Abstract: New insights into the classic Rackwitz-Fiessler random space transformation method (R-F method) are presented in this study. Following a general description of the R-F method, the geometrical relationships between the R-F method and the isoprobabilistc transformation are discussed. The forward transformation of the R-F method is then proved to follow the isoprobabilistic transformation principle, whereas its backward transformation can only be taken as a linear approximation of the isoprobabilistic transformation. Subsequently, an equivalent R-F condition is proposed, which helps to investigate the dependency variation of random variables in the R-F method. The dependence variation of the R-F method is later shown to be the same as that of the Nataf-Pearson method (N-P method). Finally, the calculation cost is compared between the dependence changed R-F method and the N-P method, finding the costs to be almost the same for each single-step iteration when optimized algorithms are used. Additionally, the differences between the dependence changed R-F method and the linearized N-P method are also discussed. Numerical examples show that the reliability index calculated using the dependence changed R-F method agrees well with that of the N-P method.