Abstract
Laguerre minimal (L-minimal) surfaces are the minimizers of the energy \(\int (H^2-K)/K d\!A\). They are a Laguerre geometric counterpart of Willmore surfaces, the minimizers of \(\int (H^2-K)d\!A\), which are known to be an entity of Möbius sphere geometry. The present paper provides a new and simple approach to L-minimal surfaces by showing that they appear as graphs of biharmonic functions in the isotropic model of Laguerre geometry. Therefore, L-minimal surfaces are equivalent to Airy stress surfaces of linear elasticity. In particular, there is a close relation between L-minimal surfaces of the spherical type, isotropic minimal surfaces (graphs of harmonic functions), and Euclidean minimal surfaces. This relation exhibits connections to geometrical optics. In this paper we also address and illustrate the computation of L-minimal surfaces via thin plate splines and numerical solutions of biharmonic equations. Finally, metric duality in isotropic space is used to derive an isotropic counterpart to L-minimal surfaces and certain Lie transforms of L-minimal surfaces in Euclidean space. The latter surfaces possess an optical interpretation as anticaustics of graph surfaces of biharmonic functions.
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Bobenko, A., Hoffmann, T., Springborn, B.A.: Minimal surfaces from circle patterns: geometry from combinatorics. Ann. Math. 164, 231–264 (2006)
Blaschke, W.: Über die geometrie von Laguerre II: Flächentheorie in Ebenenkoordinaten. Abh. Math. Sem. Univ. Hamburg 3, 195–212 (1924)
Blaschke, W.: Über die geometrie von Laguerre III: Beiträge zur Flächentheorie. Abh. Math. Sem. Univ. Hamburg 4, 1–12 (1925)
Blaschke, W.: Vorlesungen Über Differentialgeometrie, vol. 3, Springer (1929)
Bobenko, A., Pinkall, U.: Discrete isothermic surfaces. J. Reine Angew. Math. 475, 187–208 (1996)
Brell-Cokcan, S., Pottmann, H.: Tragstruktur für Freiformflächen in Bauwerken. Patent No. A1049/2006 (2006)
Bobenko, A., Schröder, P.: Discrete Willmore flow. In: Symp. Geometry Processing, pp. 101–110. Eurographics (2005)
Bobenko, A., Suris, Yu.: On organizing principles of discrete differential geometry, geometry of spheres. Russian Math. Surveys 62(1), 1–43 (2007)
Cecil, T.: Lie Sphere Geometry. Springer (1992)
Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland (1978)
Cutler, B., Whiting, E.: Constrained planar remeshing for architecture. In: Symp. Geom. Processing (2006, poster)
Duchon, J.: Splines minimizing rotation-invariant semi-norms in sobolev spaces. In: Schempp, W., and Zeller, K. (eds.) Multivariate Approximation Theory, pp. 85–100. Birkhäuser (1977)
Farouki, R.T., Hass, J.: Evaluating the boundary and covering degree of planar minkowski sums and other geometrical convolutions. J. Comput. Appl. Math. 209, 246–266 (2007)
Glymph, J., et al.: A parametric strategy for freeform glass structures using quadrilateral planar facets. In: Acadia 2002, pp. 303–321. ACM (2002)
Kommerell, K.: Strahlensysteme und Minimalflächen. Math. Ann. 70, 143–160 (1911)
König, K.: L-Minimalflächen. Mitt. Math. Ges. Hamburg, 189–203 (1926)
König, K.: L-Minimalflächen II. Mitt. Math. Ges. Hamburg, 378–382 (1928)
Koenderink, I.J., van Doorn, A.J.: Image processing done right. In: ECCV, pp. 158–172 (2002)
Liu, Y., Pottmann, H., Wallner, J., Yang, Y.-L., Wang, W.: Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graphics 25(3), 681–689 (2006)
Musso, E., Nicolodi, L.: L-minimal canal surfaces. Rend. Mat. Appl. 15, 421–445 (1995)
Musso, E., Nicolodi, L.: A variational problem for surfaces in Laguerre geometry. Trans. AMS 348, 4321–4337 (1996)
