The  absolutely nonsingular tensor is characterized by its determinant polynomial. Equivalence among absolutely nonsingular tensors with respect to a class of linear transformations, which do not change the tensor rank, is studied. For  absolutely nonsingular tensors, it is shown theoretically that certain affine geometric invariants of the constant hypersurface of a determinant polynomial are useful to detect inequivalence among absolutely nonsingular tensors. Also, numerical calculations are presented and these invariants are shown to be useful indeed for  absolutely singular tensors. The calculation of invariants by the use of t-spherical design is also introduced.