Donghe Pei

In relativistic geometric optics, massless radiation propagates along null geodesics, while caustics arise where neighboring ray labels lose resolution. We study this focusing mechanism on the lightcone of a point event in four-dimensional Minkowski spacetime. Starting from an eikonal phase, we use an affine spacelike section of the cone as a screen of normalized propagation directions. A one-parameter anisotropic emission profile on this screen generates two lightcone dual surfaces, interpreted as families of spacelike wavefront cuts selected by stationary Lorentzian contact. Their induced metrics give exact transverse area Jacobians, so the vanishing of a Jacobian identifies kinematic focusing independently of spacetime curvature. The normalized hyperbolic curvature [Formula: see text] of the emission profile controls caustic onset and local morphology. Generic real caustic branches occur for [Formula: see text], while [Formula: see text] is a degenerate threshold for the second family. Along a generic caustic, [Formula: see text] produces a stable cuspidal ridge, whereas [Formula: see text] and [Formula: see text] produce a swallowtail transition. These results separate focusing caused by the organization of null propagation directions from Ricci or Weyl focusing and provide a flat-spacetime reference for geometric-optics caustics and their extension to curved null hypersurfaces.

Generalizing both hyperbolic framed surfaces and one‐parameter families of hyperbolic framed curves, we introduce the concept of hyperbolic generalized framed surfaces and establish their relations in hyperbolic 3‐space. We provide the necessary and sufficient conditions for a smooth surface to be a hyperbolic generalized framed base surface, followed by an analysis of the singularities of hyperbolic generalized framed base surfaces. Additionally, relations among hyperbolic generalized framed surfaces, generalized framed surfaces and lightcone framed surfaces are explored. As an application of hyperbolic generalized framed surfaces, we investigate the properties of horocyclic surfaces.

1. In differential geometry, the study of surface is one of the classical research topic that has been studied by many scholars [1–5]. A canal surface is a one-parameter smooth families of sphere w...

In this paper, we provide an effective tool for studying spacelike curves on null planes in mathematical and physical research. As an application, we propose a unified method for constructing generalized null Cartan curves.

This paper gives the definition of isophote curves with singularities in Minkowski 3-space and studies their geometric properties. As a key geometric consequence of the Lambert’s cosine law, isophote curves contribute to understanding the geometric and optical characteristics of objects in Minkowski space. Based on frame curve theory, we introduce the concept of non-lightlike framed isophote curves on non-lightlike surfaces. We analyze their light vectors based on their curvature functions and investigate their relationship with non-lightlike framed slant helices. Furthermore, we study the singularities and their criterion of non-lightlike framed isophote curves.

In this paper, we define generalized null Cartan rectifying curves which may have singular points. We propose a construction method for generalized null Cartan rectifying curves, which is more general than the one for regular null rectifying curves, as it also applies to non-regular cases. Moreover, we explore the relationships between generalized null Cartan rectifying curves, generalized null Cartan helices and centrodes.

Generalizing both hyperbolic framed surfaces and one-parameter families of hyperbolic framed curves, we introduce the concept of hyperbolic generalized framed surfaces and establish their relations in hyperbolic 3-space. We provide the necessary and sufficient conditions for a smooth surface to be a hyperbolic generalized framed base surface, followed by an analysis of the singularities of hyperbolic generalized framed base surfaces. Additionally, relations between hyperbolic generalized framed surfaces, generalized framed surfaces and lightcone framed surfaces are explored. As an application of hyperbolic generalized framed surfaces, we investigate the properties of horocyclic surfaces.

By Smarandache geometry, this paper introduces a class of ruled surfaces that are generated from spherical Legendre curves in the Euclidean 3-space. The developability and minimality of the generalized Smarandache ruled surface are investigated, as well as the geometric properties of the base curve. Moreover, since Legendre curves admit singularities, the generated ruled surfaces also contain singularities. Therefore, the singularities of the generalized Smarandache ruled surface are analyzed via the theory of framed curves and framed surfaces.

We define the principal normal surfaces of generalized null Cartan curves in Minkowski 3-space. We give the necessary and sufficient conditions for the relationship between the singularities of these surfaces and the finite type of their base curves. An analogous result does not hold in Euclidean 3-space, where the singularities of the principal normal surface depend also on the torsion of the base curve.