The perception of geometry-features of airfoils is the basis in aerodynamic area for performance prediction, parameterization, aircraft inverse design, etc. There are three approaches to percept the geometric shape of an airfoil, namely manual design of airfoil geometry parameter, polynomial definition and deep learning. The first two methods can directly extract geometry-features of airfoils or polynomial equations of airfoil curves, but the number of features extracted is limited. While deep learning algorithms can extract a large number of potential features (called latent features), however, the features extracted by deep learning are lacking of explicit geometrical meaning. Motivated by the advantages of polynomial definition and deep learning, we propose a geometry-based deep learning feature extraction scheme (named Bézier-based feature extraction, BFE) for airfoils, which consists of two parts: manifold metric feature extraction and geometry-feature fusion encoder (GF encoder). Manifold metric feature extraction, with the help of the Bézier curve, captures features from tangent space of airfoil curves, and GF encoder combines airfoil coordinate data and manifold metrics together to form a novel feature representation. A public UIUC airfoil dataset is used to verify the proposed BFE. Compared with classic Auto-Encoder, the mean square error (MSE) of BFE is reduced by 17.97% ~29.14%.