 2, 10, -1, -1, -1, -1, -1, -1, -1;
  9, 2, 10, 9, 0, 2, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1;
  2, 10, 9, 2, 9, 7, 2, 7, 3, 7, 9, 4, -1, -1, -1, -1;
  8, 4, 7, 3, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  11, 4, 7, 11, 2, 4, 2, 0, 4, -1, -1, -1, -1, -1, -1, -1;
  9, 0, 1, 8, 4, 7, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1;
  4, 7, 11, 9, 4, 11, 9, 11, 2, 9, 2, 1, -1, -1, -1, -1;
  3, 10, 1, 3, 11, 10, 7, 8, 4, -1, -1, -1, -1, -1, -1, -1;
  1, 11, 10, 1, 4, 11, 1, 0, 4, 7, 11, 4, -1, -1, -1, -1;
  4, 7, 8, 9, 0, 11, 9, 11, 10, 11, 0, 3, -1, -1, -1, -1;
  4, 7, 11, 4, 11, 9, 9, 11, 10, -1, -1, -1, -1, -1, -1, -1;
  9, 5, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  9, 5, 4, 0, 8, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 5, 4, 1, 5, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  8, 5, 4, 8, 3, 5, 3, 1, 5, -1, -1, -1, -1, -1, -1, -1;
  1, 2, 10, 9, 5, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  3, 0, 8, 1, 2, 10, 4, 9, 5, -1, -1, -1, -1, -1, -1, -1;
  5, 2, 10, 5, 4, 2, 4, 0, 2, -1, -1, -1, -1, -1, -1, -1;
  2, 10, 5, 3, 2, 5, 3, 5, 4, 3, 4, 8, -1, -1, -1, -1;
  9, 5, 4, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 11, 2, 0, 8, 11, 4, 9, 5, -1, -1, -1, -1, -1, -1, -1;
  0, 5, 4, 0, 1, 5, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1;
  2, 1, 5, 2, 5, 8, 2, 8, 11, 4, 8, 5, -1, -1, -1, -1;
  10, 3, 11, 10, 1, 3, 9, 5, 4, -1, -1, -1, -1, -1, -1, -1;
  4, 9, 5, 0, 8, 1, 8, 10, 1, 8, 11, 10, -1, -1, -1, -1;
  5, 4, 0, 5, 0, 11, 5, 11, 10, 11, 0, 3, -1, -1, -1, -1;
  5, 4, 8, 5, 8, 10, 10, 8, 11, -1, -1, -1, -1, -1, -1, -1;
  9, 7, 8, 5, 7, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  9, 3, 0, 9, 5, 3, 5, 7, 3, -1, -1, -1, -1, -1, -1, -1;
  0, 7, 8, 0, 1, 7, 1, 5, 7, -1, -1, -1, -1, -1, -1, -1;
  1, 5, 3, 3, 5, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  9, 7, 8, 9, 5, 7, 10, 1, 2, -1, -1, -1, -1, -1, -1, -1;
  10, 1, 2, 9, 5, 0, 5, 3, 0, 5, 7, 3, -1, -1, -1, -1;
  8, 0, 2, 8, 2, 5, 8, 5, 7, 10, 5, 2, -1, -1, -1, -1;
  2, 10, 5, 2, 5, 3, 3, 5, 7, -1, -1, -1, -1, -1, -1, -1;
  7, 9, 5, 7, 8, 9, 3, 11, 2, -1, -1, -1, -1, -1, -1, -1;
  9, 5, 7, 9, 7, 2, 9, 2, 0, 2, 7, 11, -1, -1, -1, -1;
  2, 3, 11, 0, 1, 8, 1, 7, 8, 1, 5, 7, -1, -1, -1, -1;
  11, 2, 1, 11, 1, 7, 7, 1, 5, -1, -1, -1, -1, -1, -1, -1;
