I3809-N32: Geometric shape generation

Joint Project between Austria (FWF) and Japan (JSPS)

Abstract. Explicit classification results and representation formulae are at the core of the differential geometry of curves and surfaces - they serve to generate geometric shapes (curves or surfaces) with certain prescribed properties: for example, the classical Weierstrass representation formulae serve to generate any surface that (locally) minimizes area out of simple data. Other shape generation methods include "transformations", which transform a given shape of a certain class into another such shape, while preserving its key properties.

While such "shape generation methods" are designed to produce curves or surfaces of a particular kind out of suitable input data, it is often difficult to control other features of the generated shape by the input data - deep knowledge about the particular shapes and the generation process are required.

These shape generation methods play an important role in geometry, not just for the production of interesting shapes for design or ilustration purposes, but also to obtain a better understanding of the structure of the investigated shapes. In particular, the properties of transformations are essential for describing facetted or panelled surfaces that display similar properties as the corresponding smooth surfaces.

In this project we aim to investigate different methods to generate shapes, in particular:

• the interrelations between different shape generation methods;
• the related discretizations and, hence, discretizations of the shape generation methods;
• the applicability and scope of these shape generation methods in theory and generative art and design.

By interlinking these different aspects of shape generation we hope and expect to gain new insight and to establish new interesting methods for the geometric generation of shapes, for their use in theory as well as for their application in art or design.

Publications.

J Cho, Y Ogata (2019)

Simple factor dressings and Bianchi-Bäcklund transformations; Illinois J Math 63(4), 619-631 (2019) DOI:10.1215/00192082-7988989

Abstract

In this paper, we directly show the known equivalence of simple factor dressings of extended frames and the classical Bianchi-Bäcklund transformations for constant mean curvature surfaces. In doing so, we show how the parameters of classical Bianchi-Bäcklund transformations can be incorporated into the simple factor dressings method.

S Akamine, J Cho, Y Ogata (2020)

Analysis of timelike Thomsen surfaces; J Geom Anal 30, 731-761 (2020) DOI:10.1007/s12220-019-00166-7, EPrint arXiv:1808.09641

Abstract

Timelike Thomsen surfaces are timelike minimal surfaces that are also affine minimal. In this paper, we make use of both the Lorentz conformal coordinates and the null coordinates, and their respective representation theorems of timelike minimal surfaces, to obtain a complete global classification of these surfaces and to characterize them using a geometric invariant called lightlike curvatures. As a result, we reveal the relationship between timelike Thomsen surfaces, and timelike minimal surfaces with planar curvature lines. As an application, we give a deformation of null curves preserving the pseudo-arclength parametrization and the constancy of the lightlike curvatures.

S Akamine, M Umehara, K Yamada (2020)

Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality; Proc Amer Math Soc, Ser B 7, 17-27 (2020) DOI:10.1090/bproc/44

Abstract

Calabi's Bernstein-type theorem asserts that a zero mean curvature entire graph in Lorentz-Minkowski space $L^3$ which admits only space-like points is a space-like plane. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space $E^3$ and maximal surfaces in Lorentz-Minkowski space $L^3$, we give an improvement of this Bernstein-type theorem. More precisely, we show that a zero mean curvature entire graph in $L^3$ which does not admit time-like points (namely, a graph consists of only space-like and light-like points) is a plane.

S Akamine, A Honda, M Umehara, K Yamada (2021)

Bernstein-type theorem for zero mean curvature hypersurfaces without time-like points in Lorentz-Minkowski space; Bull Braz Math Soc, New Series 52, 175-181 (2021) DOI:10.1007/s00574-020-00196-8, EPrint arXiv:1907.01754

Abstract

Calabi and Cheng-Yau's Bernstein-type theorem asserts that an entire zero mean curvature graph in Lorentz-Minkowski $(n+1)$-space $\mathbb{R}^{n+1}_1$ which admits only space-like points is a hyperplane. Recently, the third and fourth authors proved a line theorem for hypersurfaces at their degenerate light-like points. Using this, we give an improvement of the Bernstein-type theorem, and we show that an entire zero mean curvature graph in $\mathbb{R}^{n+1}_1$ consisting only of space-like or light-like points is a hyperplane. This is a generalization of the first, third and fourth authors' previous result for $n=2$.

