Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classification of Dupin Cyclidic Cubes by Their Singularities

Jean Michel Menjanahary, Eriola Hoxhaj, Rimvydas Krasauskas

TL;DR

The paper addresses the complete Möbius classification of Dupin cyclidic (DC) systems—triply orthogonal coordinate systems in $\,\mathbb R^3$ with coordinate lines as circles or lines—via 3-linear quaternionic maps leading to DC cubes. It develops a framework based on quaternionic Bézier parametrizations, the Study quadric, and bicircular quartics to characterize singular loci, yielding four Möbius classes: spherical, offset, and two non-spherical families, A and B, distinguished by their symmetry and singularity structures. The authors construct canonical representatives using Miquel points, offset/axial constructions, and explicit control-point data, and they show how singular curves (e.g., focal bicircular quartics, ellipses, parabolas) determine the Möbius class. A key contribution is the notion of DC degrees and the demonstration that each class contains a finite, parameterized family of cubes up to Möbius equivalence, with implications for Dupin cyclide sections and classical orthogonal coordinate systems. Overall, the work unifies a rich interplay between quaternionic parametrizations, projective geometry, and Lie sphere geometry to classify all DC systems and their singularities in space, revealing deep connections to historic geometric frameworks and potential applications in geometric modeling.

Abstract

Triple orthogonal coordinate systems having coordinate lines as circles or straight lines are considered. Technically, they are represented by trilinear rational quaternionic maps and are called Dupin cyclidic cubes, naturally generalizing the bilinear rational quaternionic parametrizations of principal patches of Dupin cyclides. Dupin cyclidic cubes and their singularities are studied and classified up to Möbius equivalency in Euclidean space.

Classification of Dupin Cyclidic Cubes by Their Singularities

TL;DR

The paper addresses the complete Möbius classification of Dupin cyclidic (DC) systems—triply orthogonal coordinate systems in with coordinate lines as circles or lines—via 3-linear quaternionic maps leading to DC cubes. It develops a framework based on quaternionic Bézier parametrizations, the Study quadric, and bicircular quartics to characterize singular loci, yielding four Möbius classes: spherical, offset, and two non-spherical families, A and B, distinguished by their symmetry and singularity structures. The authors construct canonical representatives using Miquel points, offset/axial constructions, and explicit control-point data, and they show how singular curves (e.g., focal bicircular quartics, ellipses, parabolas) determine the Möbius class. A key contribution is the notion of DC degrees and the demonstration that each class contains a finite, parameterized family of cubes up to Möbius equivalence, with implications for Dupin cyclide sections and classical orthogonal coordinate systems. Overall, the work unifies a rich interplay between quaternionic parametrizations, projective geometry, and Lie sphere geometry to classify all DC systems and their singularities in space, revealing deep connections to historic geometric frameworks and potential applications in geometric modeling.

Abstract

Triple orthogonal coordinate systems having coordinate lines as circles or straight lines are considered. Technically, they are represented by trilinear rational quaternionic maps and are called Dupin cyclidic cubes, naturally generalizing the bilinear rational quaternionic parametrizations of principal patches of Dupin cyclides. Dupin cyclidic cubes and their singularities are studied and classified up to Möbius equivalency in Euclidean space.
Paper Structure (15 sections, 22 theorems, 66 equations, 14 figures, 1 table)

This paper contains 15 sections, 22 theorems, 66 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quaternions and Möbius transformations in $\mathbb R^3$
  4. The Study quadric and quaternionic--Bézier formulas
  5. Dupin cyclides: parametrizations and implicit equations
  6. Bicircular quartics
  7. Dupin cyclidic systems
  8. Spherical Dupin cyclidic systems
  9. Two dimensional DC systems
  10. Spherical DC systems
  11. Non-spherical offset Dupin cyclidic systems
  12. Case A: three real M-spheres of symmetry
  13. General non-spherical Dupin cyclidic systems
  14. Degrees of DC systems
  15. Conclusions

Key Result

Theorem 2.6

The implicit equation of a surface parametrized by bilinear DC patch with homogeneous control points $(u_i,w_i)$ is a factor of the $4 \times 4$ determinant where $[q]$ denotes the coordinate column of the quaternion $q$. The unique exception $f(x,y,z) \equiv 0$ happens only when the DC patch is on an M-sphere and all coordinate M-circles intersect at one point.

Figures (14)

  • Figure 1: Focal symmetric bicircular quartics are depicted in quarters on the first octant of the Euclidean space: 1-oval curves on the left and 2-oval curves on the right.
  • Figure 2: Four steps to build a general Dupin cyclidic cube.
  • Figure 3: Miquel point.
  • Figure 4: The 2 initial M-circles are in bold, followed by solid M-circles to fulfill the family. The dotted M-circles form the orthogonal family.
  • Figure 5: Construction of a 2-dimensional DC system using control points.
  • ...and 9 more figures

Theorems & Definitions (57)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • proof
  • Lemma 2.8
  • ...and 47 more