  A rigorous and powerful theoretical framework is proposed to obtain systems
of orthogonal functions (or shape modes) to represent optical surfaces. The
method is general so it can be applied to different initial shapes and
different polynomials. Here we present results for surfaces with circular
apertures when the first basis function (mode) is a conicoid. The system for
aspheres with rotational symmetry is obtained applying an appropriate change of
variables to Legendre polynomials, whereas the system for general freeform case
is obtained applying a similar procedure to spherical harmonics. Numerical
comparisons with standard systems, such as Forbes and Zernike polynomials, are
performed and discussed.
