| The manipulation of planar objects using linear fences is of interest in robotics and parts feeding applications. The global behavior of such systems can be characterized graphically using Brost's push stability diagram (PSD). Previously, we have shown specifically under what conditions this representation undergoes qualitative, topological transitions corresponding to globally distinct behavioral regimes. In this paper, we show these insights form a united whole when viewed from the perspective of catastrophe theory. The key result is that a planar object being pushed by a fence under the assumption of Coulomb friction is functionally equivalent to a gravitational catastrophe machine. Qualitative changes in global behavior are thus explained as catastrophes as singularities are encountered on a discriminant surface due to smooth changes in parameters. Catastrophe theory thus forms part of a computational theory of planar orientation, the aim of which is to understand such systems and make predictions about their behavior.