arXiv subjects

Ferhat Taş

Publications and source records attributed to Ferhat Taş.

4 recordsLinked to original sources

arXiv2026

Brachistochrone-ruled timelike surfaces in Newtonian and relativistic spacetimes

We introduce and study \emph{brachistochrone-ruled timelike surfaces} in Newtonian and relativistic spacetimes. Starting from the classical cycloidal brachistochrone in a constant gravitational field, we construct a Newtonian ``brachistochrone-ruled worldsheet'' whose rulings are time-minimizing trajectories between pairs of endpoints. We then generalize this construction to stationary Lorentzian spacetimes by exploiting the reduction of arrival-time functionals to Finsler- or Jacobi-type length functionals on a spatial manifold. In this framework, relativistic brachistochrones arise as geodesics of an associated Finsler structure, and brachistochrone-ruled timelike surfaces are timelike surfaces ruled by these time-minimizing worldlines. We work out explicit examples in Minkowski spacetime and in the Schwarzschild exterior: in the flat case, for a bounded-speed time functional, the brachistochrones are straight timelike lines and a simple family of brachistochrone-ruled surfaces turns out to be totally geodesic; in the Schwarzschild case, we show how coordinate-time minimization at fixed energy reduces to geodesics of a Jacobi metric on the spatial slice, and outline a numerical scheme for constructing brachistochrone-ruled timelike surfaces. Finally, we discuss basic geometric properties of such surfaces and identify natural Jacobi fields along the rulings.

gr-qc↗

arXiv2025

A Spinorial Heat Flow Framework for Geometric Degeneration on $3$-Manifolds

We study a spinor-driven formulation of geometric evolution on closed $3$-manifolds, in which the spinor field is treated as the primary dynamical variable and the Riemannian metric is induced conformally by the spinor amplitude. We introduce a spinorial heat flow governed by the squared Dirac operator, \[ \partial_t ψ= - D_{g(ψ)}^{\,2} ψ, \] where the metric $g(ψ)$ depends nonlinearly on the evolving spinor field. As a consequence, the resulting system is quasi-linear and parabolic away from the nodal set $\{ψ=0\}$, while exhibiting degenerate behavior at vanishing spinor amplitude. We show that degeneration of the induced metric corresponds analytically to nodal behavior of the spinor field, rather than to curvature blow-up of the spinor evolution itself. This observation motivates an interpretation of geometric singularities as spinorial nodal transitions, across which the spinor field remains locally bounded in a weak or weighted sense. The induced metric evolution is derived explicitly and shown to be purely conformal, capturing only the trace component of curvature evolution and containing additional gradient terms that are not controlled \emph{a priori}. Accordingly, the proposed flow should not be identified with the Ricci flow, and any analogy with curvature smoothing is understood at a heuristic level. The present work establishes a coherent analytical framework for studying geometric degeneration via spinor dynamics and highlights several open problems in degenerate parabolic theory, including rigorous existence results and the precise role of nodal structures in geometric and topological evolution.

math.DG↗

arXiv2025

Developable Ruled Surfaces Generated by the Curvature Axis of a Curve

Ruled surfaces play an important role in various types of design, architecture, manufacturing, art, and sculpture. They can be created in a variety of ways, which is a topic that has been the subject of much discussion in mathematics and engineering journals. In geometric modelling, ideas are successful if they are not too complex for engineers and practitioners to understand and not too difficult to implement, because these specialists put mathematical theories into practice by implementing them in CAD/CAM systems. Some of these popular systems such as AutoCAD, Solidworks, CATIA, Rhinoceros 3D, and others are based on simple polynomial or rational splines and many other beautiful mathematical theories that have not yet been implemented due to their complexity. Based on this philosophy, in the present work, we investigate a simple way to generate ruled surfaces whose generators are the curvature axes of curves. We show that this type of ruled surface is a developable surface and that there is at least one curve whose curvature axis is a line on the given developable surface. In addition, we discuss the classifications of developable surfaces corresponding to space curves with singularities, while these curves and surfaces are most often avoided in practical design. Our research also contributes to the understanding of the singularities of developable surfaces and, in their visualisation, proposes the use of environmental maps with a circular pattern that creates flower-like structures around the singularities.

math.DG↗

arXiv2017

On the Design and Invariants of a Ruled Surface

This paper deals with a kind of design of a ruled surface. It combines concepts from the fields of computer aided geometric design and kinematics. A dual unit spherical Bézier-like curve on the dual unit sphere (DUS) is obtained with respect the control points by a new method. So, with the aid of Study [1] transference principle, a dual unit spherical Bézier-like curve corresponds to a ruled surface. Furthermore, closed ruled surfaces are determined via control points and integral invariants of these surfaces are investigated. The results are illustrated by examples.

cs.GR↗