Journal for Geometry and Graphics 29 (2025), No. 2, 187--197
Copyright by the authors licensed under CC BY SA 4.0
Continuous Bending of Surfaces of Rotation into Helical Surfaces
Iryna Taras
National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine
iryna.taras@nung.edu.ua
Serhiy Pylypaka
University of Life and Environmental Sciences, Kyiv, Ukraine
ps55@ukr.net
Andrii Nesvidomin
University of Life and Environmental Sciences, Kyiv, Ukraine
a.nesvidomin@gmail.com
Tetiana Volina
University of Life and Environmental Sciences, Kyiv, Ukraine
t.n.zaharova@ukr.net
Oleksandra Trokhaniak
University of Life and Environmental Sciences, Kyiv, Ukraine
klendii_o@ukr.net
When a surface undergoes bending, the length of any curve on it remains unchanged. In the general case, a curve on a surface can be defined by a functional relationship between curvilinear coordinates -- that is, through an intrinsic equation. The length of such a curve can be computed using the first fundamental form. Since this length remains invariant under bending, the expression of the first fundamental form also remains unchanged. This invariance forms the foundation of the theory of surface bending. Bending of surfaces occurs under certain constraints on their deformation. For ruled surfaces -- whether developable or non-developable -- a typical constraint is the preservation of straight-line generatrices. A clear geometric example is the bending of developable surfaces while keeping their generatrices unchanged. In the case of non-developable surfaces, a non-ruled surface can be bent into a ruled one. A classical example is the bending of a surface of revolution, such as the catenoid, into a helical surface, such as the helicoid. The helicoid is a ruled surface; however, when its pitch is gradually decreased during the bending process, the surface becomes non-ruled. By continuously reducing the pitch, one can construct a one-parameter set of intermediate surfaces, making the bending process continuous. When the pitch reaches zero, the helicoid transforms into a catenoid. This example illustrates the bending of helical surfaces into surfaces of revolution. According to Bour’s theorem, when a helicoidal surface is bent into a surface of revolution, the helical lines correspond to parallels, and their orthogonal trajectories correspond to meridians. The present work explores the inverse process -- the bending of a surface of revolution into a helicoidal surface. The surface of revolution is defined via the explicit equation of its meridian. Parametric equations describing the one-parameter set of intermediate surfaces are derived, and several of these surfaces are constructed. The paper also considers the continuous bending of the catenoid into the helicoid.
Keywords: First fundamental form, meridian, bending parameter, pitch, constraint.
MSC: 53A05; 53C42.
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