The responses of visual cortical neurons are highly nonlinear functions of image stimuli. I present a geometric view of these nonlinear responses and classify them as forms of selectivity or invariance, building on a body of established work. With the sparse coding network, a well-known network model of V1 computation, I attempt to quantify selectivity and invariance by measuring the curvature of neural response surfaces in both low-dimensional subspaces and image state space. I argue that this geometric view allows the precise quantification of feature selectivity and invariance in network models in a way that provides insight into the computations necessary for object recognition, and that this view may be a useful tool for future physiological experiments.