This paper develops and evaluates a new tensor field representation to express the geometric affordance of one object over another. We expand the well known bisector surface representation to one that is weight-driven and that retains the provenance of surface points with directional vectors. We also incorporate the notion of affordance keypoints which allow for faster decisions at a point of query and with a compact and straightforward descriptor. Using a single interaction example, we are able to generalize to previously-unseen scenarios; both synthetic and also real scenes captured with RGBD sensors. We show how our interaction tensor allows for significantly better performance over alternative formulations. Evaluations also include crowdsourcing comparisons that confirm the validity of our affordance proposals, which agree on average 84% of the time with human judgments, and which is 20-40% better than the baseline methods.