The purpose of this note is to give a short proof of the theorem of Ralph in Linear Algebra Appl. 178 (1993), 249–260, on the geometry of projection maps. This theorem implies that the so-called normal maps derived from projections satisfy a general geometric criterion for bijectivity of piecewise affine maps given by Kuhn and Lowen in Linear Algebra Appl. 96 (Section 5.3), (1987), 109–129. As a consequence, one obtains Robinson's characterization of bijectivity of normal maps.