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In the first case, the tetrahedron is skinny, and we distinguish five types depending on how its vertices cluster along the line.
Think of two congruent tetrahedrons made of differently pliable material.
We can even use (3.10) to define the local spaces for any diffeomorphic image of a tetrahedron.
We shall identify these curves, which might be thought of as forming a supporting framework for the tetrahedron.
The upper bound is based on a new charging scheme which assigns each tetrahedron of minimum volume to one of its four faces.
We normalize by scaling tetrahedra to unit diameter.
The sliver is the only type of small volume tetrahedron whose circumradius over shortest edge length ratio does not grow with decreasing volume.
In particular, the analysis is based on the singularities of these tetrahedra.
Analysis that is based on the geometry of the tetrahedra have been also presented21 for a class of parallel manipulators with three legs.
This section introduces polyhedra and studies the problem of how many tetrahedra are needed to mesh them.
We assume that every triangle in the mesh is the face of a tetrahedron in the mesh.
Another benefit of a mild weight assignment is that it does not drastically change the shape of triangles and tetrahedra.
In the tetrahedron case, is one vertex and is the tetrahedron spanned by the other four vertices.
We do not consider tetrahedra with infinite elements, since they do not have a physical meaning.
We will see later that meshes of tetrahedra with small ratio also have nice combinatorial properties, such as constant size vertex stars.
