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

Author:

Ho Peter T.1,Odom Lucy H.2,Suceavă Bogdan D.3

Affiliation:

1. Department of Mathematics, California State University, Northridge, 18111 Nordho Street, Northridge, CA 91330, United States of America

2. Department of Mathematics, San Francisco State University, 1600 Holloway Ave, San Francisco, CA 94132, United States of America

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

Abstract

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.

Publisher

Walter de Gruyter GmbH

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