  9, 5, 8, 8, 5, 7, 10, 1, 3, 10, 3, 11, -1, -1, -1, -1;
  5, 7, 0, 5, 0, 9, 7, 11, 0, 1, 0, 10, 11, 10, 0, -1;
  11, 10, 0, 11, 0, 3, 10, 5, 0, 8, 0, 7, 5, 7, 0, -1;
  11, 10, 5, 7, 11, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  10, 6, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 8, 3, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  9, 0, 1, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  1, 8, 3, 1, 9, 8, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1;
  1, 6, 5, 2, 6, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  1, 6, 5, 1, 2, 6, 3, 0, 8, -1, -1, -1, -1, -1, -1, -1;
  9, 6, 5, 9, 0, 6, 0, 2, 6, -1, -1, -1, -1, -1, -1, -1;
  5, 9, 8, 5, 8, 2, 5, 2, 6, 3, 2, 8, -1, -1, -1, -1;
  2, 3, 11, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  11, 0, 8, 11, 2, 0, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1;
  0, 1, 9, 2, 3, 11, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1;
  5, 10, 6, 1, 9, 2, 9, 11, 2, 9, 8, 11, -1, -1, -1, -1;
  6, 3, 11, 6, 5, 3, 5, 1, 3, -1, -1, -1, -1, -1, -1, -1;
  0, 8, 11, 0, 11, 5, 0, 5, 1, 5, 11, 6, -1, -1, -1, -1;
  3, 11, 6, 0, 3, 6, 0, 6, 5, 0, 5, 9, -1, -1, -1, -1;
  6, 5, 9, 6, 9, 11, 11, 9, 8, -1, -1, -1, -1, -1, -1, -1;
  5, 10, 6, 4, 7, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  4, 3, 0, 4, 7, 3, 6, 5, 10, -1, -1, -1, -1, -1, -1, -1;
  1, 9, 0, 5, 10, 6, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1;
  10, 6, 5, 1, 9, 7, 1, 7, 3, 7, 9, 4, -1, -1, -1, -1;
  6, 1, 2, 6, 5, 1, 4, 7, 8, -1, -1, -1, -1, -1, -1, -1;
  1, 2, 5, 5, 2, 6, 3, 0, 4, 3, 4, 7, -1, -1, -1, -1;
  8, 4, 7, 9, 0, 5, 0, 6, 5, 0, 2, 6, -1, -1, -1, -1;
  7, 3, 9, 7, 9, 4, 3, 2, 9, 5, 9, 6, 2, 6, 9, -1;
  3, 11, 2, 7, 8, 4, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1;
  5, 10, 6, 4, 7, 2, 4, 2, 0, 2, 7, 11, -1, -1, -1, -1;
  0, 1, 9, 4, 7, 8, 2, 3, 11, 5, 10, 6, -1, -1, -1, -1;
  9, 2, 1, 9, 11, 2, 9, 4, 11, 7, 11, 4, 5, 10, 6, -1;
  8, 4, 7, 3, 11, 5, 3, 5, 1, 5, 11, 6, -1, -1, -1, -1;
  5, 1, 11, 5, 11, 6, 1, 0, 11, 7, 11, 4, 0, 4, 11, -1;
  0, 5, 9, 0, 6, 5, 0, 3, 6, 11, 6, 3, 8, 4, 7, -1;
  6, 5, 9, 6, 9, 11, 4, 7, 9, 7, 11, 9, -1, -1, -1, -1;
  10, 4, 9, 6, 4, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  4, 10, 6, 4, 9, 10, 0, 8, 3, -1, -1, -1, -1, -1, -1, -1;
