We develop a low-dimensional approximation of the set of possible deformations of smoothly deforming objects of planar topology. To this end, we propose a novel parameterization of inextensible surfaces that allows us first to effectively sample the space of all possible deformations, which is a priori very large, and then to derive the low-dimensional model using a simple dimensionality reduction technique. We incorporate the resulting models into a monocular tracking system that we use to capture complex deformations of objects such as sheets of paper or more flexible material. We also show that, even though the model was built by sampling the set of possible deformations of inextensible surfaces, it can also handle extensible ones.