Search Results (69)

This study develops a one-dimensional covariant Helmholtz model for a vector-valued wave field transported along a circular helical centerline and represented in the Frenet–Serret frame. For a helix with constant curvature $\kappa >0$ and torsion $\tau \ne 0$, the geometric coupling is described by a constant skew-symmetric connection matrix $\mathsf{\Omega }\in \mathfrak{so}\left(3\right)$. The covariant Helmholtz operator is shown to admit an exact gauge reduction to the flat componentwise Helmholtz operator through $u\left(s\right)=e^{-\mathsf{\Omega }s}y\left(s\right)$. Thus, within the one-dimensional centerline formulation, the helix preserves the operator spectrum while redistributing the observed Frenet components through parallel transport. The closed-form solutions show that a monochromatic input with wavenumber k is decomposed into a carrier and two geometric sidebands governed by the Darboux rotation rate $\lambda =\sqrt{{\kappa }^2+{\tau }^2}$. In the sub-geometric regime $k<\lambda$, the lower algebraic sideband is represented by the positive observable wavenumber $q_{-}=|k-\lambda |$, with associated scale $T_{\mathrm{beat}}=L_{-}=2\pi /q_{-}$. The lossless energy analysis proves conservation of the total averaged energy and its redistribution among the carrier and observable sidebands. A representative helical acoustic-channel design is then examined as a conceptual realization of the centerline model. Monte Carlo perturbations and additive-noise tests show that the predicted sideband locations, lower-sideband scale, and energy partition remain stable under prescribed fabrication tolerances and spectrally identifiable under weak and moderate measurement noise. Full article

In this paper, we investigate the problem of designing surface pencil pairs through Bertrand curves as iso-curvature lines in three-dimensional Euclidean space. By using the modified orthogonal frame and suitable marching-scale functions, we construct a pair of parametric surface pencils along a prescribed Bertrand pair. Necessary and sufficient conditions are derived for the Bertrand curves to be iso-curvature lines (isoparametric curves and curvature lines) on their corresponding surface pencils. The special case in which the curves also satisfy the isogeodesic condition is also discussed. In addition, several cases related to the torsion functions of the prescribed curves are analyzed. The construction is also extended to ruled surface pencil pairs, where the ruling directions are expressed with respect to the modified orthogonal frame. Finally, examples involving helical and planar Bertrand pairs are provided to show the effect of the controlling functions on the generated surface pencils. Full article

We develop a differential-geometric framework for variable-mass particles moving along non-lightlike curves with non-vanishing curvature in Minkowski 3-space ${\mathbb{E}}_1^3$, employing the Frenet–Serret apparatus adapted to a Lorentzian signature. The force is defined as the time derivative of momentum, $F=d\left(m\mathbf{v}\right)/dt$, incorporating mass variation through a Meshchersky-type reactive term; no covariant four-momentum formulation is assumed. Explicit closed-form expressions are derived for the momentum vector $\mathbf{P}\left(t\right)$, force $\mathbf{F}\left(t\right)$, and yank $\mathbf{Y}\left(t\right)=d\mathbf{F}/dt$ for three distinct causal types of regular Frenet curves: spacelike curves with a spacelike principal normal, spacelike curves with a timelike principal normal, and timelike curves. The tangential yank component carries the causal sign factor ${\delta }_{\mathbf{B}}$, reflecting the type of curve. A theorem on the evolution of kinetic energy separates the inertial contribution $mv\dot{v}$ from the reactive contribution ${\displaystyle \frac{1}{2}}\dot{m}{v}^2$ due to mass variation. A radial decomposition of the force in the osculating plane generalizes Siacci’s classical theorem to Lorentzian geometry and variable-mass systems. When the rectifying coordinate b is non-zero, a corresponding decomposition of the yank is also obtained. Three illustrative physical scenarios are discussed: rocket motion with variable mass (with potential future relevance to trajectory prediction, stability analysis, and motion-anomaly assessment in unmanned systems), a geometric analogy for orbital parameter changes, and particle motion in a magnetic monopole field. Two fully worked examples (a Lorentzian helix and a logarithmic spiral) provide explicit closed-form expressions for all geometric and dynamical quantities, accompanied by numerical plots. The results recover the Euclidean case in the appropriate signature limit. Full article

This paper develops a unified Lorentzian framework for persistent rigid motions generated by slant helices in three-dimensional Minkowski space and investigates their geometric and kinematic properties. Persistence is characterized by the constancy of the pitch of the instantaneous twist associated with a one-parameter rigid motion in the Poincaré group $\mathrm{ISO}(2,1)$. Interpreting curves in the motion group as trajectories of rigid motions, we study Frenet–Serret and adapted frame motions determined by slant helices under different causal characters. Necessary and sufficient conditions are established for these frame motions to generate persistent Lorentzian motions. An explicit intrinsic relationship between the pitches of Frenet–Serret and adapted frame motions is obtained in terms of the geodesic curvature of the spherical image of the principal normal indicatrix, showing that persistence is governed by intrinsic curve invariants and is independent of the chosen moving frame. The geometric structure of persistent motions is further clarified through associated ruled surfaces. In particular, the pitch of a persistent motion is shown to coincide with the distribution parameter of the ruled surface associated with the corresponding frame motion. Illustrative examples are presented for different causal configurations. These results extend classical Euclidean theory of persistent rigid motions to the Lorentzian setting and provide a unified framework connecting curve theory, frame geometry, ruled surfaces, and Lorentzian kinematics. Full article

