Characterization of Rectifying Curves by Their Involutes and Evolutes

Marilena Jianu, Sever Achimescu, Leonard Dăuş, Adela Mihai, Olimpia-Alice Roman and Daniel Tudor
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Marilena Jianu: Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania
Sever Achimescu: Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania
Leonard Dăuş: Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania
Adela Mihai: Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania
Olimpia-Alice Roman: Faculty of Hydrotechnics, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania
Daniel Tudor: Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania

Mathematics, 2021, vol. 9, issue 23, 1-11

Abstract: A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves, namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves; we present an illustrative example of a rectifying curve obtained as the evolute of a spherical helix. We also express the curvature and the torsion of a rectifying spherical curve and give necessary and sufficient conditions for a curve and its involute to be both rectifying curves.

Keywords: rectifying curves; involutes; evolutes (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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