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Isometric Surfaces in Isotropic 3-Space

Christian Müller, Helmut Pottmann

TL;DR

This work defines a rigorous notion of isometric surfaces in isotropic 3-space $I^3$ by pairing top-view congruence with an isotropic Gauss curvature condition, yielding nontrivial isotropic deformations. It develops both the continuous and discrete theories, introducing metric dualities, contact-element formalisms, and Minkowski-sum duals, and then demonstrates isotropic analogues of key Euclidean results such as the associated family of minimal surfaces, Bour’s theorem, and Minding isometries. A central construct is the isotropic Darboux wreath, a six-surface network of interrelated infinitesimally flexible surfaces connected through velocity, rotation, translation diagrams and metric duality, with special cases including isotropic linear Weingarten and Voss nets. The study also links isotropic geometry to statics via Airy stress functions, providing a static-mechanical interpretation of the diagrams and suggesting practical pathways for designing flexible isotropic nets and transferring insights to Euclidean meshes through optimization. Overall, the paper establishes a comprehensive framework for isotropic isometries, their infinitesimal and discrete realizations, and their applications to mechanics and geometric design.

Abstract

While the notion of isometric deformations of surfaces is straightforward for surfaces with Euclidean metric, a corresponding notion in isotropic space has been missing. By making Gauss' Theorema Egregium a necessary condition we develop a sensible notion of isometric surfaces in isotropic space. The well-known examples in Euclidean space, like isometries within the associated family of minimal surfaces, Bour's theorem, and Minding isometries, find their natural analogues in isotropic space. We also include an extensive treatment of infinitesimal flexibility, or infinitesimal deformation, of surfaces. We prove results for the isotropic displacement diagrams in analogy to its well-known counterparts in Euclidean space culminating in the existence of an isotropic Darboux wreath consisting of six surfaces. We show several interesting relations for special parametrizations involving Koenigs and Voss nets of smooth and discrete surfaces within the Darboux wreath and we encounter surfaces of constant Gaussian and mean curvature. At several occasions, we point to connections to statics as the isotropic space is a natural language to describe the Airy stress function.

Isometric Surfaces in Isotropic 3-Space

TL;DR

This work defines a rigorous notion of isometric surfaces in isotropic 3-space by pairing top-view congruence with an isotropic Gauss curvature condition, yielding nontrivial isotropic deformations. It develops both the continuous and discrete theories, introducing metric dualities, contact-element formalisms, and Minkowski-sum duals, and then demonstrates isotropic analogues of key Euclidean results such as the associated family of minimal surfaces, Bour’s theorem, and Minding isometries. A central construct is the isotropic Darboux wreath, a six-surface network of interrelated infinitesimally flexible surfaces connected through velocity, rotation, translation diagrams and metric duality, with special cases including isotropic linear Weingarten and Voss nets. The study also links isotropic geometry to statics via Airy stress functions, providing a static-mechanical interpretation of the diagrams and suggesting practical pathways for designing flexible isotropic nets and transferring insights to Euclidean meshes through optimization. Overall, the paper establishes a comprehensive framework for isotropic isometries, their infinitesimal and discrete realizations, and their applications to mechanics and geometric design.

Abstract

While the notion of isometric deformations of surfaces is straightforward for surfaces with Euclidean metric, a corresponding notion in isotropic space has been missing. By making Gauss' Theorema Egregium a necessary condition we develop a sensible notion of isometric surfaces in isotropic space. The well-known examples in Euclidean space, like isometries within the associated family of minimal surfaces, Bour's theorem, and Minding isometries, find their natural analogues in isotropic space. We also include an extensive treatment of infinitesimal flexibility, or infinitesimal deformation, of surfaces. We prove results for the isotropic displacement diagrams in analogy to its well-known counterparts in Euclidean space culminating in the existence of an isotropic Darboux wreath consisting of six surfaces. We show several interesting relations for special parametrizations involving Koenigs and Voss nets of smooth and discrete surfaces within the Darboux wreath and we encounter surfaces of constant Gaussian and mean curvature. At several occasions, we point to connections to statics as the isotropic space is a natural language to describe the Airy stress function.

Paper Structure

This paper contains 49 sections, 39 theorems, 104 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions and overview
  4. Basic Concepts in Isotropic Geometry
  5. The isotropic space $I^3$
  6. Metric duality and parallel points
  7. Contact elements
  8. Graph representation and its dual
  9. Contact elements of a surface
  10. Minkowski sum and its dual
  11. Curvatures of surfaces in $I^3$
  12. Change of curvatures under metric duality
  13. Curvatures in terms of the support function
  14. Curvatures of Minkowski sums and their duals
  15. Gaussian curvature and isotropic area of duals
  16. ...and 34 more sections

Key Result

Lemma 4

The metric dualities $\delta, \nu$ map two points $p_1, p_2$ with isotropic distance $l$ to two planes with intersection angle $l$ and vice versa:

Figures (13)

  • Figure 1: (a)Metric isometry of surfaces is not a sensible notion in isotropic geometry. As distances are measured in the top view any two surfaces are isometric to each other in just the metrical sense if their top view is congruent. (b-c) The isotropic unit sphere $S$ (of parabolic type) is a Euclidean paraboloid (b). The Gauss image $\sigma(g)$ of a surface $g$(c) to $S$ is obtained by the correspondence of parallel tangent planes. The ratio of the areas of the top views of the red domains around a point $p$ converges to the isotropic Gaussian curvature at $p$ as the diameter of the domain goes to zero.
  • Figure 2: Left: Illustration of the isotropic support function $h(p, q)$ which measures the distance on the $z$-axis between the origin and the intersection of the tangent plane at $(p, q)$ with the $z$-axis. Right: A surface $F(u, v) = (u, v, \frac{1}{2} (u^2 - v^3))$ and its metric dual $\delta(F)$ parametrized by $(u, -\frac{3}{2} v^2, \frac{1}{2} (u^2 - 2 v^3))$. Note that the Gaussian image of $F$ is not regular resulting in a cuspidal edge at $\delta(F)$.
  • Figure 3: Three types of Minkowski sums illustrated by curves. Left:Euclidean Minkowski sum $F_1 + F_2$ of two curves $F_1$ (red) and $F_2$ (green) obtained by adding points with parallel tangent planes and parallel normal vectors. Center:Isotropic Minkowski sum $F_1 + F_2$ of two curves $F_1$ (red) and $F_2$ (green). The support function of the Minkowski sum is the sum of the support functions of $F_1$ and $F_2$. Right: The sum of functions is the dual of the Minkowski sum in the setting of point-parallelism.
  • Figure 4: Associated family of isotropic minimal surfaces. For the sake of "paper-space-economy", the isotropic direction in this image is horizontal instead of the usual vertical direction.
  • Figure 5: A rotational surface (left) together with two isotropic isometric helical surface. Note that the meridian curve of the helical surface on the right is strict monotonically increasing (since $\varepsilon_2(v) = 1$). The meridian curve of the helical surface in the center is not strict monotonically increasing (since $\varepsilon_1(v) \neq \text{const}$).
  • ...and 8 more figures

Theorems & Definitions (90)

  • Definition 1
  • Definition 2
  • Example 3
  • Lemma 4
  • Definition 5
  • Definition 6
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 80 more