Journal for Geometry and Graphics 25 (2021), No. 2, 163--170
Copyright Heldermann Verlag 2021
Parametric Equations of a Spatial Curve as a Function of Length of the Arc with Given Dependences of Curvature and Angle of Ascent
Serhiy Pylypaka
University of Life and Environmental Sciences, Kyiv, Ukraine
ps55@ukr.net
Tetiana Kresan
University of Life and Environmental Sciences, Kyiv, Ukraine
tanyakresan@i.ua
Oleksandra Trokhaniak
University of Life and Environmental Sciences, Kyiv, Ukraine
klendii_o@ukr.net
Iryna Taras
National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine
i.taras@nung.edu.ua
Ivan Demchuk
Agrotechnical Institute, National University of Life and Environmental Sciences, Nizhyn, Ukraine
alex_ks@mail.ru
The shape of a flat curve is completely determined by its natural equation that is the dependence of the curvature on the length of the arc. From the natural equation it is possible to pass to parametric equations of a curve which allow constructing a curve on the plane. The shape of the spatial curve is determined by two natural equations: the dependence of the torsion of the curvature at the length of the arc is combined with the dependence of the curvature. However, there is no simple transition from the natural equations of the spatial curve to the parametric ones. To reproduce the curve in space, it is necessary to solve a system of differential equations using numerical methods. We propose to replace the dependence of the torsion with the dependence of the angle of ascent on the length of the arc. In this case similarly to a flat curve, it is possible to write the parametric equations of the spatial curve. Examples are given, spatial curves are constructed, and partial case for a slope curve is considered.
Keywords: Spatial curve, natural and parametric equations, angle of ascent, curvature, torsion, slope curve.
MSC: 53A05; 65D17.
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