  10, 0, 1, 10, 6, 0, 6, 4, 0, -1, -1, -1, -1, -1, -1, -1;
  8, 3, 1, 8, 1, 6, 8, 6, 4, 6, 1, 10, -1, -1, -1, -1;
  1, 4, 9, 1, 2, 4, 2, 6, 4, -1, -1, -1, -1, -1, -1, -1;
  3, 0, 8, 1, 2, 9, 2, 4, 9, 2, 6, 4, -1, -1, -1, -1;
  0, 2, 4, 4, 2, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  8, 3, 2, 8, 2, 4, 4, 2, 6, -1, -1, -1, -1, -1, -1, -1;
  10, 4, 9, 10, 6, 4, 11, 2, 3, -1, -1, -1, -1, -1, -1, -1;
  0, 8, 2, 2, 8, 11, 4, 9, 10, 4, 10, 6, -1, -1, -1, -1;
  3, 11, 2, 0, 1, 6, 0, 6, 4, 6, 1, 10, -1, -1, -1, -1;
  6, 4, 1, 6, 1, 10, 4, 8, 1, 2, 1, 11, 8, 11, 1, -1;
  9, 6, 4, 9, 3, 6, 9, 1, 3, 11, 6, 3, -1, -1, -1, -1;
  8, 11, 1, 8, 1, 0, 11, 6, 1, 9, 1, 4, 6, 4, 1, -1;
  3, 11, 6, 3, 6, 0, 0, 6, 4, -1, -1, -1, -1, -1, -1, -1;
  6, 4, 8, 11, 6, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  7, 10, 6, 7, 8, 10, 8, 9, 10, -1, -1, -1, -1, -1, -1, -1;
  0, 7, 3, 0, 10, 7, 0, 9, 10, 6, 7, 10, -1, -1, -1, -1;
  10, 6, 7, 1, 10, 7, 1, 7, 8, 1, 8, 0, -1, -1, -1, -1;
  10, 6, 7, 10, 7, 1, 1, 7, 3, -1, -1, -1, -1, -1, -1, -1;
  1, 2, 6, 1, 6, 8, 1, 8, 9, 8, 6, 7, -1, -1, -1, -1;
  2, 6, 9, 2, 9, 1, 6, 7, 9, 0, 9, 3, 7, 3, 9, -1;
  7, 8, 0, 7, 0, 6, 6, 0, 2, -1, -1, -1, -1, -1, -1, -1;
  7, 3, 2, 6, 7, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  2, 3, 11, 10, 6, 8, 10, 8, 9, 8, 6, 7, -1, -1, -1, -1;
  2, 0, 7, 2, 7, 11, 0, 9, 7, 6, 7, 10, 9, 10, 7, -1;
  1, 8, 0, 1, 7, 8, 1, 10, 7, 6, 7, 10, 2, 3, 11, -1;
  11, 2, 1, 11, 1, 7, 10, 6, 1, 6, 7, 1, -1, -1, -1, -1;
  8, 9, 6, 8, 6, 7, 9, 1, 6, 11, 6, 3, 1, 3, 6, -1;
  0, 9, 1, 11, 6, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  7, 8, 0, 7, 0, 6, 3, 11, 0, 11, 6, 0, -1, -1, -1, -1;
  7, 11, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  7, 6, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  3, 0, 8, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 1, 9, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  8, 1, 9, 8, 3, 1, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1;
  10, 1, 2, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  1, 2, 10, 3, 0, 8, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1;
  2, 9, 0, 2, 10, 9, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1;
  6, 11, 7, 2, 10, 3, 10, 8, 3, 10, 9, 8, -1, -1, -1, -1;
  7, 2, 3, 6, 2, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  7, 0, 8, 7, 6, 0, 6, 2, 0, -1, -1, -1, -1, -1, -1, -1;
