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Journals / International Electronic Journal of Geometry / 2026 / Volume: 19 Issue: 2

Tubular Hypersurfaces with Respect to the Quasi-frame in Four-dimensional Euclidean Space

Dört Boyutlu Öklid Uzayında Quasi-Çatıya Göre Tubular Hiperyüzeyler

Başak Yağbasan0000-0003-4067-3034
Cumali Ekici(ESKİŞEHİR OSMANGAZİ ÜNİVERSİTESİ)0000-0002-3247-5727
Hatice Tozak0000-0003-1383-0276
Pages
—
Publication Date
2026-10-25
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

These examples include comprehensive symbolic computations and visualizations of the directional tubular hypersurfaces rendered in various projection subspaces, thereby elucidating both the construction methodology and the inherent geometric interpretations of the directional tubular hypersurfaces.

📌 3 Key Takeaways

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Objective & Hypothesis:This study focuses on the differential geometric analysis of a newly defined class of hypersurfaces within four-dimensional Euclidean space $\mathbb{E}^{4}$.
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Key Findings & Data:These hypersurfaces are intrinsically generated as the envelope of a one-parameter family of spheres with constant radius, whose centers trace a space curve with a quasi-frame a generalized orthonormal frame specifically adapted to the curve.
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Conclusion & Impact:These examples include comprehensive symbolic computations and visualizations of the directional tubular hypersurfaces rendered in various projection subspaces, thereby elucidating both the construction methodology and the inherent geometric interpretations of the directional tubular hypersurfaces.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Hypersurface · Tubular surface · quasi-frame

Abstract

This study focuses on the differential geometric analysis of a newly defined class of hypersurfaces within four-dimensional Euclidean space $\mathbb{E}^{4}$. These hypersurfaces are intrinsically generated as the envelope of a one-parameter family of spheres with constant radius, whose centers trace a space curve with a quasi-frame a generalized orthonormal frame specifically adapted to the curve. Initially, general parametrizations are systematically established for directional canal hypersurfaces, which encompass directional tubular hypersurfaces as a generalized case, and separately for directional tubular hypersurfaces in the four-dimensional Euclidean space $\mathbb{E}^{4}$. Subsequently, the fundamental geometric properties of directional tubular hypersurface are meticulously investigated. Specifically, the first and second fundamental forms, the unit normal vector field, the shape operator, and including Gaussian curvature and mean curvature are explicitly computed. Finally, this theoretical formulation is exemplified and validated through detailed illustrative examples. These examples include comprehensive symbolic computations and visualizations of the directional tubular hypersurfaces rendered in various projection subspaces, thereby elucidating both the construction methodology and the inherent geometric interpretations of the directional tubular hypersurfaces.

Özet

Bu çalışma, dört boyutlu Öklid uzayında tanımlanan yeni bir hiperyüzeyin diferansiyel geometrik analizine odaklanmaktadır. Söz konusu hiperyüzeyler, ortonormal adapte çatı olan quasi-çatı ile türetilmiş bir merkez uzay eğrisi boyunca hareket eden sabit yarıçaplı tek parametreli bir küre ailesinin zarfı olarak oluşturulmuştur. İlk olarak, dört boyutlu Öklid uzayında yönlü tubular hiperyüzeylerin genel hali olan yönlü kanal hiperyüzeyler için sistematik olarak genel parametrizasyonu elde edilmiş, bu yapının özel bir durumu olarak ayrıca yönlü tubular hiperyüzeylerin parametrizasyonu tanıtılmıştır. Ardından, yönlü tubular hiperyüzeylerin temel geometrik özellikleri ayrıntılı biçimde incelenmiştir. Bu kapsamda, birinci ve ikinci temel formlar, birim normal vektör alanı, şekil operatörü ile Gauss eğriliği ve ortalama eğrilik açık olarak hesaplanmıştır. Son olarak, geliştirilen kuramsal teorilerin somutlaştırılması ve doğrulanması ayrıntılı örnekler üzerinden gösterilmiştir. Bu örneklerde, yönlü tubular hiperyüzeylere ilişkin kapsamlı sembolik hesaplamalara ve çeşitli izdüşüm projeksiyon uzaylarında elde edilen görselleştirmelere yer verilmiş, böylece hem yönlü tubular hiperyüzeylerin kuramsal metodolojisi hem de bu hiperyüzeylerin geometrik yorumları ortaya konmuştur.

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Cite This Article

Yağbasan, B., Ekici, C., Tozak, H. (2026). Tubular Hypersurfaces with Respect to the Quasi-frame in Four-dimensional Euclidean Space. International Electronic Journal of Geometry, Volume: 19 Issue: 2. https://doi.org/10.36890/iejg.1940183

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