Before revision:  We present an O(n \log n) algorithm for trajectories in R^1 with n vertices and a (1 + \epsilon)-approximation algorithm for trajectories in R^2 for identifying all such stay points.
After revision:  We present an O(n \log n) algorithm for trajectories in R^1 with n vertices and a (1 + \epsilon)-approximation algorithm for trajectories in R^2 to identify all such stay points.