  2, 7, 6, 2, 3, 7, 0, 1, 9, -1, -1, -1, -1, -1, -1, -1;
  1, 6, 2, 1, 8, 6, 1, 9, 8, 8, 7, 6, -1, -1, -1, -1;
  10, 7, 6, 10, 1, 7, 1, 3, 7, -1, -1, -1, -1, -1, -1, -1;
  10, 7, 6, 1, 7, 10, 1, 8, 7, 1, 0, 8, -1, -1, -1, -1;
  0, 3, 7, 0, 7, 10, 0, 10, 9, 6, 10, 7, -1, -1, -1, -1;
  7, 6, 10, 7, 10, 8, 8, 10, 9, -1, -1, -1, -1, -1, -1, -1;
  6, 8, 4, 11, 8, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  3, 6, 11, 3, 0, 6, 0, 4, 6, -1, -1, -1, -1, -1, -1, -1;
  8, 6, 11, 8, 4, 6, 9, 0, 1, -1, -1, -1, -1, -1, -1, -1;
  9, 4, 6, 9, 6, 3, 9, 3, 1, 11, 3, 6, -1, -1, -1, -1;
  6, 8, 4, 6, 11, 8, 2, 10, 1, -1, -1, -1, -1, -1, -1, -1;
  1, 2, 10, 3, 0, 11, 0, 6, 11, 0, 4, 6, -1, -1, -1, -1;
  4, 11, 8, 4, 6, 11, 0, 2, 9, 2, 10, 9, -1, -1, -1, -1;
  10, 9, 3, 10, 3, 2, 9, 4, 3, 11, 3, 6, 4, 6, 3, -1;
  8, 2, 3, 8, 4, 2, 4, 6, 2, -1, -1, -1, -1, -1, -1, -1;
  0, 4, 2, 4, 6, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  1, 9, 0, 2, 3, 4, 2, 4, 6, 4, 3, 8, -1, -1, -1, -1;
  1, 9, 4, 1, 4, 2, 2, 4, 6, -1, -1, -1, -1, -1, -1, -1;
  8, 1, 3, 8, 6, 1, 8, 4, 6, 6, 10, 1, -1, -1, -1, -1;
  10, 1, 0, 10, 0, 6, 6, 0, 4, -1, -1, -1, -1, -1, -1, -1;
  4, 6, 3, 4, 3, 8, 6, 10, 3, 0, 3, 9, 10, 9, 3, -1;
  10, 9, 4, 6, 10, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  4, 9, 5, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 8, 3, 4, 9, 5, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1;
  5, 0, 1, 5, 4, 0, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1;
  11, 7, 6, 8, 3, 4, 3, 5, 4, 3, 1, 5, -1, -1, -1, -1;
  9, 5, 4, 10, 1, 2, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1;
  6, 11, 7, 1, 2, 10, 0, 8, 3, 4, 9, 5, -1, -1, -1, -1;
  7, 6, 11, 5, 4, 10, 4, 2, 10, 4, 0, 2, -1, -1, -1, -1;
  3, 4, 8, 3, 5, 4, 3, 2, 5, 10, 5, 2, 11, 7, 6, -1;
  7, 2, 3, 7, 6, 2, 5, 4, 9, -1, -1, -1, -1, -1, -1, -1;
  9, 5, 4, 0, 8, 6, 0, 6, 2, 6, 8, 7, -1, -1, -1, -1;
  3, 6, 2, 3, 7, 6, 1, 5, 0, 5, 4, 0, -1, -1, -1, -1;
  6, 2, 8, 6, 8, 7, 2, 1, 8, 4, 8, 5, 1, 5, 8, -1;
  9, 5, 4, 10, 1, 6, 1, 7, 6, 1, 3, 7, -1, -1, -1, -1;
  1, 6, 10, 1, 7, 6, 1, 0, 7, 8, 7, 0, 9, 5, 4, -1;
  4, 0, 10, 4, 10, 5, 0, 3, 10, 6, 10, 7, 3, 7, 10, -1;
  7, 6, 10, 7, 10, 8, 5, 4, 10, 4, 8, 10, -1, -1, -1, -1;
  6, 9, 5, 6, 11, 9, 11, 8, 9, -1, -1, -1, -1, -1, -1, -1;
