This surface type is ''not'' unique up to isometry or even up to similarity transformations, but rather, depends upon multiple
nonzero parameters <math>a,b,c</math>. If we're considering the surface up to rigid isometries, the parameters are unique up to permutations. If we're considering the surface up to similarity transformations, the parameters are unique up to the action of permutations ''and'' projective equivalence.
The surface, however, ''is'' unique up to affine transformations, which include transformations that do not preserve the geometric structure.