When two elements share an edge it is important for them to be able to determine if their local coordinate system at the shared edge are aligned in the same direction. For instance, the mode shapes used in the C0 methods must be continuous across element boundaries. If the local coordinate systems are not aligned in the same direction then edges modes of odd polynomial order will have different signs and the odd edge modes of one of the edges will need to be multiplied by -1.
This condition becomes more complicated in three-dimensions when two elements share a face. In this case it is not automatic that their coordinate systems on the common face will line up. Considering the triangular faces of the tetrahedron, pyramid and prism we see that there is a vertex on each triangular face that the coordinate system for that face radiates from. We will call this vertex the face origin as it is similar to a polar coordinate origin. The alignment constraint necessitates that when two triangular faces meet their origin vertices must coincide. Initially, it is not obvious how to satisfy this constraint for a mesh consisting of just tetrahedral elements. We outline two algorithms that will satisfy this constraint. The first is based on the topology of the mesh. We will only use the connections between elements to determine how we should orientate elements. In the second method we will assume that each unique vertex in the mesh will have been given a number. This second method works under some loose conditions but is extremely easy to implement and is very local in its nature.
It is useful to observe that one of the vertices of a tetrahedron is the face origin vertex for the three faces sharing that vertex. We will call this the local top vertex. Then there is one more face origin vertex on the remaining face which we call the local base vertex. These are shown in figure 2.7.
