Trilinear mappings appear naturally when performing spatial isogeometric discretizations of degree p = 1. Among them, birational mappings are characterized by the property that both the mapping and the associated inverse mapping are rational and thus easy to evaluate. These mappings have recently been analyzed by Busé et al. [1]. Among other results, the authors provide a classification of these mappings over the field of complex numbers. The parameter lines of trilinear mappings form three two-parameter systems of straight lines, and thus it is promising to analyze these mappings with the tools provided by the field of line geometry, which is a classical branch of higher geometry, see [2] for a recent survey. Indeed, in the birational case, the three systems of lines form space-filling line congruences associated with rational mappings that can be used to parameterize certain algebraic surfaces [3]. Moreover, the three systems are closely related, and based on these observations we will present a geometric discussion of the results of Busé et al. [1] together with a more detailed analysis of the classification over the field of real numbers. Joint work with Pablo González-Mazón and Josef Schicho. REFERENCES [1] Busé, L., González-Mazón, P., and Schicho, J.: Tri-linear birational maps in dimension three. Mathematics of Computation, 92 (2023), 1837-1866 [2] Pottmann, H., and Wallner, J.: Computational Line Geometry, Springer-Verlag Berlin Heidelberg, 2001. [3] Jüttler, B., and Rittenschober, K.: Using line congruences for parameterizing special alge- braic surfaces. In Mathematics of Surfaces: 10th IMA International Conference, Springer-Verlag Berlin Heidelberg, 2003, 223-243.