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The main aim of this paper is to investigate the nature of invariancy of rectifying curve under conformal transformation and obtain a sufficient condition for which such a curve remains conformally invariant. It is shown that the normal component and the geodesic curvature of the rectifying curve is homothetic invariant.

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... Both of these problems were discussed in [7,9,10]. Later in [11], we generalize the notion of study by conformal transformation, wherein we study the conformal transformation of rectifying curves lying on smooth surfaces. In this paper [11], in addition to various geometric invariants, we obtain a sufficient condition with respect to which a rectifying curve retains its nature under conformal motion. ...

... Later in [11], we generalize the notion of study by conformal transformation, wherein we study the conformal transformation of rectifying curves lying on smooth surfaces. In this paper [11], in addition to various geometric invariants, we obtain a sufficient condition with respect to which a rectifying curve retains its nature under conformal motion. Now, the first hand possible studies depending upon the position vector field and the conformality will be the investigation of osculating and normal curves. ...

The main intention of the paper is to investigate an osculating curve under the conformal map. We obtain a sufficient condition for the conformal invariance of an osculating curve. We also find an equivalent system of a geodesic curve under the conformal transformation(motion) and show its invariance under isometry and homothetic motion.

... Afterwards, we generalized the notion of study by the conformal transformation. The invariant properties of rectifying and osculating curves under a conformal transformation are studied in [10,11]. Now, in this paper, we try to investigate the following: ...

... 10.[9]Let J : S →S be an isometry and β be a normal curve on S. The for the tangential component of β, we havẽβ ·T − (β · T) = η κ (κ n − κ n ) (av ′ + bu ′ )and is invariant if and only if the position vector of β is in the normal direction or the normal curvature is invariant.Proposition 1. Let J be a conformal map between two smooth surfaces S andS and let β(s) be a parameterized curve on S such thatβ(s) = J • β(s) is conformal parameterized image of β onS. ...

In this paper, we investigate the geometric invariant properties of a normal curve on a smooth immersed surface under conformal transformation. We obtain an invariant-sufficient condition for the conformal image of a normal curve. We also find the deviations of normal and tangential components of the normal curve under the same motion. The results in [9] are claimed as special cases of this paper.

... Both of these problems were discussed in [7,9,10]. Later in [11], we generalize the notion of study by conformal transformation, wherein we study the conformal transformation of rectifying curves lying on smooth surfaces. In this paper( [11]), in addition to various geometric invariants, we obtain a sufficient condition with respect to which a rectifying curve retains its nature under conformal motion. ...

... Later in [11], we generalize the notion of study by conformal transformation, wherein we study the conformal transformation of rectifying curves lying on smooth surfaces. In this paper( [11]), in addition to various geometric invariants, we obtain a sufficient condition with respect to which a rectifying curve retains its nature under conformal motion. Now, the first hand possible studies depending upon the position vector field and the conformality will be the investigation of osculating and normal curves. ...

... Both of these problems were discussed in [7,9,10]. Later in [11], we generalize the notion of study by conformal transformation, wherein we study the conformal transformation of rectifying curves lying on smooth surfaces. In this paper( [11]), in addition to various geometric invariants, we obtain a sufficient condition with respect to which a rectifying curve retains its nature under conformal motion. ...

... Later in [11], we generalize the notion of study by conformal transformation, wherein we study the conformal transformation of rectifying curves lying on smooth surfaces. In this paper( [11]), in addition to various geometric invariants, we obtain a sufficient condition with respect to which a rectifying curve retains its nature under conformal motion. Now, the first hand possible studies depending upon the position vector field and the conformality will be the investigation of osculating and normal curves. ...

The main intention of the paper is to investigate an osculating curve under the conformal map. We obtain a sufficient condition for the conformal invariance of an osculating curve. We also find an equivalent system of a geodesic curve under the conformal transformation(motion) and show its invariance under isometry and homothetic motion.

First, we study rectifying curves via the dilation of unit speed curves on the unit sphere S^2 in the Euclidean 3-space E^3. Then we obtain a necessary and sufficient condition for which the centrode d(s) of a unit speed curve in E^3 is rectifying curve to improve a main result of [ B.-Y. Chen and F. Dillen, Rectifying curves as centrodes and extremal curves. Bull. Inst. Math. Acad. Sinica 33 (2005), no. 2, 77-90 ]. Finally, we prove that if a unit speed curve in E^3 is neither a planar curve nor a helix, then its dilated centrode b(s)=r(s)d(s) with dilation factor r(s) is always a rectifying curve, where r(s) is the radius of curvature of the unit speed curve.

The aim of this paper is to investigate a sufficient condition for the invariance of a normal curve on a smooth immersed surface under isometry. We also find the deviations of the tangential and normal components of the curve with respect to the given isometry.

The main motive of the paper is to look on rectifying and osculating curves on a smooth surface. In this paper we find the normal and geodesic curvature for a rectifying curve on a smooth surface and we also prove that geodesic curvature is invariant under the isometry of surfaces such that rectifying curves remain. We find a sufficient condition for which an osculating curve on a smooth surface remains invariant under isometry of surfaces and also we prove that the component of the position vector of an osculating curve α(s) on a smooth surface along any tangent vector to the surface at α(s) is invariant under such isometry.

The main intention of the paper is to investigate an osculating curve under the conformal map. We obtain a sufficient condition for the conformal invariance of an osculating curve. We also find an equivalent system of a geodesic curve under the conformal transformation(motion) and show its invariance under isometry and homothetic motion.

The main objective of the present paper is to investigate a sufficient condition for which a rectifying curve on a smooth surface remains invariant under isometry of surfaces, and also it is shown that under such an isometry the component of the position vector of a rectifying curve on a smooth surface along the normal to the surface is invariant.

We characterize conformal mapping between two surfaces, S and S∗, based on Gaussian curvature before and after motion. An explicit representation of the Gaussian curvature after conformal mapping is presented in terms of Riemann-Christoffel tensor and Ricci tensor and their derivatives. Based on changes in surface curvature, we are able to estimate the stretching of non-rigid motion during conformal mapping via a polynomial approximation.

Rectfying curve as centrode and extremal curve

- Chen

Normal curves on a smooth immersed surface

- Shaikh