  3, 6, 11, 0, 6, 3, 0, 5, 6, 0, 9, 5, -1, -1, -1, -1;
  0, 11, 8, 0, 5, 11, 0, 1, 5, 5, 6, 11, -1, -1, -1, -1;
  6, 11, 3, 6, 3, 5, 5, 3, 1, -1, -1, -1, -1, -1, -1, -1;
  1, 2, 10, 9, 5, 11, 9, 11, 8, 11, 5, 6, -1, -1, -1, -1;
  0, 11, 3, 0, 6, 11, 0, 9, 6, 5, 6, 9, 1, 2, 10, -1;
  11, 8, 5, 11, 5, 6, 8, 0, 5, 10, 5, 2, 0, 2, 5, -1;
  6, 11, 3, 6, 3, 5, 2, 10, 3, 10, 5, 3, -1, -1, -1, -1;
  5, 8, 9, 5, 2, 8, 5, 6, 2, 3, 8, 2, -1, -1, -1, -1;
  9, 5, 6, 9, 6, 0, 0, 6, 2, -1, -1, -1, -1, -1, -1, -1;
  1, 5, 8, 1, 8, 0, 5, 6, 8, 3, 8, 2, 6, 2, 8, -1;
  1, 5, 6, 2, 1, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  1, 3, 6, 1, 6, 10, 3, 8, 6, 5, 6, 9, 8, 9, 6, -1;
  10, 1, 0, 10, 0, 6, 9, 5, 0, 5, 6, 0, -1, -1, -1, -1;
  0, 3, 8, 5, 6, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  10, 5, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  11, 5, 10, 7, 5, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  11, 5, 10, 11, 7, 5, 8, 3, 0, -1, -1, -1, -1, -1, -1, -1;
  5, 11, 7, 5, 10, 11, 1, 9, 0, -1, -1, -1, -1, -1, -1, -1;
  10, 7, 5, 10, 11, 7, 9, 8, 1, 8, 3, 1, -1, -1, -1, -1;
  11, 1, 2, 11, 7, 1, 7, 5, 1, -1, -1, -1, -1, -1, -1, -1;
  0, 8, 3, 1, 2, 7, 1, 7, 5, 7, 2, 11, -1, -1, -1, -1;
  9, 7, 5, 9, 2, 7, 9, 0, 2, 2, 11, 7, -1, -1, -1, -1;
  7, 5, 2, 7, 2, 11, 5, 9, 2, 3, 2, 8, 9, 8, 2, -1;
  2, 5, 10, 2, 3, 5, 3, 7, 5, -1, -1, -1, -1, -1, -1, -1;
  8, 2, 0, 8, 5, 2, 8, 7, 5, 10, 2, 5, -1, -1, -1, -1;
  9, 0, 1, 5, 10, 3, 5, 3, 7, 3, 10, 2, -1, -1, -1, -1;
  9, 8, 2, 9, 2, 1, 8, 7, 2, 10, 2, 5, 7, 5, 2, -1;
  1, 3, 5, 3, 7, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 8, 7, 0, 7, 1, 1, 7, 5, -1, -1, -1, -1, -1, -1, -1;
  9, 0, 3, 9, 3, 5, 5, 3, 7, -1, -1, -1, -1, -1, -1, -1;
  9, 8, 7, 5, 9, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  5, 8, 4, 5, 10, 8, 10, 11, 8, -1, -1, -1, -1, -1, -1, -1;
  5, 0, 4, 5, 11, 0, 5, 10, 11, 11, 3, 0, -1, -1, -1, -1;
  0, 1, 9, 8, 4, 10, 8, 10, 11, 10, 4, 5, -1, -1, -1, -1;
  10, 11, 4, 10, 4, 5, 11, 3, 4, 9, 4, 1, 3, 1, 4, -1;
  2, 5, 1, 2, 8, 5, 2, 11, 8, 4, 5, 8, -1, -1, -1, -1;
  0, 4, 11, 0, 11, 3, 4, 5, 11, 2, 11, 1, 5, 1, 11, -1;
  0, 2, 5, 0, 5, 9, 2, 11, 5, 4, 5, 8, 11, 8, 5, -1;
  9, 4, 5, 2, 11, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  2, 5, 10, 3, 5, 2, 3, 4, 5, 3, 8, 4, -1, -1, -1, -1;
