The movements of the spherical mechanisms take an important place in robotic studies. In this paper, design and trajectory motion equations joints were presented by methods sequential SLERP, fast SLERP with De-Moivre, modular interpolation that we firstly defined. Additionally, Serret Frenet frame curvatures geometrical curve for locations computed, then, several numerical examples on produce 2...

In this article, we examine the relationship between Darboux frames along parameter curves and frame of base curve ruled surface. We derive equations quaternionic shape operators, which can rotate tangent vectors around normal vector, find corresponding rotation matrices. Using these Gauss curvature mean explore how properties are found by use Frenet instead generator vectors. provide illustrat...

The role of differential geometry in describing a curve can not be denied. The differential forms defined for Bézier curves which are widely used in computer aided geometric design, plays a significant role in classification and image processing of curves. For this reason, the definitions such as Serret-Frenet frame, curvature and torsion which are described for Bézier curves are very important...

We inspect the geometry of proteins by identifying their backbones as framed polygons. We find that the left-handed helix region of the Ramachandran map for non-glycyl residues corresponds to an isolated and highly localized sector in the orientation of the Cβ carbons, when viewed in a Frenet frame that is centered at the corresponding Cα carbons. We show that this localization in the orientati...

In this paper, we study the problem of constructing a family of surfaces from a given spatial geodesic curve. We derive a parametric representation for a surface pencil whose members share the same geodesic curve as an isoparametric curve. By utilizing the Frenet trihedron frame along the given geodesic, we express the surface pencil as a linear combination of the components of this local coord...

We propose an algorithm for generating a moving coordinate frame on a space curve based on the concept of parallel transport. Such algorithms can be used for creating ribbons, tubes, and camera orientations that are smoothly varying and controlled by the curve geometry itself. The more familiar Frenet frame approach su ers from ambiguity and sudden orientation changes when the curve straightens...

in this paper, the problem of optimal path following for a high speed planing boat is addressed. first, a nonlinear mathematical model of the boat’s dynamics is derived and then the serret-frenet frame is presented to facilitate the path following control design. to satisfy the constraints on the states and the input controls of the boat's nonlinear dynamics and minimize both the cross tracking...

In the first part of this paper [6], a method to determine Frenet apparatus of the space-like curves in Minkowski space-time is presented. In this work, the mentioned method is developed for the time-like curves in Minkowski space-time. Additionally, an example of presented method is illustrated. Keywords—Frenet Apparatus, Time-like Curves, Minkowski Space-time.

This article is devoted to the rotation minimizing frames that are associated with spatial curves. Firstly we summarize some results concerning the differential geometry of the sweeping surfaces which are generated by these frames (the so–called profile or moulding surfaces). In the second part of the article we describe a rational approximation scheme. This scheme is based on the use of spatia...