in this paper, biharmonic slant helices are studied according to bishop frame in the heisenberg group heis3. we give necessary and sufficient conditions for slant helices to be biharmonic. the biharmonic slant helices arecharacterized in terms of bishop frame in the heisenberg group heis3. we give some characterizations for tangent bishop spherical images of b-slant helix. additionally, we illu...

We derive a complete set of invariants for a formal Bishop surface near a point of complex tangent with a vanishing Bishop invariant under the action of formal transformations. We prove that the modular space of Bishop surfaces with a vanishing Bishop invariant and with a fixed Moser invariant s < ∞ is of infinite dimension. We also prove that the equivalence class of the germ of a generic real...

The objective of the study in this paper is to define M₁,M₂-direction curves and M₁,M₂-donor curves of the spacelike curve γ via the Bishop frame of type-2 in E₁³. We obtained the necessary and sufficient conditions when the associated curves to be slant helices and general helices via the Bishop frame of type-2 in E₁³. After defining the spherical indicatrices of the associated curves, we obta...

Real data are often characterized by high dimensional feature vectors. However, such data contain redundant information that may not be beneficial for analysis algorithms. As such, feature transformation arises in related fields of study, including geoscientific applications, as a means to capture the few characteristics that are useful for pattern analysis algorithms. In this study, we investi...

Environment-mapped rendering of Lambertian isotropic surfaces is common, and a popular technique is to use a quadratic spherical harmonic expansion. This compact irradiance map representation is widely adopted in interactive applications like video games. However, many materials are anisotropic, and shading is determined by the local tangent direction, rather than the surface normal. Even for v...

In this paper, we will be concerned with the local biholomorphic properties of a real n-manifold M in C. At a generic point, such a manifold basically has the nature of the standard R in C. Near a complex tangent, however, the consideration can be much more complicated and the manifold may acquire a nontrivial local hull of holomorphy and many other biholomorphic invariants. The study of such a...