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function [cutpos, cutvalue, facedata, elemid, nodeid] = qmeshcut(elem, node, value, cutat, varargin)
%
% [cutpos,cutvalue,facedata,elemid,nodeid]=qmeshcut(elem,node,value,cutat)
%
% fast tetrahedral mesh slicer
%
% author:Qianqian Fang, <q.fang at neu.edu>
%
% input:
% elem: integer array with dimensions of NE x 4, each row contains
% the indices of all the nodes for each tetrahedron
% node: node coordinates, 3 columns for x, y and z respectively
% value: a scalar array with the length of node numbers, can have
% multiple columns
% cutat: cutat can have different forms:
% if cutat is a 3x3 matrix, it defines a plane by 3 points:
% cutat=[x1 y1 z1;x2 y2 z2;x3 y3 z3]
% if cutat is a vector of 4 element, it defines a plane by
% a*x+b*y+c*z+d=0 and cutat=[a b c d]
% if cutat is a single scalar, it defines an isosurface
% inside the mesh at value=cutat
% if cutat is a string, it defines an implicit surface
% at which the cut is made. it must has form expr1=expr2
% where expr1 expr2 are expressions made of x,y,z,v and
% constants
% if cutat is a cell in the form of {'expression',scalar},
% the expression will be evaluated at each node to
% produce a new value, then an isosurface is produced
% at the levelset where new value=scalar; the
% expression can contain constants and x,y,z,v
%
% output:
% cutpos: all the intersections of mesh edges by the cutat
% cutpos is similar to node, containing 3 columns for x/y/z
% cutvalue: interpolated values at the intersections, with row number
% being the num. of the intersections, column number being the
% same as "value".
% facedata: define the intersection polygons in the form of patch "Faces"
% elemid: the index of the elem in which each intersection polygon locates
% nodeid: 3 column array, first two columns are the node indices that
% each intersection position is interpolated between, and the
% last column is a weight (0-1) for the first node (that for
% the 2nd node is 1-weight).
%
% without any output, qmeshcut generates a cross-section plot
%
% the outputs of this subroutine can be easily plotted using
%
% % if value has a length of node:
% patch('Vertices',cutpos,'Faces',facedata,'FaceVertexCData',cutvalue,'FaceColor','interp');
%
% % if value has a length of elem:
% patch('Vertices',cutpos,'Faces',facedata,'CData',cutvalue,'FaceColor','flat');
%
% -- this function is part of iso2mesh toolbox (http://iso2mesh.sf.net)
%
% get the coefficients of the cutat equation: ax+by+cz+d=0
if (nargin < 4)
error('qmeshcut requires at least 4 inputs');
end
if (size(value, 1) ~= size(node, 1) && size(value, 1) ~= size(elem, 1) && ~isempty(value))
error('the length of value must be either that of node or elem');
end
if (isempty(value))
cutvalue = [];
end
if (ischar(cutat) || (iscell(cutat) && length(cutat) == 2 && ischar(cutat{1})))
x = node(:, 1);
y = node(:, 2);
z = node(:, 3);
if (ischar(cutat))
expr = regexp(cutat, '(.+)=(.+)', 'tokens', 'once'); % regexp(cutat,'=','split');
if (length(expr) ~= 2)
error('single expression must contain a single "=" sign');
end
dist = eval(expr{1}) - eval(expr{2});
else
dist = eval(cutat{1}) - cutat{2};
end
if (all(dist <= 0))
asign = -double(dist < 0);
asign(asign == 0) = 1;
else
asign = double(dist > 0);
asign(asign == 0) = -1;
end
elseif (numel(cutat) == 9 || numel(cutat) == 4)
if (numel(cutat) == 9)
[a, b, c, d] = getplanefrom3pt(cutat);
else
coeff = num2cell(cutat(:));
[a, b, c, d] = deal(coeff{:});
end
% compute which side of the cutat for all nodes in the mesh
co = repmat([a b c], size(node, 1), 1);
dist = sum((co .* node)') + d;
asign = dist;
asign(find(asign >= 0)) = 1;
asign(find(asign < 0)) = -1;
else
