The following problem is studied: determine families of curves in the upper half-plane that meet the \(x\)-axis in their endpoints at constant angles and evolve by curvature motion. The author investigates this problem from an analytic point of view, providing an existence and uniqueness result for self-similar solutions and studying their asymptotic behaviour. In particular, he shows that the curves shrink to a point in a self-similar manner when the initial curve is a graph, and he also investigates their eventual concavity.