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Some aspects of normal curve on smooth surface under isometry

Singh Kuljeet, Sharma Sandeep, Bhardwaj Arun Kumar

Demonstratio Mathematica

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2024

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Vol. 57, nr 1

art. no. 20240017

EN

The normal curve is a space curve that plays an important role in the field of differential geometry. This research focuses on analyzing the properties of normal curves on smooth immersed surfaces, considering their invariance under isometric transformations. The primary contribution of this article is to explore the requirements for the image of a normal curve that preserves its invariance under isometric transformations. In this article, we investigate the invariant condition for the component of the position vector of the normal curves under isometry and compute the expression for the normal and geodesic curvature of such curves. Moreover, it has been investigated that the geodesic curvature and Christoffel symbols remain unchanged under the isometry of surfaces in R3.

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

Singh Kuljeet, Sharma Sandeep, Lone Mohamd Saleem

Demonstratio Mathematica

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2024

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Vol. 57, nr 1

art. no. 20230145

EN

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. 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.

3

Eye disease segmentation using hybrid neural encoder decoder based Unet hybrid inception

Bali Le Akanksha, Singh Kuljeet, Mansotra Vibhakar

Computer Science

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2024

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T. 25 (4)

579–-620

EN

Diabetic retinopathy (DR) is one of the major causes of vision problems worldwide. With proper treatment, early diagnosis of DR can prevent the progression of the disease. In this paper, we present a combinative method using U-Net with a modified Inception architecture for the diagnosis of both the diseases. The proposed method is based on deep neural architecture formalising encoder decoder modelling with convolutional architectures namely Inception and Residual Connection. The performance of the proposed model was validated on the IDRid 2019 contest dataset. Experiments demonstrate that the modified Inception deep feature extractor improves DR classification with a classification accuracy of 99.34% in IDRid across classes with comparison to Resnet. The paper Benchmark tests the dataset with proposed model of Hybrid Dense-EDUHI: Encoder Decoder based U-Net Hybrid Inception model with 15 fold cross validation. The paper in details discusses the various metrics of the proposed model with various visualisation and multifield validations.