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Volume 9
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10.3390/math9233077
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Article

Characterization of Rectifying Curves by Their Involutes and Evolutes

by
Marilena Jianu
1,
Sever Achimescu
1,
Leonard Dăuş
1,
Adela Mihai
1,2,*,
Olimpia-Alice Roman
3 and
Daniel Tudor
1
1
Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania
2
Interdisciplinary Doctoral School, Transilvania University of Braşov, 500036 Braşov, Romania
3
Faculty of Hydrotechnics, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania
*
Author to whom correspondence should be addressed.
Mathematics 2021, 9(23), 3077; https://doi.org/10.3390/math9233077
Submission received: 7 November 2021 / Revised: 20 November 2021 / Accepted: 26 November 2021 / Published: 29 November 2021
(This article belongs to the Special Issue Differential Geometry: Theory and Applications Part II)
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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 rectifying curves; involutes; evolutes

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MDPI and ACS Style

Jianu, M.; Achimescu, S.; Dăuş, L.; Mihai, A.; Roman, O.-A.; Tudor, D. Characterization of Rectifying Curves by Their Involutes and Evolutes. Mathematics 2021, 9, 3077. https://doi.org/10.3390/math9233077

AMA Style

Jianu M, Achimescu S, Dăuş L, Mihai A, Roman O-A, Tudor D. Characterization of Rectifying Curves by Their Involutes and Evolutes. Mathematics. 2021; 9(23):3077. https://doi.org/10.3390/math9233077

Chicago/Turabian Style

Jianu, Marilena, Sever Achimescu, Leonard Dăuş, Adela Mihai, Olimpia-Alice Roman, and Daniel Tudor. 2021. "Characterization of Rectifying Curves by Their Involutes and Evolutes" Mathematics 9, no. 23: 3077. https://doi.org/10.3390/math9233077

APA Style

Jianu, M., Achimescu, S., Dăuş, L., Mihai, A., Roman, O.-A., & Tudor, D. (2021). Characterization of Rectifying Curves by Their Involutes and Evolutes. Mathematics, 9(23), 3077. https://doi.org/10.3390/math9233077

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