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Rolling Spheres and the Willmore Energy.

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Title: Rolling Spheres and the Willmore Energy.
Authors: Knöppel, Felix1 (AUTHOR), Pinkall, Ulrich1 (AUTHOR), Schröder, Peter2 (AUTHOR), Soliman, Yousuf3 (AUTHOR) yousufs@sidefx.com
Source: Discrete & Computational Geometry. Jul2026, Vol. 76 Issue 1, p177-218. 42p.
Subjects: Spheres, Conformal geometry, Spherical geometry, Conformal invariants, Discrete geometry
Abstract: In this paper we provide a new interpretation of the Willmore energy as the curvature of a quaternionic connection that has a clear geometric interpretation in terms of mean curvature spheres rolling over the surface. Building on this interpretation, we show that the Möbius invariant discretization of the Willmore energy introduced by Bobenko can be interpreted as the curvature of a discrete connection defined by rolling the edge circumspheres. We describe how a choice of discrete mean curvature spheres produces a Möbius invariant discrete Willmore energy. In this way, we define a new discrete Willmore energy for surfaces built out of spherical pieces. To facilitate the computations we describe a new quaternionic description of the conformal three-sphere, along with realizations of the spaces of circles, spheres, and point pairs in. It is obtained by specializing the quaternionic projective model of the conformal four-sphere. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Jul2026, Vol. 76 Issue 1, p177-218. 42p.
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  Data: In this paper we provide a new interpretation of the Willmore energy as the curvature of a quaternionic connection that has a clear geometric interpretation in terms of mean curvature spheres rolling over the surface. Building on this interpretation, we show that the Möbius invariant discretization of the Willmore energy introduced by Bobenko can be interpreted as the curvature of a discrete connection defined by rolling the edge circumspheres. We describe how a choice of discrete mean curvature spheres produces a Möbius invariant discrete Willmore energy. In this way, we define a new discrete Willmore energy for surfaces built out of spherical pieces. To facilitate the computations we describe a new quaternionic description of the conformal three-sphere, along with realizations of the spaces of circles, spheres, and point pairs in. It is obtained by specializing the quaternionic projective model of the conformal four-sphere. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – SubjectFull: Spherical geometry
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              Text: Jul2026
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              Y: 2026
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