  Quadratic surfaces gain more and more attention among the Geometric Algebra
community and some frameworks were proposed in order to represent, transform,
and intersect these quadratic surfaces. As far as the authors know, none of
these frameworks support all the operations required to completely handle these
surfaces. Some frameworks do not allow the construction of quadratic surfaces
from control points when others do not allow to transform these quadratic
surfaces. However , if we consider all the frameworks together, then all the
required operations over quadratic are covered. This paper presents a
unification of these frameworks that enables to represent any quadratic
surfaces either using control points or from the coefficients of its implicit
form. The proposed approach also allows to transform any quadratic surfaces and
to compute their intersection and to easily extract some geometric properties .
