Shapes of tetrahedra

Abstract.

The equivalence classes of triangles and tetrahedra with respect to the group of the space dilatations and translations can be expressed by quaternions and ordered pairs of quaternions, respectively. These quaternions and ordered pairs of quaternions are called space shapes of triangles and shapes of tetrahedra. Using shapes, we discuss the similarity of two tetrahedra and obtain a common way for description of affine invariants of three collinear points and four coplanar points in the Euclidean space. We also examine a two-parameter set of tetrahedra with the same centroid.