In this paper, firstly the ruled surface drawn by Darboux vector is expressed as a quaternion. Then, spatial quaternionic definition of striction curve given and integral invariants are calculated. Finally, which corresponds to dual derived with help quaternions obtained.

In this paper, we give the parametric equation of Bishop frame for a timelike sweeping surface with unit speed curve in Minkowski 3-space. We introduce new geometric invariant to explain properties and local singularities surface. derive sufficient necessary conditions be developable ruled Afterwards, take advantage singularity theory classification Furthermore, some representative examples sho...

(i) is the most important one. We showed that extensions < = are classified by >@?A& 1 B DCE GF ( . The element 0 corresponded to a splitting. If one element is a non-zero multiple of the other, they correspond to the same , although different extensions. As an application, we proved: Proposition. Every rank 2 locally free sheaf on ? is a direct sum of invertible sheaves. I can’t remember if I ...

In this work, we present a new Bishop frame for the conjugate curve of in 3-dimensional Lie group G3. With help frame, derive parametric representation sweeping surface and show that curves on are curvature lines. We then examine local singularities convexity establish sufficient necessary conditions it to be developable ruled surface. Additionally, provide detailed explanations examples its ap...

Visual events in computer vision are studied from the perspective of algebraic geometry. Given a sufficiently general curve or surface in 3-space, we consider the image or contour curve that arises by projecting from a viewpoint. Qualitative changes in that curve occur when the viewpoint crosses the visual event surface. We examine the components of this ruled surface, and observe that these co...

We define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere, and show how this enables us to construct, from one such surface, a sequence of such surfaces. We also use the transforms to show how to associate to such a surface a corresponding ruled minimal Lagrangian submanifold of complex projective 3-space, which gives the converse of a construction...

In this paper, we prove that quantum layers over a surface which is ruled outside a compact set, nonplanar but asymptotically flat, admit a ground state for the Dirichlet Laplacian. The work here also represents some technical progress toward resolving the conjecture that a quantum layer over any nonplanar, asymptotically flat surface with integrable Gauss curvature must possess ground state. ©...

This paper investigated the rotation minimizing frames that are related to space curves and sweeping surfaces traced by these in three-dimensional Lie group. Then, sufficient necessary conditions for surface be a developable ruled were obtained. In particular, we mostly focused on study of resulting is cylinder, cone, or tangent surface. Meanwhile, support results paper, some illustrative examp...