Mannheim-d'Ocagne-Koenderink type formulas for asymptotic directions.
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| Title: | Mannheim-d'Ocagne-Koenderink type formulas for asymptotic directions. |
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| Authors: | FUKUI, Toshizumi1, HONDA, Atsufumi2, UMEHARA, Masaaki3 |
| Source: | Proceedings of the Japan Academy, Series A: Mathematical Sciences. Jun2026, Vol. 102 Issue 6, p29-35. 7p. |
| Subjects: | Gaussian curvature, Mathematical singularities, Surface geometry, Euclidean metric, Curvature, Mathematical formulas |
| Abstract: | We consider a surface embedded in the Euclidean 3-space and fix a tangential vector v at a given point p on the surface. We first review the history of the formula obtained by Mannheim, d'Ocagne and Koenderink, which asserts that the Gaussian curvature of the surface at p can be obtained if one knows "the normal curvature at p with respect to v" and "the curvature at p of the contour line Γ of the surface" with respect to the orthogonal projection induced by v. However, this formula breaks down when v is an asymptotic direction. In such cases, we present analogues of the formula that involve an invariant associated with cusp singularities of Γ. [ABSTRACT FROM AUTHOR] |
| Copyright of Proceedings of the Japan Academy, Series A: Mathematical Sciences is the property of Japan Academy 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 |
| FullText | Links: – Type: pdflink Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 194285003 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Mannheim-d'Ocagne-Koenderink type formulas for asymptotic directions. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22FUKUI%2C+Toshizumi%22">FUKUI, Toshizumi</searchLink><relatesTo>1</relatesTo><br /><searchLink fieldCode="AR" term="%22HONDA%2C+Atsufumi%22">HONDA, Atsufumi</searchLink><relatesTo>2</relatesTo><br /><searchLink fieldCode="AR" term="%22UMEHARA%2C+Masaaki%22">UMEHARA, Masaaki</searchLink><relatesTo>3</relatesTo> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Proceedings+of+the+Japan+Academy%2C+Series+A%3A+Mathematical+Sciences%22">Proceedings of the Japan Academy, Series A: Mathematical Sciences</searchLink>. Jun2026, Vol. 102 Issue 6, p29-35. 7p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Gaussian+curvature%22">Gaussian curvature</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+singularities%22">Mathematical singularities</searchLink><br /><searchLink fieldCode="DE" term="%22Surface+geometry%22">Surface geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Euclidean+metric%22">Euclidean metric</searchLink><br /><searchLink fieldCode="DE" term="%22Curvature%22">Curvature</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+formulas%22">Mathematical formulas</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We consider a surface embedded in the Euclidean 3-space and fix a tangential vector v at a given point p on the surface. We first review the history of the formula obtained by Mannheim, d'Ocagne and Koenderink, which asserts that the Gaussian curvature of the surface at p can be obtained if one knows "the normal curvature at p with respect to v" and "the curvature at p of the contour line Γ of the surface" with respect to the orthogonal projection induced by v. However, this formula breaks down when v is an asymptotic direction. In such cases, we present analogues of the formula that involve an invariant associated with cusp singularities of Γ. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Proceedings of the Japan Academy, Series A: Mathematical Sciences is the property of Japan Academy 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: BibEntity: Identifiers: – Type: doi Value: 10.3792/pjaa.102.006 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 7 StartPage: 29 Subjects: – SubjectFull: Gaussian curvature Type: general – SubjectFull: Mathematical singularities Type: general – SubjectFull: Surface geometry Type: general – SubjectFull: Euclidean metric Type: general – SubjectFull: Curvature Type: general – SubjectFull: Mathematical formulas Type: general Titles: – TitleFull: Mannheim-d'Ocagne-Koenderink type formulas for asymptotic directions. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: FUKUI, Toshizumi – PersonEntity: Name: NameFull: HONDA, Atsufumi – PersonEntity: Name: NameFull: UMEHARA, Masaaki IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 03862194 Numbering: – Type: volume Value: 102 – Type: issue Value: 6 Titles: – TitleFull: Proceedings of the Japan Academy, Series A: Mathematical Sciences Type: main |
| ResultId | 1 |
