An important task in trajectory analysis is clustering. The results of a clustering are often summarized by a single representative trajectory and an associated size of each cluster. We study the problem of computing a suitable representative of a set of similar trajectories. To this end we define a central trajectory \(\mathcal{C}\), which consists of pieces of the input trajectories, switches from one entity to another only if they are within a small distance of each other, and such that at any time \(t\), the point \(\mathcal{C}(t)\) is as central as possible. We measure centrality in terms of the radius of the smallest disk centered at \(\mathcal{C}(t)\) enclosing all entities at time \(t\), and discuss how the techniques can be adapted to other measures of centrality. We first study the problem in \(\R^1\), where we show that an optimal central trajectory \(\mathcal{C}\) representing \(n\) trajectories, each consisting of \(\tau\) edges, has complexity \(\Theta(\tau n^2)\) and can be computed in \(O(\tau n^2 \log n)\) time. We then consider trajectories in \(\mathbb{R}^d\) with \(d \geq 2\), and show that the complexity of \(\mathcal{C}\) is at most \(O(\tau n^{5/2})\) and can be computed in \(O(\tau n^3)\) time.