Since the development of 3D computational heat transfer in the early 1970s, construction of the lines of vector fields has been a fundamental visualisation technique. In addition to the usual velocity lines, the lines of transport vectors for mass, energy and entropy can be especially relevant to heat and mass transfer. The use of scalar stream functions and vector potentials to ensure that these lines satisfy flux conservation is discussed in this paper by tracing the history of their use for streamline construction. A flux conservative method that uses an energy vector potential for constructing energy transport lines is described and its use for constructing 3D heat lines is demonstrated for natural convection in a cavity. (C) 2009 Elsevier Ltd. All rights reserved.