Sturmian words are infinite words that have exactlyn+1 factors of length n for every positive integer n. A Sturmian word sα,p is also definedas a coding over a two-letter alphabet of the orbit of point ρ under the action of the irrational rotation Rα : x → x + α (mod 1).A substitution fixes a Sturmian word if and only if it is invertible.The main object of the present paper is to investigate Rauzy fractals associated with two-letter invertible substitutions. As an application, we give an alternative geometric proof of Yasutomi's characterizationof all pairs (α,p) such that sα,p is a fixedpoint of some non-trivial substitution.