The Boolean model is a random set process in which random shapes are positioned according to outcomes of a Poisson process. Two- and three-dimensional versions of the model characterize structures of certain heterogeneous materials. Linear transects of the Boolean model produce a one-dimensional Boolean model that summarizes some material properties. Two functions from linear transects, clump-length and lineal path distributions, provide information on material phase connectivity. Computation of these distributions is notoriously difficult. We provide a discrete approximation to the one-dimensional convex-grain Boolean model that yields stable, linear-time, recursive algorithms to approximate these functions. Computer simulations demonstrate accuracy and speed.