Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.
📌 3 Key Takeaways
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Objective & Hypothesis:In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric.
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Key Findings & Data:Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ \mathbb{G} $.
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Conclusion & Impact:Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Natural mate · conjugate mate · helix
Abstract
In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ \mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ \mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.
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