Ruled Laguerre minimal surfaces

Skopenkov, Mikhail; Pottmann, Helmut

doi:10.1007/s00209-011-0953-0

Cited by 6 publications

(4 citation statements)

References 30 publications

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“…As a by-product, the Ttransformation establishes an isometric correspondence between zero mean curvature spacelike surfaces in a (degenerate) isotropic 3-space and zero mean curvature spacelike surfaces in a time-oriented lightcone of R 4 1 . For a brief introduction to isotropic geometry we refer to [37,39]. 1 if M is locally isometric to some minimal surface in R 3 or to some maximal surface in R 3 1 .…”

Section: Laguerre Deformation and Lawson Correspondencementioning

confidence: 99%

“…One should consider that, already in the fundamental work of Blaschke and Thomsen [5], Möbius and Laguerre surface geometries were developed in parallel, as subgeometries of Lie sphere geometry. Another motivation is that several classical topics in Laguerre geometry, including Laguerre minimal surfaces and Laguerre isothermic surfaces, have recently received much attention in the theory of integrable systems, in discrete differential geometry, and in the applications to geometric computing and architectural geometry [6,7,8,25,26,27,28,31,36,37,39].…”

Section: Introductionmentioning

confidence: 99%

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Holomorphic differentials and Laguerre deformation of surfaces

Musso

Nicolodi

2016

Math. Z.

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A Laguerre geometric local characterization is given of L-minimal surfaces and Laguerre deformations (T-transforms) of L-minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential. This is used to prove that, via their Laguerre Gauss maps, the T-transforms of L-minimal isothermic surfaces have constant mean curvature in some translate of hyperbolic 3-space or de Sitter 3-space in Minkowski 4-space, or have mean curvature zero in some translate of a time-oriented lightcone in Minkowski 4-space. As an application, we show that various instances of the Lawson isometric correspondence can be viewed as special cases of the T-transformation of L-isothermic surfaces with holomorphic quartic differential.Comment: 25 page

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Section: Laguerre Deformation and Lawson Correspondencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Holomorphic differentials and Laguerre deformation of surfaces

Musso

Nicolodi

2016

Math. Z.

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“…(Sometimes this even leads to confusion: e.g., osculating circles of a generic curve are nested but all tangent to the curve; their envelope is the curve itself or empty depending on the choice of definition. In view of that notice that [38,Lemma 7] remains true for nested circles and should be applied in case (3) of the proof of Theorem 4 there. )…”

Section: Motivationmentioning

confidence: 99%

Characterizing envelopes of moving rotational cones and applications in CNC machining

Skopenkov

Bartoň

et al. 2020

Computer Aided Geometric Design

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Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a oneparametric family of congruent rotational cones. As limit cases, we also address ruled surfaces and their offsets. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces and ruled surfaces, respectively. The derivation includes results on local approximations of a surface by cones of revolution, which are expressed by contact order in the space of planes. To this purpose, the isotropic model of Laguerre geometry is used as there rotational cones correspond to curves (isotropic circles) and higher order contact is computed with respect to the image of the input surface in the isotropic model. Therefore, one studies curve-surface contact that is conceptually simpler than the surface-surface case. We show that, in a generic case, there exist at most six positions of a fixed rotational cone that have third order contact with the input surface. These results are themselves of interest in geometric computing, for example in cutter selection and positioning for flank CNC machining.

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“…Laguerre geometry is a classical sphere geometry that has its origins in the work of E. Laguerre in the mid 19th century and that had been extensively studied in the 1920s by Blaschke and Thomsen [6,7]. The study of surfaces in Laguerre geometry is currently still an active area of research [2,37,38,42,43,44,47,49,53,54,55] and several classical topics in Laguerre geometry, such as Laguerre minimal surfaces and Laguerre isothermic surfaces and their transformation theory, have recently received much attention in the theory of integrable systems [46,47,57,60,62], in discrete differential geometry, and in the applications to geometric computing and architectural geometry [8,9,10,58,59,61].…”

Section: Introductionmentioning

confidence: 99%

Surfaces in Laguerre geometry

Musso

Nicolodi

2019

Banach Center Publ.

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This exposition gives an introduction to the theory of surfaces in Laguerre geometry and surveys some results, mostly obtained by the authors, about three important classes of surfaces in Laguerre geometry, namely L-isothermic, L-minimal, and generalized L-minimal surfaces. The quadric model of Lie sphere geometry is adopted for Laguerre geometry and the method of moving frames is used throughout. As an example, the Cartan-Kähler theorem for exterior differential systems is applied to study the Cauchy problem for the Pfaffian differential system of L-minimal surfaces. This is an elaboration of the talks given by the authors at IMPAN, Warsaw, in September 2016. The objective was to illustrate, by the subject of Laguerre surface geometry, some of the topics presented in a series of lectures held at IMPAN by G. R. Jensen on Lie sphere geometry and by B. McKay on exterior differential systems.2000 Mathematics Subject Classification. 53A35, 53C42.

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Ruled Laguerre minimal surfaces

Cited by 6 publications

References 30 publications

Holomorphic differentials and Laguerre deformation of surfaces

Holomorphic differentials and Laguerre deformation of surfaces

Characterizing envelopes of moving rotational cones and applications in CNC machining

Surfaces in Laguerre geometry

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