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Spherical image analysis of origami and kirigami.

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Title: Spherical image analysis of origami and kirigami.
Authors: Walker, M. G.1 (AUTHOR) martin.g.walker@durham.ac.uk, Dias, M. A.2 (AUTHOR)
Source: Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences. 5/1/2026, Vol. 482 Issue 2337, p1-20. 20p.
Subjects: Origami, Gauss maps, Geometric analysis, Kinematics, Paper arts, Polyhedra, Gaussian curvature
Abstract: Analysing origami structures typically consists of computing the dihedral relationships between facets, for which bar-and-hinge models and spatial linkages are widely used. An alternative geometric viewpoint is provided by the spherical image: the map of surface points to the unit sphere via their normal vectors. Although classical in differential geometry, its application in origami has mostly been confined to four-crease vertices, leaving its broader potential largely unexplored. In this study, we provide an overview of the mathematical foundations of the spherical image and its application to general polyhedral vertices. We then specialize this approach to the kinematic analysis of origami. We demonstrate how the discrete Gaussian curvature encodes a key geometric constraint independent of the dihedral angles. The spherical image approach is then applied to the analysis of the four-crease and symmetric waterbomb vertices. We extend spherical image analysis to the case of non-rigid origami, including stretching deformations. Finally, we consider the consequences of a cut on the spherical image to apply the approach to kirigami and clarify the definitions of the e-cone and k-cone. The spherical image offers a versatile, and intuitive, means to analyse the kinematics of origami and kirigami while also providing valuable geometric insights. [ABSTRACT FROM AUTHOR]
Copyright of Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences is the property of Royal Society 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.)
Database: Engineering Source
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DbLabel: Engineering Source
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  Data: Spherical image analysis of origami and kirigami.
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  Data: <searchLink fieldCode="DE" term="%22Origami%22">Origami</searchLink><br /><searchLink fieldCode="DE" term="%22Gauss+maps%22">Gauss maps</searchLink><br /><searchLink fieldCode="DE" term="%22Geometric+analysis%22">Geometric analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Kinematics%22">Kinematics</searchLink><br /><searchLink fieldCode="DE" term="%22Paper+arts%22">Paper arts</searchLink><br /><searchLink fieldCode="DE" term="%22Polyhedra%22">Polyhedra</searchLink><br /><searchLink fieldCode="DE" term="%22Gaussian+curvature%22">Gaussian curvature</searchLink>
– Name: Abstract
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  Data: Analysing origami structures typically consists of computing the dihedral relationships between facets, for which bar-and-hinge models and spatial linkages are widely used. An alternative geometric viewpoint is provided by the spherical image: the map of surface points to the unit sphere via their normal vectors. Although classical in differential geometry, its application in origami has mostly been confined to four-crease vertices, leaving its broader potential largely unexplored. In this study, we provide an overview of the mathematical foundations of the spherical image and its application to general polyhedral vertices. We then specialize this approach to the kinematic analysis of origami. We demonstrate how the discrete Gaussian curvature encodes a key geometric constraint independent of the dihedral angles. The spherical image approach is then applied to the analysis of the four-crease and symmetric waterbomb vertices. We extend spherical image analysis to the case of non-rigid origami, including stretching deformations. Finally, we consider the consequences of a cut on the spherical image to apply the approach to kirigami and clarify the definitions of the e-cone and k-cone. The spherical image offers a versatile, and intuitive, means to analyse the kinematics of origami and kirigami while also providing valuable geometric insights. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences is the property of Royal Society 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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RecordInfo BibRecord:
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    Identifiers:
      – Type: doi
        Value: 10.1098/rspa.2025.0820
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      – Code: eng
        Text: English
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        PageCount: 20
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    Subjects:
      – SubjectFull: Origami
        Type: general
      – SubjectFull: Gauss maps
        Type: general
      – SubjectFull: Geometric analysis
        Type: general
      – SubjectFull: Kinematics
        Type: general
      – SubjectFull: Paper arts
        Type: general
      – SubjectFull: Polyhedra
        Type: general
      – SubjectFull: Gaussian curvature
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      – TitleFull: Spherical image analysis of origami and kirigami.
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            NameFull: Dias, M. A.
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            – D: 01
              M: 05
              Text: 5/1/2026
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              Y: 2026
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