Geodesic
Application of spaces of subspheres to conformal invariants of curves and canal surfaces
Rémi Langevin1 ; Jun O'Hara2 ; Shigehiro Sakata2
1Institut de Mathématiques de Bourgogne UMR CNRS 5584 Université de Bourgogne 21078 Dijon, France
2Department of Mathematics Tokyo Metropolitan University Tokyo 192-0397, Japan
Annales Polonici Mathematici, Tome 108 (2013) no. 2, pp. 109-131
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We review some techniques from the Möbius geometry of curves and surfaces in the 3-sphere, consider canal surfaces using their characteristic circles, and express the conformal curvature, and conformal torsion, of a vertex-free space curve in terms of its corresponding curve of osculating circles, and osculating spheres, respectively. We accomplish all of this strictly within the framework of Möbius geometry, and compare our results with the literature. Finally, we show how our formulation allows for the re-expression of the conformal invariants in terms of standard Euclidean invariants.
DOI : 10.4064/ap108-2-1
Keywords: review techniques bius geometry curves surfaces sphere consider canal surfaces using their characteristic circles express conformal curvature conformal torsion vertex free space curve terms its corresponding curve osculating circles osculating spheres respectively accomplish strictly within framework bius geometry compare results literature finally formulation allows re expression conformal invariants terms standard euclidean invariants

Rémi Langevin  1   ; Jun O'Hara  2   ; Shigehiro Sakata  2

1 Institut de Mathématiques de Bourgogne UMR CNRS 5584 Université de Bourgogne 21078 Dijon, France
2 Department of Mathematics Tokyo Metropolitan University Tokyo 192-0397, Japan 
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 conformal invariants of curves and canal surfaces
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Rémi Langevin; Jun O'Hara; Shigehiro Sakata. Application of spaces of subspheres to
 conformal invariants of curves and canal surfaces. Annales Polonici Mathematici, Tome 108 (2013) no. 2, pp. 109-131. doi: 10.4064/ap108-2-1

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