Within an aerodynamic shape optimization framework, an efficient shape parameterization and deformation scheme is critical to allow flexible deformation of the surface with the maximum possible design space coverage. Numerous approaches have been developed for the geometric representation of airfoils. A fundamental approach is considered here from the geometric perspective; and a method is presented to allow the derivation of efficient, generic, and orthogonal airfoil geometric design variables. This is achieved by the mathematical decomposition of a training library. The resulting geometric modes are independent of a parameterization scheme, surface and volume mesh, and flow solver; thus, they are generally applicable. However, these modes are dependent on the training library, and so a benchmark performance measure, called the airfoil technology factor, has also been incorporated into the scheme to allow intelligent metric-based filtering, or design space reduction, of the training library to ensure efficient airfoil deformation modes are extracted. Results are presented for several geometric shape recovery problems, using two optimization approaches, and it is shown that these mathematically extracted degrees of freedom perform particularly well in all cases, showing excellent design space coverage. These design variables are also shown to outperform those based on other widely used approaches; the Hicks-Henne "bump" functions and a linear (deformative) approximation to Sobieczky's parametric section parameterization are considered.