On Chasles' Property of the Helicoid in Tri-Twisted Real Ambient Space
An elementary property of the helicoid is that at every point of the surface the following condition holds: cot θ = C · d; where d is the distance between an arbitrary point to the helicoid axis, and θ is the angle between the normal and the helicoid’s axis. This rigidity property was discovered by...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Sciendo
2015-06-01
|
| Series: | Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/auom-2015-0030 |
| _version_ | 1811329866717462528 |
|---|---|
| author | Ho Peter T. Odom Lucy H. Suceavă Bogdan D. |
| author_facet | Ho Peter T. Odom Lucy H. Suceavă Bogdan D. |
| author_sort | Ho Peter T. |
| collection | DOAJ |
| description | An elementary property of the helicoid is that at every point of the surface the following condition holds: cot θ = C · d; where d is the distance between an arbitrary point to the helicoid axis, and θ is the angle between the normal and the helicoid’s axis. This rigidity property was discovered by M. Chasles in the first half of the XIXth century. Starting from this property, we give a characterization of the so-called tri-twisted metrics on the real three dimensional space with the property that a given helicoid satisfies the classical invariance condition. Similar studies can be pursued in other geometric contexts. Our most general result presents a property of surfaces of rotation observing an invariance property suggested by the analogy with Chasles’s property. |
| first_indexed | 2024-04-13T15:51:31Z |
| format | Article |
| id | doaj.art-f26ac40024a9471e98b83d863521cada |
| institution | Directory Open Access Journal |
| issn | 1844-0835 |
| language | English |
| last_indexed | 2024-04-13T15:51:31Z |
| publishDate | 2015-06-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica |
| spelling | doaj.art-f26ac40024a9471e98b83d863521cada2022-12-22T02:40:50ZengSciendoAnalele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica1844-08352015-06-0123212113210.1515/auom-2015-0030On Chasles' Property of the Helicoid in Tri-Twisted Real Ambient SpaceHo Peter T.0Odom Lucy H.1Suceavă Bogdan D.2Department of Mathematics, California State University, Northridge, 18111 Nordho Street, Northridge, CA 91330, United States of AmericaDepartment of Mathematics, San Francisco State University, 1600 Holloway Ave, San Francisco, CA 94132, United States of AmericaDepartment of Mathematics, California State University at Fullerton, 800 N. State College Blvd., Fullerton, CA, 92834-6850, United States of AmericaAn elementary property of the helicoid is that at every point of the surface the following condition holds: cot θ = C · d; where d is the distance between an arbitrary point to the helicoid axis, and θ is the angle between the normal and the helicoid’s axis. This rigidity property was discovered by M. Chasles in the first half of the XIXth century. Starting from this property, we give a characterization of the so-called tri-twisted metrics on the real three dimensional space with the property that a given helicoid satisfies the classical invariance condition. Similar studies can be pursued in other geometric contexts. Our most general result presents a property of surfaces of rotation observing an invariance property suggested by the analogy with Chasles’s property.https://doi.org/10.1515/auom-2015-0030helicoidcatenoidsubmanifoldhelicestwisted metricsisometric immersionrotation surface |
| spellingShingle | Ho Peter T. Odom Lucy H. Suceavă Bogdan D. On Chasles' Property of the Helicoid in Tri-Twisted Real Ambient Space Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica helicoid catenoid submanifold helices twisted metrics isometric immersion rotation surface |
| title | On Chasles' Property of the Helicoid in Tri-Twisted Real Ambient Space |
| title_full | On Chasles' Property of the Helicoid in Tri-Twisted Real Ambient Space |
| title_fullStr | On Chasles' Property of the Helicoid in Tri-Twisted Real Ambient Space |
| title_full_unstemmed | On Chasles' Property of the Helicoid in Tri-Twisted Real Ambient Space |
| title_short | On Chasles' Property of the Helicoid in Tri-Twisted Real Ambient Space |
| title_sort | on chasles property of the helicoid in tri twisted real ambient space |
| topic | helicoid catenoid submanifold helices twisted metrics isometric immersion rotation surface |
| url | https://doi.org/10.1515/auom-2015-0030 |
| work_keys_str_mv | AT hopetert onchaslespropertyofthehelicoidintritwistedrealambientspace AT odomlucyh onchaslespropertyofthehelicoidintritwistedrealambientspace AT suceavabogdand onchaslespropertyofthehelicoidintritwistedrealambientspace |