Surface Pencil Couple with Bertrand Couple as Joint Principal Curves in Galilean 3-Space

A principal curve on a surface plays a paramount role in reasonable implementations. A curve on a surface is a principal curve if its tangents are principal directions. Using the Serret–Frenet frame, the surface pencil couple can be expressed as linear combinations of the components of the local fra...

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Bibliographic Details
Published inAxioms Vol. 12; no. 11; p. 1022
Main Authors Alluhaibi, Nadia, Abdel-Baky, Rashad A.
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.10.2023
Subjects
Asymptomatic
Euclidean space
Geometry
marching-scale functions
Serret–Frenet formulae
sotropic normal
Tangents
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Summary:A principal curve on a surface plays a paramount role in reasonable implementations. A curve on a surface is a principal curve if its tangents are principal directions. Using the Serret–Frenet frame, the surface pencil couple can be expressed as linear combinations of the components of the local frames in Galilean 3-space G3. With these parametric representations, a family of surfaces using principal curves (curvature lines) are constructed, and the necessary and sufficient condition for the given Bertrand couple to be the principal curves on these surfaces are derived in our approach. Moreover, the necessary and sufficient condition for the given Bertrand couple to satisfy the principal curves and the geodesic requirements are also analyzed. As implementations of our main consequences, we expound upon some models to confirm the method.
ISSN:2075-1680
2075-1680
DOI:10.3390/axioms12111022