We revisit Doob's transformation for the Boundary crossing problem. This relates the hitting time of a moving curve by a standard Brownian motion to the hitting time of a fixed level by a generalized Ornstein-Uhlenbeck process. Some underlying mappings are introduced and studied which lead to a new classification of concave boundaries. From another viewpoint, the parabolic and square root curves are modified to construct two new real-parametrized families of curves for which the boundary crossing density is calculated explicitly.