Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space
In this paper, we study the surfaces of osculating circles, which are the sets of all osculating circles at all points of regular curves. Since the surfaces of osculating circles may be singular, it is necessary to investigate the singular points of these surfaces. However, traditional methods and t...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/17/3714 |
| _version_ | 1797582200472862720 |
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| author | Kemeng Liu Zewen Li Donghe Pei |
| author_facet | Kemeng Liu Zewen Li Donghe Pei |
| author_sort | Kemeng Liu |
| collection | DOAJ |
| description | In this paper, we study the surfaces of osculating circles, which are the sets of all osculating circles at all points of regular curves. Since the surfaces of osculating circles may be singular, it is necessary to investigate the singular points of these surfaces. However, traditional methods and tools for analyzing singular properties have certain limitations. To solve this problem, we define the framed surfaces of osculating circles in the Euclidean 3-space. Then, we discuss the types of singular points using the theory of framed surfaces and show that generic singular points of the surfaces consist of cuspidal edges and cuspidal cross-caps. |
| first_indexed | 2024-03-10T23:17:32Z |
| format | Article |
| id | doaj.art-8f60c80621a34e849c52ea6cc0c890c2 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T23:17:32Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-8f60c80621a34e849c52ea6cc0c890c22023-11-19T08:31:07ZengMDPI AGMathematics2227-73902023-08-011117371410.3390/math11173714Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean SpaceKemeng Liu0Zewen Li1Donghe Pei2School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, ChinaSchool of Mathematics and Statistics, Northeast Normal University, Changchun 130024, ChinaSchool of Mathematics and Statistics, Northeast Normal University, Changchun 130024, ChinaIn this paper, we study the surfaces of osculating circles, which are the sets of all osculating circles at all points of regular curves. Since the surfaces of osculating circles may be singular, it is necessary to investigate the singular points of these surfaces. However, traditional methods and tools for analyzing singular properties have certain limitations. To solve this problem, we define the framed surfaces of osculating circles in the Euclidean 3-space. Then, we discuss the types of singular points using the theory of framed surfaces and show that generic singular points of the surfaces consist of cuspidal edges and cuspidal cross-caps.https://www.mdpi.com/2227-7390/11/17/3714surfaces of osculating circlesframed surfacessingularities |
| spellingShingle | Kemeng Liu Zewen Li Donghe Pei Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space Mathematics surfaces of osculating circles framed surfaces singularities |
| title | Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space |
| title_full | Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space |
| title_fullStr | Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space |
| title_full_unstemmed | Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space |
| title_short | Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space |
| title_sort | singular surfaces of osculating circles in three dimensional euclidean space |
| topic | surfaces of osculating circles framed surfaces singularities |
| url | https://www.mdpi.com/2227-7390/11/17/3714 |
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