Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3

Abstract

An osculating curve is a type of curve in space that holds significance in the study of differential geometry. In this article, we investigate certain geometric invariants of osculating curves on smooth and regularly immersed surfaces under conformal transformations in Euclidean space R3{{\mathbb{R}}}^{3}. The primary objective of this article is to explore conditions sufficient for the conformal invariance of the osculating curve under both conformal transformations and isometries. We also compute the tangential and normal components of the osculating curves, demonstrating that they remain invariant under the isometry of the surfaces in R3{{\mathbb{R}}}^{3}.

Keywords

• osculating curves
• conformal transformation
• homothetic transformation
• isometry of surfaces
• normal and tangential components
• 53a04
• 53a05
• 53a15
• 53a55