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########################################################################
##
## Copyright (C) 2007-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## This file is part of Octave.
##
## Octave is free software: you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {@var{idx} =} tsearchn (@var{x}, @var{t}, @var{xi})
## @deftypefnx {} {[@var{idx}, @var{p}] =} tsearchn (@var{x}, @var{t}, @var{xi})
## Find the simplexes enclosing the given points.
##
## @code{tsearchn} is typically used with @code{delaunayn}:
## @code{@var{t} = delaunayn (@var{x})} returns a set of simplexes @code{t},
## then @code{tsearchn} returns the row index of @var{t} containing each point
## of @var{xi}. For points outside the convex hull, @var{idx} is NaN.
##
## If requested, @code{tsearchn} also returns the barycentric coordinates
## @var{p} of the enclosing simplexes.
##
## @seealso{delaunay, delaunayn, tsearch}
## @end deftypefn
function [idx, p] = tsearchn (x, t, xi)
if (nargin != 3)
print_usage ();
endif
if (columns (x) != columns (xi))
error ("tsearchn: number of columns of X and XI must match");
endif
if (max (t(:)) > rows (x))
error ("tsearchn: triangulation T must not access points outside X");
endif
if (nargout <= 1 && columns (x) == 2) # pass to the faster tsearch.cc
idx = tsearch (x(:,1), x(:,2), t, xi(:,1), xi(:,2));
return;
endif
nt = rows (t);
[m, n] = size (x);
mi = rows (xi);
idx = NaN (mi, 1);
p = NaN (mi, n + 1);
ni = [1:mi].';
for i = 1 : nt # each simplex in turn
T = x(t(i, :), :); # T is the current simplex
P = xi(ni, :); # P is the set of points left to calculate
## Convert to barycentric coords: these are used to express a point P
## as P = Beta * T
## where T is a simplex.
##
## If 0 <= Beta <= 1, then the linear combination is also convex,
## and the point P is inside the simplex T, otherwise it is outside.
## Since the equation system is underdetermined, we apply the constraint
## sum (Beta) == 1 to make it unique up to scaling.
##
## Note that the code below is vectorized over P, one point per row.
b = (P - T(end,:)) / (T(1:end-1,:) - T(end,:));
b(:, end+1) = 1 - sum (b, 2);
## The points xi are inside the current simplex if
## (all (b >= 0) && all (b <= 1)). As sum (b,2) == 1, we only need to
## test all(b>=0).
inside = all (b >= -1e-12, 2); # -1e-12 instead of 0 for rounding errors
idx(ni(inside)) = i;
p(ni(inside), :) = b(inside, :);
ni = ni(! inside);
endfor
endfunction
%!shared x, tri
%! x = [-1,-1;-1,1;1,-1];
%! tri = [1, 2, 3];
%!test
%! [idx, p] = tsearchn (x,tri,[-1,-1]);
%! assert (idx, 1);
%! assert (p, [1,0,0], 1e-12);
%!test
%! [idx, p] = tsearchn (x,tri,[-1,1]);
%! assert (idx, 1);
%! assert (p, [0,1,0], 1e-12);
%!test
%! [idx, p] = tsearchn (x,tri,[1,-1]);
%! assert (idx, 1);
%! assert (p, [0,0,1], 1e-12);
%!test
%! [idx, p] = tsearchn (x,tri,[-1/3,-1/3]);
%! assert (idx, 1);
%! assert (p, [1/3,1/3,1/3], 1e-12);
%!test
%! [idx, p] = tsearchn (x,tri,[1,1]);
%! assert (idx, NaN);
%! assert (p, [NaN, NaN, NaN]);
## Test input validation
%!error <Invalid call> tsearchn ()
%!error <Invalid call> tsearchn (1)
%!error <Invalid call> tsearchn (1, 2)
%!error <number of columns of X and XI must match> tsearchn ([1,2], 3, 4)
%!error <T must not access points outside X> tsearchn (1, 2, 3)
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