<abstract xmlns="http://www.w3.org/1999/xhtml"><p> 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.</p></abstract>

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.

On Chasles' Property of the Helicoid in Tri-Twisted Real Ambient Space

Department of Mathematics, California State University, Northridge, 18111 Nordho Street, Northridge, CA 91330,

Department of Mathematics, San Francisco State University, 1600 Holloway Ave, San Francisco, CA 94132,

Department of Mathematics, California State University at Fullerton, 800 N. State College Blvd., Fullerton, CA, 92834-6850,

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"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...

On Chasles' Property of the Helicoid in Tri-Twisted Real Ambient Space

Data publikacji: 22 kwi 2017

Zakres stron: 121 - 132

Otrzymano: 17 wrz 2013

Przyjęty: 02 gru 2013

DOI: https://doi.org/10.1515/auom-2015-0030

Słowa kluczowe
Helicoid, catenoid, submanifold, helices, twisted metrics, isometric immersion, rotation surface

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

On Chasles' Property of the Helicoid in Tri-Twisted Real Ambient Space

Data publikacji: 22 kwi 2017

Zakres stron: 121 - 132

Otrzymano: 17 wrz 2013

Przyjęty: 02 gru 2013

DOI: https://doi.org/10.1515/auom-2015-0030

Słowa kluczoweHelicoid, catenoid, submanifold, helices, twisted metrics, isometric immersion, rotation surface

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Słowa kluczowe
Helicoid, catenoid, submanifold, helices, twisted metrics, isometric immersion, rotation surface