Characterizing the developable surfaces with curves whose position vectors lie in the planes spanned by Darboux frame

Takeshi Takahashi ()
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Takeshi Takahashi: University of Tsukuba

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 2, 456-466

Abstract: Abstract Curves on surface are called osculating Darboux curve and rectifying Darboux curve if its position vectors always lies in the osculating Darboux plane and rectifying Darboux plane which are spanned by Darboux frame. We study the case that osculating Darboux curve and rectifying Darboux curve are geodesic, line of curvature or asymptotic curve. In this case, we determine each curve is related to rectifying curve, spherical curve and planer curve. Using this result, we also give the classification of a special developable surface under the condition of the existence of osculating Darboux curve as a geodesic or a line of curvature. As a result, in this case, we determine these developable surfaces become conical surfaces.

Keywords: Darboux frame; Osculating Darboux curve; Rectifying Darboux curve; Developable surface; Conical surface; Rectifying curve; 53A04; 53A05 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00267-0

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