S Akamine, H Fujino (2021)

Reflection principle for lightlike line segments on maximal surfaces. Ann Glob Anal Geom 59, 93-108 (2021) DOI:10.1007/s10455-020-09743-4

Abstract

As in the case of minimal surfaces in the Euclidean 3-space, the reflection principle for maximal surfaces in the Lorentz-Minkowski 3-space asserts that if a maximal surface has a spacelike line segment $L$, the surface is invariant under the $180^\circ$-rotation with respect to $L$. However, such a reflection property does not hold for lightlike line segments on the boundaries of maximal surfaces in general. In this paper, we show some kind of reflection principle for lightlike line segments on the boundaries of maximal surfaces when lightlike line segments are connecting shrinking singularities. As an application, we construct various examples of periodic maximal surfaces with lightlike lines from tessellations of $\mathbb{R}^2$.

J Cho, W Rossman (2020)

Discrete isothermicity in Moebius subgeometries; in Yang (ed): An introduction to Discrete Differential Geometry, Anam Lecture Notes 2, 37-81 (2020)

Abstract

We give an elementary description of Möbius geometry using a Minkowski space model, primarily in low dimensions with comments about generalizing to higher dimensions. We then give an application to the discretization of isothermic surfaces in three dimensional spaceforms.

J Cho, W Rossman, T Seno (2021)

Discrete mKdV equation via Darboux transformation; Math Phys Anal Geom 24, 25 (2021) DOI:10.1007/s11040-021-09398-y

Abstract

We introduce an efficient route to obtaining the discrete potential mKdV equation emerging from a particular discrete motion of discrete planar curves.

J Cho, W Rossman, S-D Yang (2021)

Discrete Minimal Nets with Symmetries; in: Hoffmann, Kilian, Leschke, Martin (eds), Minimal Surfaces: Integrable Systems and Visualisation, m:iv 2017-19; Springer Proceedings in Mathematics & Statistics 349. DOI:10.1007/978-3-030-68541-6_3, EPrint arXiv:2105.08102

Abstract

In this paper, we extend the notion of Schwarz reflection principle for smooth minimal surfaces to the discrete analogues for minimal surfaces, and use it to create global examples of discrete minimal nets with high degree of symmetry.

J Cho, K Leschke, Y Ogata (2022)

Generalised Bianchi permutability for isothermic surfaces; Ann Glob Anal Geom 61, 799-829 (2022) DOI:10.1007/s10455-022-09833-5

Abstract

Isothermic surfaces are surfaces which allow a conformal curvature line parametrisation. They form an integrable system, and Darboux transforms of isothermic surfaces obey Bianchi permutability: for two distinct spectral parameters, the corresponding Darboux transforms have a common Darboux transform which can be computed algebraically. In this paper, we discuss two-step Darboux transforms with the same spectral parameter, and show that these are obtained by a Sym-type construction: All two-step Darboux transforms of an isothermic surface are given, without further integration, by parallel sections of the associated family of the isothermic surface, either algebraically or by differentiation against the spectral parameter.

J Cho, W Rossman, T Seno (2022)

Infinitesimal Darboux transformation and semi-discrete MKDV equation; Nonlinearity 35(4), 2134 (2022) DOI:10.1088/1361-6544/ac591f

Abstract

We connect certain continuous motions of discrete planar curves resulting in semi-discrete potential Korteweg-de Vries (mKdV) equation with Darboux transformations of smooth planar curves. In doing so, we define infinitesimal Darboux transformations that include the aforementioned motions, and also give an alternate geometric interpretation for establishing the semi-discrete potential mKdV equation.

J Dubois, U Hertrich-Jeromin, G Szewieczek (2022)

Notes on flat fronts in hyperbolic space; J Geom 113, 20 (2022) DOI:10.1007/s00022-022-00628-4

Abstract

We give a short introduction to discrete flat fronts in hyperbolic space and prove that any discrete flat front in the mixed area sense admits a Weierstrass representation.

U Hertrich-Jeromin, G Szewieczek (2022)

Discrete cyclic systems and circle congruences; Annali di Matematica 201, 2797-2824 (2022) DOI:10.1007/s10231-022-01219-5

Abstract

We discuss integrable discretizations of 3-dimensional cyclic systems, that is, orthogonal coordinate systems with one family of circular coordinate lines. In particular, the underlying circle congruences are investigated in detail and characterized by the existence of a certain flat connection. Within the developed framework, discrete cyclic systems with a family of discrete flat fronts in hyperbolic space and discrete cyclic systems, where all coordinate surfaces are discrete Dupin cyclides, are investigated.

D Polly (2022 PhD thesis)

Channel linear Weingarten surfaces in smooth and discrete differential geometry; PhD thesis, TU Wien (2022)

Abstract

We consider channel linear Weingarten surfaces in spaceforms of arbitrary curvature and their discretisation in the sense of integrable discrete dierential geometry. We use Lie sphere geometry to unify the dierent ambient spaces.

