Using work of Colmez, we give a quick algorithm for obtaining a clean fundamental domain for the action on R-+(3) of the totally positive units of a totally real cubic field. The fundamental domain consists of two infinite solid cones in R-3, one generated by 1. epsilon(1) and epsilon(1)epsilon(2), the other by 1, epsilon(2) and epsilon(1)epsilon(2). Here epsilon(1), epsilon(2) are certain fundamental totally positive units, included in R-+(3) by the usual geometric embedding, which we show to be easily computable from any set of fundamental units of k. Similar cones were found by Thomas and Vasquez in 1980, and by Halbritter and Pohst in 2000, but their methods did not result in practical algorithms. (C) 2012 Elsevier Inc. All rights reserved.