Code covered by the BSD License

# geom3d

### David Legland (view profile)

19 Jun 2009 (Updated )

Library to handle 3D geometric primitives: create, intersect, display, and make basic computations

### Editor's Notes:

This file was selected as MATLAB Central Pick of the Week

intersectLineMesh3d(line, vertices, faces, varargin)
```function [points, pos, faceInds] = intersectLineMesh3d(line, vertices, faces, varargin)
%INTERSECTLINEMESH3D Intersection points of a 3D line with a mesh
%
%   INTERS = intersectLineMesh3d(LINE, VERTICES, FACES)
%   Compute the intersection points between a 3D line and a 3D mesh defined
%   by vertices and faces.
%
%   [INTERS POS INDS] = intersectLineMesh3d(LINE, VERTICES, FACES)
%   Also returns the position of each intersection point on the input line,
%   and the index of the intersected faces.
%   If POS > 0, the point is also on the ray corresponding to the line.
%
%   Example
%   intersectLineMesh3d
%
%   meshes3d, triangulateFaces, intersectLineTriangle3d
%

% ------
% Author: David Legland
% e-mail: david.legland@grignon.inra.fr
% Created: 2011-12-20,    using Matlab 7.9.0.529 (R2009b)
% Copyright 2011 INRA - Cepia Software Platform.

tol = 1e-12;
if ~isempty(varargin)
tol = varargin{1};
end

% ensure the mesh has triangular faces
tri2Face = [];
if iscell(faces) || size(faces, 2) ~= 3
[faces, tri2Face] = triangulateFaces(faces);
end

% find triangle edge vectors
t0  = vertices(faces(:,1), :);
u   = vertices(faces(:,2), :) - t0;
v   = vertices(faces(:,3), :) - t0;

% triangle normal
n   = normalizeVector3d(vectorCross3d(u, v));

% direction vector of line
dir = line(4:6);

% vector between triangle origin and line origin
w0 = bsxfun(@minus, line(1:3), t0);

a = -dot(n, w0, 2);
b = dot(n, repmat(dir, size(n, 1), 1), 2);

valid = abs(b) > tol & vectorNorm3d(n) > tol;

% compute intersection point of line with supporting plane
% If pos < 0: point before ray
% IF pos > |dir|: point after edge
pos = a ./ b;

% coordinates of intersection point
points = bsxfun(@plus, line(1:3), bsxfun(@times, pos, dir));

%% test if intersection point is inside triangle

% normalize direction vectors of triangle edges
uu  = dot(u, u, 2);
uv  = dot(u, v, 2);
vv  = dot(v, v, 2);

% coordinates of vector v in triangle basis
w   = points - t0;
wu  = dot(w, u, 2);
wv  = dot(w, v, 2);

% normalization constant
D = uv.^2 - uu .* vv;

% test first coordinate
s = (uv .* wv - vv .* wu) ./ D;
% ind1 = s < 0.0 | s > 1.0;
ind1 = s < -tol | s > (1.0 + tol);
points(ind1, :) = NaN;
pos(ind1) = NaN;

% test second coordinate, and third triangle edge
t = (uv .* wu - uu .* wv) ./ D;
% ind2 = t < 0.0 | (s + t) > 1.0;
ind2 = t < -tol | (s + t) > (1.0 + tol);
points(ind2, :) = NaN;
pos(ind2) = NaN;

% keep only interesting points
inds = ~ind1 & ~ind2 & valid;
points = points(inds, :);

pos = pos(inds);
faceInds = find(inds);

% convert to face indices of original mesh
if ~isempty(tri2Face)
faceInds = tri2Face(faceInds);
end
```