In the smooth category, we prove that all non-tubular channel linear Weingarten surfaces are rotational within their ambient spaceforms. Further we give explicit parametrisations of all rotational surfaces with constant Gauss curvature in terms of Jacobi elliptic functions. Thus, we obtain a transparent classication of all channel linear Weingarten surfaces in spaceforms with non-negative curvature as well as a large subclass of channel linear Weingarten surfaces in hyperbolic spaceforms up to parallel transformation. We further point out how similar arguments to the ones presented in this text lead to explicit parametrisations of constant mean curvature surfaces.

In the discrete category, we advance the theory of the recently dened class of discrete channel surfaces [U Hertrich-Jeromin, W Rossman, G Szewieczek: Discrete channel surfaces; Math Z 294, 747-767 (2020)]. Further, we prove that all strongly non-tubular discrete channel linear Weingarten surfaces are rotational within their ambient spaceforms. Also we outline how this will lead to a classication of large classes of channel linear Weingarten surfaces, similar to the results of the smooth theory.

F Burstall, J Cho, U Hertrich-Jeromin, M Pember, W Rossman (2023)

Discrete $\Omega$-nets and Guichard nets via discrete Koenigs nets; Proc London Math Soc 126(2), 790-836 (2023) DOI:10.1112/plms.12499

Abstract

We provide a convincing discretisation of Demoulin's $\Omega$-surfaces along with their specialisations to Guichard and isothermic surfaces with no loss of integrable structure.

J Cho, M Pember, G Szewieczek (2023)

Constrained elastic curves and surfaces with spherical curvature lines; Indiana Univ Math J 72(5), 2059-2099 (2023) DOI:10.1512/iumj.2023.72.9487, EPrint arXiv:2104.11058

Abstract

In this paper, we consider surfaces with one or two families of spherical curvature lines. We show that every surface with a family of spherical curvature lines can locally be generated by a pair of initial data: a suitable curve of Lie sphere transformations and a spherical Legendre curve. We the provide conditions on the initial data for which such a surface is Lie applicable, an integrable class of of surfaces that includes cmc and pseudospherical surfaces. In particular, we show that a Lie applicable surface with exactly one family of spherical curvature lines must be generated by the lift of a constrained elastic curve in some space form. In view of this goal, we give a Lie sphere geometric characterisation of constrained elastic curves via polynomial conserved quantities of a certain family of connections.

U Hertrich-Jeromin, M Pember, D Polly (2023)

Channel linear Weingarten surfaces in space forms; Beitr Algebra Geom 64, 969-1009 (2023) DOI:10.1007/s13366-022-00664-w, EPrint arXiv:2105.00702

Abstract

Channel linear Weingarten surfaces in space forms are investigated in a Lie sphere geometric setting, which allows for a uniform treatment of different ambient geometries. We show that any channel linear Weingarten surface in a space form is isothermic and, in particular, a surface of revolution in its ambient space form. We obtain explicit parametrisations for channel surfaces of constant Gauss curvature in space forms, and thereby for a large class of linear Weingarten surfaces up to parallel transformation.

M Pember, D Polly, M Yasumoto (2023)

Discrete Weierstrass-Type Representations; Discrete Comput Geom 70, 816-844 (2023) DOI:10.1007/s00454-022-00439-z, EPrint arXiv:2105.06774

Abstract

Discrete Weierstrass-type representations yield a construction method in discrete differential geometry for certain classes of discrete surfaces. We show that the known discrete Weierstrass-type representations of certain surface classes can be viewed as applications of the $\Omega$-dual transform to lightlike Gauss maps in Laguerre geometry. From this construction, further Weierstrass-type representations arise. As an application of the techniques we develop, we show that all discrete linear Weingarten surfaces of Bryant or Bianchi type locally arise via Weierstrass-type representations from discrete holomorphic maps.

S Akamine, H Fujino (2024)

Duality of boundary value problems for minimal and maximal surfaces; Commun Anal Geom 32(4), 1057-1094 (2024) DOI:10.4310/CAG.241015230035, EPrint arXiv:1909.00975

Abstract

In 1966, Jenkins and Serrin gave existence and uniqueness results for infinite boundary value problems of minimal surfaces in the Euclidean space, and after that such solutions have been studied by using the univalent harmonic mapping theory. In this paper, we show that there exists a one-to-one correspondence between solutions of infinite boundary value problems for minimal surfaces and those of lightlike line boundary value problems for maximal surfaces in the Lorentz-Minkowski spacetime. We also investigate some symmetry relations associated with the above correspondence together with their conjugations, and observe function theoretical aspects of the geometry of these surfaces. Finally, a reflection property along lightlike line segments on boundaries of maximal surfaces is discussed.