Why are "freeBoundary" surface meshes from tetrahedrons not oriented consistently?

Why are "freeBoundary" surface meshes from tetrahedrons not oriented consistently?

Answer 1

[MathWorks Support Team]

The triangles returned from the "freeBoundary" function inherit the orientation of the tetrahedrons they originally belonged to.

Orientation of the tetrahedrons is correct when:

(v12 x v13) * v14 > 0

where

vij is the vector between Points i and j : Pj - Pi

i.e. the mathematical volume of the tetrahedron is positive.

Preprocessing the tetrahedrons fixes the inconsistently oriented mesh.

A function to achieve this for an incoming tetrahedron triangulation is:

function [triRet, reversed] = repairTet(tri)
%repair tetrahedron definition 
% reverse the tetrahedron connectivity definition when they are
% defined inconsistently

nTetra=size(tri.ConnectivityList,1);

volumes=zeros(nTetra,1); % allocation

for j=1:nTetra
    pts=tri.Points(tri.ConnectivityList(j,:),:);
    vecs=pts(2:4,:)-pts(1,:);
    volumes(j)=1/6*cross(vecs(1,:),vecs(2,:))*(vecs(3,:)');
end

% identify wrongly defined tetrahedrons
reversed=(volumes<0);

temp=tri.ConnectivityList(reversed,:);
tNew=tri.ConnectivityList;
tNew(reversed,:)=temp(:,[1 2 4 3]); % notice the swapping of columns

% create the new tetrahedrons (members of triangulations are read-only)
triRet=triangulation(tNew,tri.Points);
end

The surface mesh returned from the "freeBoundary" function then contains outwards pointing normal vectors.