In 1942 Freudenthal showed that a simplex in Euclidean space can be subdivided such that the quality (well-shapedness of the simplex, quantified in terms of e.g. fatness) of the simplices in the subdivision is lower bounded. This answered a question of Brouwer. Recently, Brunck discussed the same problem for simplices in two dimensional spaces of constant curvature and provided a closely related construction. In this note we exhibit the close relation between the two constructions and observe that, using radial projection, Freudenthal's construction and result immediately generalize to simplices in spaces of constant non-zero curvature of arbitrary dimension. This note focuses on displaying classical results that may be unfamiliar to people in the field that do not speak German, rather than building new theory.