Most conventional methods of feature extraction for pattern recognition do not pay sufficient attention to the inherent geometric properties of data, even where data have characteristic spatial features. In this study, we introduce geometric algebra to systematically extract invariant geometric features from spatial data given in a vector space. Geometric algebra is a multidimensional generalization of complex numbers and of quaternions, and can accurately describe oriented spatial objects and relations between them. We further propose a combination of several geometric features using Gaussian mixture models. We demonstrate our new method by classification of hand-written digits and alphabetic characters.