In the 1920’s Marston Morse developed what is now known as Morse theory trying to study the topology of the space of closed curves on S2 ([7], [5]). We propose to attack a very similar problem, which 80 years later remains open, about the topology of the space of closed curves on S2 which are locally convex (i.e., without inflection points). One of the main difficulties is the absence of the co...

In this study, the parametric equation of hypersurfaces passing through spacelike and timelike curves that has non-null Frenet vectors in E_1^4 Lorentz Minkowski space was expressed with help curve is frame. Furthermore, hypersurface families were created by giving necessary sufficient conditions so on common isoasymptotic hypersurface. Then, these obtained deviation marching-scale functions si...

Curvy roads are a particular type of urban road scenario, wherein the curvature centerline changes drastically. This paper is focused on trajectory planning task for autonomous driving curvy road. The prevalent on-road planners in Frenet frame cannot impose accurate restrictions curvature, thus easily making resultant trajectories beyond ego vehicle’s kinematic capability. Regarding Cartesian f...

‎In the present study‎, ‎we consider a regular curve in Galilean‎ ‎$4$-space $mathbb{G}_{4}$ whose position vector is written as a‎ ‎linear combination of its Frenet vectors‎. ‎We characterize such‎ ‎curves in terms of their curvature functions‎. ‎Further‎, ‎we obtain‎ ‎some results of rectifying‎, ‎constant ratio‎, ‎$T$-constant and‎ ‎$N$-constant curves in $mathbb{G}_{4}$‎.

In this paper, we introduce original definitions of Smarandache ruled surfaces according to Frenet-Serret frame a curve in E 3 . It concerns TN-Smarandache surface, TB-Smarandache and NB-Smarandache surface. We investigate theorems that give necessary sufficient conditions for those...

We develop a transfer matrix formalism to visualize the framing of discrete piecewise linear curves in three-dimensional space. Our approach is based on the concept of an intrinsically discrete curve. This enables us to more effectively describe curves that in the limit where the length of line segments vanishes approach fractal structures in lieu of continuous curves. We verify that in the cas...

Standard approaches to the computation of stereo correspondence have difficulty when scene structure does not lie in or near the frontal-parallel plane, in part because an orientation disparity as well as a positional disparity is introduced. We propose a correspondence algorithm based on differential and projective geometry, and inspired by neurobiology, that takes explicit advantage of both d...

The submanifold Dirac operator has been studied for this decade, which is closely related to Frenet-Serret and generalized Weierstrass relations. In this article, we will give a submanifold Dirac operator defined over a surface immersed in E with U(1)-gauge field as torsion in the sense of the Frenet-Serret relation, which also has data of immersion of the surface in E. MSC2000: 34L40, 53A05, 5...