  5, 10, 2, 5, 2, 4, 4, 2, 0, -1, -1, -1, -1, -1, -1, -1;
  3, 10, 2, 3, 5, 10, 3, 8, 5, 4, 5, 8, 0, 1, 9, -1;
  5, 10, 2, 5, 2, 4, 1, 9, 2, 9, 4, 2, -1, -1, -1, -1;
  8, 4, 5, 8, 5, 3, 3, 5, 1, -1, -1, -1, -1, -1, -1, -1;
  0, 4, 5, 1, 0, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  8, 4, 5, 8, 5, 3, 9, 0, 5, 0, 3, 5, -1, -1, -1, -1;
  9, 4, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  4, 11, 7, 4, 9, 11, 9, 10, 11, -1, -1, -1, -1, -1, -1, -1;
  0, 8, 3, 4, 9, 7, 9, 11, 7, 9, 10, 11, -1, -1, -1, -1;
  1, 10, 11, 1, 11, 4, 1, 4, 0, 7, 4, 11, -1, -1, -1, -1;
  3, 1, 4, 3, 4, 8, 1, 10, 4, 7, 4, 11, 10, 11, 4, -1;
  4, 11, 7, 9, 11, 4, 9, 2, 11, 9, 1, 2, -1, -1, -1, -1;
  9, 7, 4, 9, 11, 7, 9, 1, 11, 2, 11, 1, 0, 8, 3, -1;
  11, 7, 4, 11, 4, 2, 2, 4, 0, -1, -1, -1, -1, -1, -1, -1;
  11, 7, 4, 11, 4, 2, 8, 3, 4, 3, 2, 4, -1, -1, -1, -1;
  2, 9, 10, 2, 7, 9, 2, 3, 7, 7, 4, 9, -1, -1, -1, -1;
  9, 10, 7, 9, 7, 4, 10, 2, 7, 8, 7, 0, 2, 0, 7, -1;
  3, 7, 10, 3, 10, 2, 7, 4, 10, 1, 10, 0, 4, 0, 10, -1;
  1, 10, 2, 8, 7, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  4, 9, 1, 4, 1, 7, 7, 1, 3, -1, -1, -1, -1, -1, -1, -1;
  4, 9, 1, 4, 1, 7, 0, 8, 1, 8, 7, 1, -1, -1, -1, -1;
  4, 0, 3, 7, 4, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  4, 8, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  9, 10, 8, 10, 11, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  3, 0, 9, 3, 9, 11, 11, 9, 10, -1, -1, -1, -1, -1, -1, -1;
  0, 1, 10, 0, 10, 8, 8, 10, 11, -1, -1, -1, -1, -1, -1, -1;
  3, 1, 10, 11, 3, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  1, 2, 11, 1, 11, 9, 9, 11, 8, -1, -1, -1, -1, -1, -1, -1;
  3, 0, 9, 3, 9, 11, 1, 2, 9, 2, 11, 9, -1, -1, -1, -1;
  0, 2, 11, 8, 0, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  3, 2, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  2, 3, 8, 2, 8, 10, 10, 8, 9, -1, -1, -1, -1, -1, -1, -1;
  9, 10, 2, 0, 9, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  2, 3, 8, 2, 8, 10, 0, 1, 8, 1, 10, 8, -1, -1, -1, -1;
  1, 10, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  1, 3, 8, 9, 1, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 9, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  0, 3, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1;
  -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 ] + 1;

endfunction