The study of linear fractals has gained a great deal from the study of quadratic fractals, despite important differences. Methods for classifying points in the complement of a fractal shape were originally developed for quadratic fractals, to provide insight
into their underlying dynamics. These methods were later modified for use with linear fractals. This paper reconsiders one such classication, called escape time, and presents a new algorithm for its computation that is significantly faster and conceptually simpler. Previous methods worked backwards, by mapping pixels into classified regions, whereas the
new forward algorithm uses an "escape buffer" to map classified regions onto pixels. The efficiency of the escape buer is justified by a careful analysis of its performance on linear fractals with various properties.