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Real Line Congruences of Trilinear Birational Maps

Bert Jüttler, Pablo Mazón, Josef Schicho

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.

Real Line Congruences of Trilinear Birational Maps

Abstract

Trilinear mappings appear naturally when performing spatial isogeometric discretizations of degree . 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.
Paper Structure (16 sections, 4 theorems, 43 equations, 4 figures)

This paper contains 16 sections, 4 theorems, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Duality
  4. Plücker coordinates of lines in 3D space
  5. The Klein quadric
  6. Real line congruences
  7. Parametric line congruences of trilinear maps
  8. Preliminary Results
  9. Real Focal Varieties
  10. Real Linear Congruences
  11. Classification of Real Line Congruences
  12. Real line congruences of trilinear birational maps of type $(1,1,1)$
  13. Real line congruences of trilinear birational maps of type $(1,1,2)$
  14. Real Line Congruences of Trilinear Birational Maps of Type $(1,2,2)$
  15. Real Line Congruences of Trilinear Birational Maps of Type $(2,2,2)$
  16. ...and 1 more sections

Key Result

Lemma 9

Let $X$ be a line congruence. Moreover, let $C\subset\mathbb{P}_\mathbb{C}^3$ be a curve such that $C\cap x\not= \emptyset$ for every $x\in X$. Then, one of the following holds: If additionally $X$ is real, then $\overline{V}$ is also a focal variety of $X$.

Figures (4)

  • Figure 1: Focal lines (thick lines) and line congruences (thin lines) for type $(1,1,1)$, class 1.
  • Figure 2: Focal curves (thick lines) and line congruences (thin lines) for type $(1,1,2)$, class 1.
  • Figure 3: Focal lines (thick lines) and line congruences (thin lines) for type $(1,2,2)$, class a1.
  • Figure 4: Focal curves (thick lines) and line congruences (thin lines) for type $(2,2,2)$, class 1.

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Remark 7
  • Definition 8
  • Lemma 9
  • proof
  • ...and 18 more