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Minimal Surfaces via Complex Quaternions

Amedeo Altavilla, Hans-Peter Schröcker, Zbyněk Šír, Jan Vršek

TL;DR

The paper introduces a novel complex-quaternionic framework for minimal surfaces, encoding isothermal minimal data through Φ = χ L χ^{-1} with a fixed null quaternion L and leveraging the Sylvester equation to characterize conjugacy classes. It generalizes the classical Weierstraß–Enneper representation by using complex quaternions to generate polynomial and rational minimal surfaces via Φ = λ A L A^c, linking to Pythagorean Hodograph curves and their algebraic parametrizations. The work provides a unified algebraic approach that recovers and extends PH-based constructions, enabling explicit Enneper patches and catenoid examples, and offers practical interpolation methods for Enneper patches. These developments pave the way for computational geometry and geometric modeling applications, including CAD/graphics, by bridging abstract quaternionic analysis with concrete minimal-surface patches.

Abstract

Minimal surfaces play a fundamental role in differential geometry, with applications spanning physics, material science, and geometric design. In this paper, we explore a novel quaternionic representation of minimal surfaces, drawing an analogy with the well-established theory of Pythagorean Hodograph (PH) curves. By exploiting the algebraic structure of complex quaternions, we introduce a new approach to generating minimal surfaces via quaternionic transformations. This method extends classical Weierstraß-Enneper representations and provides insights into the interplay between quaternionic analysis, PH curves, and minimal surface geometry. Additionally, we discuss the role of the Sylvester equation in this framework and demonstrate practical examples, including the construction of Enneper surface patches. The findings open new avenues in computational geometry and geometric modeling, bridging abstract algebraic structures with practical applications in CAD and computer graphics.

Minimal Surfaces via Complex Quaternions

TL;DR

The paper introduces a novel complex-quaternionic framework for minimal surfaces, encoding isothermal minimal data through Φ = χ L χ^{-1} with a fixed null quaternion L and leveraging the Sylvester equation to characterize conjugacy classes. It generalizes the classical Weierstraß–Enneper representation by using complex quaternions to generate polynomial and rational minimal surfaces via Φ = λ A L A^c, linking to Pythagorean Hodograph curves and their algebraic parametrizations. The work provides a unified algebraic approach that recovers and extends PH-based constructions, enabling explicit Enneper patches and catenoid examples, and offers practical interpolation methods for Enneper patches. These developments pave the way for computational geometry and geometric modeling applications, including CAD/graphics, by bridging abstract quaternionic analysis with concrete minimal-surface patches.

Abstract

Minimal surfaces play a fundamental role in differential geometry, with applications spanning physics, material science, and geometric design. In this paper, we explore a novel quaternionic representation of minimal surfaces, drawing an analogy with the well-established theory of Pythagorean Hodograph (PH) curves. By exploiting the algebraic structure of complex quaternions, we introduce a new approach to generating minimal surfaces via quaternionic transformations. This method extends classical Weierstraß-Enneper representations and provides insights into the interplay between quaternionic analysis, PH curves, and minimal surface geometry. Additionally, we discuss the role of the Sylvester equation in this framework and demonstrate practical examples, including the construction of Enneper surface patches. The findings open new avenues in computational geometry and geometric modeling, bridging abstract algebraic structures with practical applications in CAD and computer graphics.

Paper Structure

This paper contains 8 sections, 8 theorems, 88 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Toolbox
  3. Minimal Surfaces and the Weierstraß-Enneper Representation
  4. Quaternions and Complex Quaternions
  5. PH Curves and Minimal Surfaces
  6. The Sylvester Equation
  7. Minimal Surfaces
  8. Enneper Patches

Key Result

Theorem 2.2

All spatial polynomial (resp. rational) PH curves $\gamma(t)$ are obtained by integration of where $A(t)\in \mathbb{H}[t]$ is a quaternion valued polynomial and $\lambda(t)$ is a polynomial (resp. rational) function. In the case of rational $\lambda(t)$ all the residues of eq:LAiA must be zero.

Figures (4)

  • Figure 1: Catenoid in standard and in isothermal parametrization.
  • Figure 2: A rational minimal surface.
  • Figure 3: Enneper's surface in isothermal parametrization.
  • Figure 4: Enneper surface patch to Example \ref{['ex:1']}.

Theorems & Definitions (26)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Example 3.2
  • Proposition 3.3: Proposition 5.3 altavillaLAA
  • Remark 3.4
  • Corollary 4.1
  • Example 4.2: Catenoid
  • ...and 16 more