In the present article, we consider a parametric surface generated by Frenet frame of curve, and study minimality condition for surface. As result, give characterizations helicoid catenoid. Finally show some examples minimal surfaces circle helix.

A method to recognize polarization constellations is proposed. Described method is based on analysis of torsion and curvature of three-dimensional curve. The three-dimensional curve is defined by states of polarization on the Poincare sphere. Key-Words: Fiber optics; Polarization light; Birefringence; Poincare sphere; Space curve; Frenet frame

In this paper, we introduce the adapted frame of framed curves and give relations between Frenet type curve in four-dimensional Euclidean space. Moreover, define normal We obtain some characterizations terms their curvature functions. Furthermore, necessary sufficient condition for a to be curve.

We investigate the properties of a special kind of frame, which we call the Euler–Rodrigues frame (ERF), defined on the spatial Pythagorean-hodograph (PH) curves. It is a frame that can be naturally constructed from the PH condition. It turns out that this ERF enjoys some nice properties. In particular, a close examination of its angular velocity against a rotation-minimizing frame yields a cha...

The paper studies invariants for curves in three dimensional equiform Galilean geometry. We obtain Lie algebra structure space and concrete expressions of curve using Fels-Olver’s moving frame method. corresponding relationship between is received, the Frenet formula also showed.

New optical methods now allow experimentalists to track the trajectories of Lagrangian tracer particles in fluid flows at high Reynolds numbers. Independently, quaternions are used in the aerospace and computer graphics industries to track the paths of objects undergoing three-axis rotations. It is shown here that quaternions are a natural way of selecting an appropriate ortho-normal quaternion...

In this paper, we attain the problem of constructing hypersurfaces from a given geodesic curve in 4D Euclidean space E4. Using Serret–Frenet frame curve, express hypersurface as linear combination and analyze necessary sufficient conditions for that to be geodesic. We illustrate method by presenting some examples.

Through a 3D reconstruction of the human back surface using structured light techniques, we study the properties of spine curve by means of a set of parameters related to measures commonly applied in medicine. In this way, descriptors for measuring the abnormalities in the projections of the front and sagittal planes can be computed. We build the spine curve in 3D and analyse the behaviour of t...

We present an augmented reality application which introduces differential geometry in educational dynamic geometry software. New functionality such as a Frenet frame, center and circle of curvature in arbitrary curve points, and others were implemented. Dynamic geometry allows to study differential geometric properties under movement. Using this tool we developed examples which enable teachers ...