Real Line Congruences of Trilinear Birational Maps

TL;DR

This paper classifies real line congruences derived from trilinear birational maps, which are important in spatial isogeometric discretizations, using tools from line geometry to understand their structure and relationships.

Contribution

It provides a classification over the real numbers of the parametric line congruences associated with trilinear birational maps, extending previous complex classifications.

Findings

Classification of real line congruences from trilinear birational maps

Identification of three related line families forming space-filling congruences

Insights into the geometric structure of these mappings

Abstract

Trilinear mappings appear naturally when performing spatial isogeometric discretizations of degree $p = 1$. Among them, birational maps are characterized by the property that both the mapping and the associated inverse map are rational and thus easy to evaluate. These mappings have recently been analyzed, and a classification over the field of complex numbers has been obtained. The parameter lines of trilinear mappings form three two-parameter families 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. Indeed, in the birational case, the three families of lines form space-filling line congruences associated with rational mappings that can be used to parameterize certain algebraic surfaces. Moreover, the three systems are closely related. In this paper, we present a classification, over the field of real numbers, of the parametric line congruences arising from trilinear birational maps.