Palmer, B.: Remarks on a variational problem in Laguerre geometry. Rend. Mat. Appl. 19, 281–293 (1999)
Pottmann, H., Brell-Cokcan, S., Wallner, J.: Discrete surfaces for architectural design. In: Lyche, T., Merrien, J.L., Schumaker, L.L. (eds.) Curves and Surfaces: Avignon 2006, pp. 213–234. Nashboro Press (2007)
Peternell, M.: Developable surface fitting to point clouds. Comput. Aided Geom. Design 21(8), 785–803 (2004)
Pottmann, H., Leopoldseder, S.: Geometries for CAGD. In: Farin, G., Hoschek, J., Kim, M.-S. (eds.) Handbook of 3D Modeling, pp. 43–73. Elsevier (2002)
Pottmann, H., Liu, Y.: Discrete surfaces of isotropic geometry with applications in architecture. In: Martin, R., Sabin, M., Winkler, J. (eds.) The Mathematics of Surfaces, pp. 341–363. Lecture Notes in Computer Science 4647. Springer (2007)
Pottmann, H., Liu, Y., Wallner, J., Bobenko, A., Wang, W.: Geometry of multi-layer freeform structures for architecture. ACM Trans. Graphics 26(3), 1–11 (2007)
Pottmann, H., Opitz, K.: Curvature analysis and visualization for functions defined on Euclidean spaces or surfaces. Comput. Aided Geom. Design 11, 655–674 (1994)
Peternell, M., Pottmann, H.: A Laguerre geometric approach to rational offsets. Comput. Aided Geom. Design 15, 223–249 (1998)
Pottmann, H., Peternell, M.: Applications of Laguerre geometry in CAGD. Comput. Aided Geom. Design 15, 165–186 (1998)
Peternell, M., Steiner, T.: Minkowski sum boundary surfaces of 3d objects. Graph. Models (2007)
Pottmann, H., Wallner, J.: The focal geometry of circular and conical meshes. Adv. Comput. Math (2008, to appear). doi:10.1007/s10444-007-9045-4
Sachs, H.: Isotrope Geometrie des Raumes. Vieweg (1990)
Schober, H.: Freeform glass structures. In: Glass processing days 2003. Glass Processing Days, Tampere (Fin.), pp. 46–50 (2003)
Sampoli, M.L., Peternell, M., Jüttler, B.: Rational surfaces with linear normals and their convolutions with rational surfaces. Comput. Aided Geom. Design 23, 179–192 (2006)
Strubecker, K.: Differentialgeometrie des isotropen Raumes I: Theorie der Raumkurven. Sitzungsber. Akad. Wiss. Wien, Abt. IIa 150, 1–53 (1941)
Strubecker, K.: Differentialgeometrie des isotropen Raumes II: Die Flächen konstanter Relativkrümmung K=rt–s 2. Math. Z. 47, 743–777 (1942)
Strubecker, K.: Differentialgeometrie des isotropen Raumes III: Flächentheorie. Math. Z. 48, 369–427 (1942)
Strubecker, K.: Airy’sche Spannungsfunktion und isotrope Differentialgeometrie. Math. Z. 78, 189–198 (1962)
Strubecker, K.: Über die isotropen Gegenstücke der Minimalfläche von Scherk. J. Reine Angew. Math. 293/294, 22–51 (1977)
Strubecker, K.: Duale Minimalflächen des isotropen Raumes. Rad JAZU 382, 91–107 (1978)
Strubecker, K.: Über das isotrope Gegenstück \(z={3\over2}{J}(x+iy)^{2/3}\) der Minimalfläche von Enneper. Abh. Math. Sem. Univ. Hamburg 44, 152–174 (1975/1976)
Wallner, J., Pottmann, H.: Infinitesimally flexible meshes and discrete minimal surfaces. Monatsh. Math. 153, 347–365 (2008)
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Communicated by Rida Farouki.
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Pottmann, H., Grohs, P. & Mitra, N.J. Laguerre minimal surfaces, isotropic geometry and linear elasticity. Adv Comput Math 31, 391 (2009). https://doi.org/10.1007/s10444-008-9076-5
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DOI: https://doi.org/10.1007/s10444-008-9076-5
Keywords
- Differential geometry
- Laguerre geometry
- Laguerre minimal surface
- Isotropic geometry
- Linear elasticity
- Airy stress function
- Biharmonic function
- Thin plate spline
- Geometrical optics