An Explicit Isometric Reduction of the Unit Sphere into an Arbitrarily Small Ball

Evangelis Bartzos 1 Vincent Borrelli 2 Roland Denis 3 Francis Lazarus 4, * Damien Rohmer 5, 6, 7 Boris Thibert 8
* Corresponding author
1 ERGA - Lab of Geometric and Algebraic Algorithms
2 AGL - Algèbre, géométrie, logique
ICJ - Institut Camille Jordan [Villeurbanne]
3 ICJ - Institut Camille Jordan [Villeurbanne]
4 GIPSA-AGPIG - AGPIG
GIPSA-DIS - Département Images et Signal
5 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
6 IMAGINE - Intuitive Modeling and Animation for Interactive Graphics & Narrative Environments
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
7 CPE - Department of Computer Science [Lyon]
8 CVGI - Calcul des Variations, Géométrie, Image
LJK - Laboratoire Jean Kuntzmann
Abstract : Spheres are known to be rigid geometric objects: they cannot be deformed isometrically, i.e. while preserving the length of curves, in a twice differentiable way. An unexpected result by J. Nash (Ann. of Math. 60: 383-396, 1954) and N. Kuiper (Indag. Math. 17: 545-555, 1955) shows that this is no longer the case if one requires the deformations to be only continuously differentiable. A remarkable consequence of their result makes possible the isometric reduction of a unit sphere inside an arbitrarily small ball. In particular, if one views the Earth as a round sphere the theory allows to reduce its diameter to that of a terrestrial globe while preserving geodesic distances. Here we describe the first explicit construction and visualization of such a reduced sphere. The construction amounts to solve a non-linear PDE with boundary conditions. The resulting surface consists of two unit spherical caps joined by a C 1 fractal equatorial belt. An intriguing question then arises about the transition between the smooth and the C 1 fractal geometries. We show that this transition is similar to the one observed when connecting a Koch curve to a line segment.
Keywords : Isometric embedding Convex integration Sphere reduction Boundary conditions
Document type :
Journal articles
Complete list of metadatas

Cited literature [17 references]  Display  Hide  Download
https://hal.inria.fr/hal-01647062
Contributor : Damien Rohmer <>
Submitted on : Friday, November 24, 2017 - 10:37:45 AM
Last modification on : Wednesday, November 20, 2019 - 3:18:18 AM

File

sphere_reduite_focm2017.pdf
Files produced by the author(s)

Identifiers

Collections

CNRS | X | EC-LYON | LIX | INSMI | LJK | X-LIX | PERSYVAL-LAB | INRIA | UNIV-ST-ETIENNE | INSA-LYON | UGA | UNIV-PARIS-SACLAY | PARISTECH | ICJ | GIPSA | LJK_GI | LJK-GI-CVGI | GIPSA-AGPIG | GIPSA-DIS | UNIV-LYON1 | X-DEP-INFO | INSA-GROUPE | ICJ-AGL | UDL

Citation

Evangelis Bartzos, Vincent Borrelli, Roland Denis, Francis Lazarus, Damien Rohmer, et al.. An Explicit Isometric Reduction of the Unit Sphere into an Arbitrarily Small Ball. Foundations of Computational Mathematics, Springer Verlag, 2018, 18 (4), pp.1015-1042. ⟨10.1007/s10208-017-9360-1⟩. ⟨hal-01647062⟩

Export

BibTeX TEI DC DCterms EndNote

Share

Metrics

Record views

1757

Files downloads

309

Contact

CCSD