Boundaries are the timestamps at which a class in a time series changes. Recently, representation-based boundary detection has gained popularity, but its emphasis on consecutive distance difference backfires, especially when the changes are gradual. In this paper, we propose a boundary detection method, RECURVE, based on a novel change metric, the curvature of a representation trajectory, to accommodate both gradual and abrupt changes. Here, a sequence of representations in the representation space is interpreted as a trajectory, and a curvature at each timestamp can be computed. Using the theory of random walk, we formally show that the mean curvature is lower near boundaries than at other points. Extensive experiments using diverse real-world time-series datasets confirm the superiority of RECURVE over state-of-the-art methods.