Smooth pursuit eye movements hold information about the health, activity and situation of people, but to date there has been no efficient method for their automated detection. In this work we present a method to tackle the problem, based on machine learning. At the core of our method is a novel set of shape features that capture the characteristic shape of smooth pursuit movements over time. The features individually represent incomplete information about smooth pursuits but are combined in a machine learning approach. In an evaluation with eye movements collected from 18 participants, we show that our method can detect smooth pursuit movements with an accuracy of up to 92%, depending on the size of the feature set used for their prediction. Our results have twofold significance. First, they demonstrate a method for smooth pursuit detection in mainstream eye tracking, and secondly they highlight the utility of machine learning for eye movement analysis.