Journal for Geometry and Graphics 28 (2024), No. 1, 073--085
Copyright by the authors licensed under CC BY SA 4.0
Creating Ruled Surfaces Using the Base Curve of the Frenet Trihedron
Serhiy Pylypaka
University of Life and Environmental Sciences of Ukraine, Kyiv, Ukraine
ps55@ukr.net
Andrii Nesvidomin
University of Life and Environmental Sciences of Ukraine, Kyiv, Ukraine
a.nesvidomin@gmail.com
Tetiana Volina
University of Life and Environmental Sciences of Ukraine, Kyiv, Ukraine
t.n.zaharova@ukr.net
Alexandra Trokhaniak
University of Life and Environmental Sciences of Ukraine, Kyiv, Ukraine
klendii_o@ukr.net
Iryna Taras
National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine
iryna.taras@nung.edu.ua
A ruled surface is formed by moving a straight line (as a generator) along a base curve. If the straight line's direction remains constant in a fixed coordinate system, the resulting surface will be cylindrical. In the general case, the direction of the generator can be described by a unit vector in the projections on the coordinate axis and will vary depending on the point on the base curve. This surface can be defined by a grid of both rectangular and oblique coordinate lines. Additionally, a ruled surface can be either developable or non-developable. In the general case, the surface is associated with an oblique grid of coordinate lines and is non-developable. In order to obtain specific cases, it is necessary to impose restrictions on the direction of the unit vector of the straight line.
The article describes a method for creating ruled surfaces using the Frenet trihedron of the base curve. It explores cases where a straight line rotates within the accompanying trihedron of the base curve or remains fixed in place. The article also establishes conditions for non-developable surfaces when the straight line rotates. It provides the parametrization of the ruled surfaces and their first quadratic forms based on the Frenet trihedron. Additionally, the article presents examples of ruled surfaces and provides visual representations.
Keywords: Frenet formulas, curvature, torsion, vector surface equation, first quadratic form.
MSC: 53A05
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