if (size(value, 1) ~= size(node, 1))
error('must use nodal value list when cutting mesh at an isovalue');
end
dist = value - cutat;
if (all(dist <= 0))
asign = -double(dist < 0);
asign(asign == 0) = 1;
else
asign = double(dist > 0);
asign(asign == 0) = -1;
end
end
% get all the edges of the mesh
esize = size(elem, 2);
if (esize == 4)
edges = [elem(:, [1, 2]); elem(:, [1, 3]); elem(:, [1, 4])
elem(:, [2, 3]); elem(:, [2, 4]); elem(:, [3, 4])];
elseif (esize == 3)
edges = [elem(:, [1, 2]); elem(:, [1, 3]); elem(:, [2, 3])];
elseif (esize == 10)
edges = [elem(:, [1, 5]); elem(:, [1, 8]); elem(:, [1, 7])
elem(:, [2, 5]); elem(:, [2, 6]); elem(:, [2, 9])
elem(:, [3, 6]); elem(:, [3, 7]); elem(:, [3, 10])
elem(:, [4, 8]); elem(:, [4, 9]); elem(:, [4, 10])];
end
% find all edges with two ends at the both sides of the plane
edgemask = sum(asign(edges), 2);
cutedges = find(edgemask == 0);
% edgemask=prod(asign(edges)');
% cutedges=find(edgemask<0);
% calculate the distances of the two nodes, and use them as interpolation weight
cutweight = dist(edges(cutedges, :));
totalweight = diff(cutweight');
% caveat: if an edge is co-planar to the cutat, then totalweight will be 0
% and dividing zero will cause trouble for cutweight
cutweight = abs(cutweight ./ repmat(totalweight(:), 1, 2));
% calculate the cross-cut position and the interpolated values
nodeid = edges(cutedges, :);
nodeid(:, 3) = cutweight(:, 2);
cutpos = node(edges(cutedges, 1), :) .* repmat(cutweight(:, 2), [1 3]) + ...
node(edges(cutedges, 2), :) .* repmat(cutweight(:, 1), [1 3]);
if (size(value, 1) == size(node, 1))
if (iscell(cutat) || ischar(cutat) || numel(cutat) == 9 || numel(cutat) == 4)
cutvalue = value(edges(cutedges, 1), :) .* repmat(cutweight(:, 2), [1 size(value, 2)]) + ...
value(edges(cutedges, 2), :) .* repmat(cutweight(:, 1), [1 size(value, 2)]);
elseif (numel(cutat) == 1)
cutvalue = ones(size(cutpos, 1), 1) * cutat;
end
end
% organize all cross-cuts into patch facedata format
emap = zeros(size(edges, 1), 1);
emap(cutedges) = 1:length(cutedges);
if (esize == 10)
emap = reshape(emap, [size(elem, 1), 12]); % 10-node element
else
emap = reshape(emap, [size(elem, 1), esize * (esize - 1) / 2]); % C^n_2
end
faceid = find(any(emap, 2) == 1);
facelen = length(faceid);
% cross-cuts can only be triangles or quadrilaterals for tetrahedral mesh
% (co-plannar mesh needs to be considered)
etag = sum(emap > 0, 2); % emap && etag are of length size(elem,1)
if (esize == 3) % surface mesh cut by a plane
linecut = find(etag == 2);
lineseg = emap(linecut, :)';
facedata = reshape(lineseg(find(lineseg)), [2, length(linecut)])';
elemid = linecut;
if (size(value, 1) == size(elem, 1) && ~exist('cutvalue', 'var'))
cutvalue = value(elemid, :);
end
return
end
tricut = find(etag == 3);
quadcut = find(etag == 4);
elemid = [tricut(:); quadcut(:)];
if (size(value, 1) == size(elem, 1) && ~exist('cutvalue', 'var'))
cutvalue = value(elemid, :);
end
% fast way (vector-form) to get all triangles
tripatch = emap(tricut, :)';
tripatch = reshape(tripatch(find(tripatch)), [3, length(tricut)])';
% fast way to get all quadrilaterals in convexhull form (avoid using
% convhulln)
quadpatch = emap(quadcut, :)';
quadpatch = reshape(quadpatch(find(quadpatch)), [4, length(quadpatch)])';
% combine the two sets to create the final facedata
% using the matching-tetrahedra algorithm as shown in
% https://visualization.hpc.mil/wiki/Marching_Tetrahedra
facedata = [tripatch(:, [1 2 3 3]); quadpatch(:, [1 2 4 3])];
% plot your results with the following command
if (nargout == 0)
patch('Vertices', cutpos, 'Faces', facedata, 'FaceVertexCData', cutvalue, 'facecolor', 'interp', varargin{:});
end
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