A novel method we refer to as the tie-line field (TLF) method has been developed to globally determine the tie lines of any three-component two-phase coexistence region by fitting electron-spin resonance (ESR) spectra obtained from compositions on the coexistence curve and within the coexistence region. The TLF method is illustrated by applying it to the liquid-ordered (L-o) and liquid-disordered (L-d) phase coexistence region of the lipid system brain-sphingomyelin/dioleoylphosphatidylcholine/cholesterol (SPM/DOPC/chol), for which an estimate of a tie-line was previously obtained by an earlier method also using ESR spectra. The essential aspect of the TLF method is the unique parametrization of the coexistence region called a "ruled surface". The use of the ruled surface enables one to guarantee that tie lines do not cross, as required by the phase rule, whereas previous methods lack this important constraint. It also makes simultaneous use of the full data set in determining the TLF and leads to a more efficient experimental design than previously used. The method is first tested out on synthetic data sets, then least-squares fitting of the ESR spectra with the parametrized model results in a tie-line field consistent with other known information on this lipid system. The best-fit tie-line field consists of the set of tie lines which are not exactly parallel; they exhibit a gradual change in slope with the largest slope within the coexistence region connecting the coexistence curve compositions with the highest and lowest cholesterol concentrations. The results are compared with those from more constrained methods of representing the tie-line fields as well as with the previous tie-line determination for the SPM/DOPC/chol system. An accurate determination of the tie-line field of phase coexistence regions in lipid systems is a necessary step in determining coexisting lipid compositions to serve as models of cell plasma membranes.