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In this study, we investigate the focal surfaces associated with translation surfaces in Euclidean 3-space from the viewpoint of differential geometry. We begin by defining the translation surface generated by two planar curves and derive the corresponding focal surfaces using the framework of the Frenet frame. Analytical conditions are obtained under which the focal surfaces exhibit minimality or flatness. Several theorems are proven to classify the focal images, supported by illustrative examples. The results provide insights into the curvature structure of translation surfaces and contribute to the broader understanding of their geometric behavior.

This study explores parallel hypersurfaces in four-dimensional Euclidean space E[sup.4] , deriving explicit expressions for their Gaussian and mean curvatures in terms of the curvature functions of the base hypersurface. We identify conditions under which these parallel hypersurfaces are flat or minimal. The theory is applied to several key hypersurfaces, including rotational hypersurfaces, hyperspheres, catenoidal hypersurfaces, and helicoidal hypersurfaces, with detailed curvature computations and visualizations. These results not only extend classical curvature relations into higher-dimensional spaces but also offer valuable insights into curvature transformations, with practical applications in both theoretical and computational geometry.

The spherical product of two curves, composed of a total of n components, gives rise to spherical product surfaces in Euclidean space En, frequently resulting in surfaces of revolution, including superquadrics, which often exhibit inherent symmetry. When (n−1)-planar curves are considered, this construction enables the generation of hypersurfaces in n-dimensional spaces. Building upon this geometric framework, we conduct the first-ever investigation of spherical product hypersurfaces in the context of Minkowski geometry. We define these hypersurfaces in four-dimensional Minkowski space E14 and derive explicit expressions for their Gaussian and mean curvatures. We also determine the conditions under which such hypersurfaces are flat or minimal. Furthermore, we reinterpret certain hyperquadrics as specific instances of spherical product hypersurfaces in E14, supported by visual illustrations. Finally, we extend the construction to arbitrary-dimensional Minkowski spaces, providing a unified formulation for spherical product hypersurfaces across higher-dimensional Lorentzian geometries.

This study explores parallel hypersurfaces in four-dimensional Euclidean space E4, deriving explicit expressions for their Gaussian and mean curvatures in terms of the curvature functions of the base hypersurface. We identify conditions under which these parallel hypersurfaces are flat or minimal. The theory is applied to several key hypersurfaces, including rotational hypersurfaces, hyperspheres, catenoidal hypersurfaces, and helicoidal hypersurfaces, with detailed curvature computations and visualizations. These results not only extend classical curvature relations into higher-dimensional spaces but also offer valuable insights into curvature transformations, with practical applications in both theoretical and computational geometry.

In this study, some curvature conditions of quaternionic curves are obtained by writing the position vectors of quaternionic curves as linear combinations of new vector fields, named as D i  (1 ≤ i ≤ 4), that are produced from Frenet frame fields of the quaternionic curves. Also, the relations between D i D j ‐quaternionic curves and the rectifying quaternionic curves and the osculating quaternionic curves are presented. Moreover, the results obtained are illustrated with an example in which the new vector fields are examined.

In this paper, we study spinor Bishop equations of curves in E^3. We research the spinor formulations of curves according to Bishop frames in E^3. Also, the relation between spinor formulations of Bishop frames and Frenet frame are expressed.

Bu tez 5 bölümden oluşmaktadır.İlk bölümde spinorlar ve uygulama alanları ile ilgili kısa bir literatür bilgisi verilmektedir. İkinci bölümde bazı temel kavram ve özellikler verilmektedir. Üçüncü bölümde spinorlar ortonormal taban yardımıyla tanıtılmıştır. 4. bölümde ise 3 E Öklid uzayında eğriler hakkında bilgi verilmiş ve Frenet türev denklemlerinin spinorlar cinsinden ifadesi verilmiştir .Son bölümde ise yüzey üzerindeki eğrilerin spinor gösterimi Darboux türev denklemleri cinsinden verilmekte ve Frenet çatısının spinor gösterimi ile Darboux çatısının spinor gösterimi karşılaştırılmaktadır.

Here, we focus on focal surfaces of a tubular surface in Euclidean 3-space E^3: Firstly, we give the tubular surfaces with respect to Frenet and Darboux frames. Then, we define focal surfaces of these tubular surfaces. We get some results for these types of surfaces to become flat and we show that there is no minimal focal surface of a tubular surface in E^3. We give some examples for these type surfaces. Further, we show that u-parameter curves cannot be asymptotic curves and we obtain some results about v-parameter curves of the focal surface M^*.