This paper develops a unified superelliptic quaternionic framework for the generation and differential geometric analysis of curves and surfaces in affine three-space. Classical quaternionic methods provide an effective algebraic representation of rotations; however, they do not directly incorporate the radial deformation and anisotropic geometric behavior arising from superelliptic structures. To overcome this limitation, we combine quaternion multiplication with the superelliptic metric induced by the Gielis superformula and introduce a systematic construction of space curves and surfaces through superelliptic quaternion-valued functions. The proposed approach represents direction and radius curves as superelliptic quaternions and generates geometric objects by quaternionic rotation followed by projective normalization. This construction extends classical quaternion-based curve and surface generation by allowing rotational motion and superelliptic deformation to be handled within the same algebraic setting. Beyond geometric construction, the framework also provides explicit tools for differential geometric analysis. In particular, we derive the superelliptic Frenet frame associated with a curve and obtain formulations for curvature and torsion in terms of superelliptic quaternion functions. The theory is further extended to parametrized surfaces, where Gaussian curvature and mean curvature are expressed through the corresponding superelliptic quaternionic representation. The results demonstrate that superelliptic quaternions offer a flexible and mathematically coherent structure for linking rotation, deformation, geometric generation, and invariant computation. Therefore, the proposed framework contributes to differential geometry and geometric modeling by providing a unified method for constructing and analyzing a broad class of superelliptic curves and surfaces. Full article

This study investigates the robustness of Frenet–Serret curvature ($\kappa$) and torsion ($\tau$) estimates derived from noisy discretely-sampled three-dimensional space curves, with emphasis on the comparative performance of cubic spline and cubic Hermite interpolation methods. Accurate estimation of these geometric invariants is essential for reliable analysis of curves arising in signal processing and shape reconstruction; yet, the higher-order derivatives required for their computation exhibit pronounced sensitivity to measurement noise. We examine curves constructed through a Hilbert transform-based parameterization of the form $\mathrm{r}\left(t\right)=\left(X\left(t\right),A\left(t\right)sin\varphi \left(t\right),g\left(t\right)\right),$ where discrete samples are contaminated with additive white Gaussian noise at varying signal-to-noise ratios. Reconstruction is performed using cubic spline interpolation, which ensures global $C^2$ continuity, as well as cubic Hermite spline interpolation, which provides $C^1$ continuity with local tangent control. Frenet frame computations are then applied via regularized finite difference schemes. To characterize noise amplification theoretically, we derive the Curvature Stability Index (CSI) and Torsion Stability Index (TSI) as first-order variance bounds under the delta method. While these indices formalize the derivative-order dependence of noise sensitivity, Monte Carlo simulations reveal that empirical variance exceeds theoretical predictions by factors of ${10}^4$ to ${10}^6$, indicating dominance of nonlinear error propagation. Nevertheless, the indices establish that torsion instability arises fundamentally from third-order derivative structure rather than ground-truth magnitude. Numerical experiments across three geometric regimes constant-invariant helices, variable-curvature helices, and planar curves with identically zero torsion demonstrate that the ratio of the torsion root mean square error to curvature root mean square error consistently ranges from $6.5$ to $9.8$. This disparity persists even in the degenerate planar case, where $\tau \equiv 0$ analytically, confirming that torsion sensitivity is an intrinsic property of the Frenet–Serret formulation. Across all configurations, cubic spline reconstruction yields lower Monte Carlo mean RMSE and reduced empirical variance compared to Hermite spline, providing superior stability for derivative-based invariant estimation. Full article

The Gielis superformula is a powerful parametric tool that generates an infinite variety of natural and organic curves and surfaces through a compact set of parameters. However, classical differential geometry has lacked a unified framework for analyzing their curvature, torsion, and intrinsic geometric properties. This study addresses this gap by developing a novel superelliptic geometric framework that integrates the superformula with the differential geometry of curves and surfaces. We define the superelliptic inner and cross products, the star derivative, and the superelliptic Frenet frame to extend Euclidean and Riemannian interpretations of curvature and torsion to a more flexible parametric structure. The framework provides a uniform geometric characterization of all Gielis curves and surfaces in an intrinsic sense with respect to the proposed superelliptic metric, rather than relying on their classical Euclidean parametric representations; singular cases (e.g., $n_1<2$), which correspond to non-smooth or corner-like behavior in the Euclidean setting due to degeneracies in the radial function $r\left(t\right)$, are regularized within this framework, since the induced metric maps such Gielis-type curves to intrinsically circular geometries with constant superelliptic curvature. This unifies the entire family under a common, robust foundation while preserving orthonormality and differentiability. This superelliptic approach offers a consistent and computationally tractable model that bridges mathematical abstraction with real-world morphology, with the superformula serving as a representative example of the framework’s broad generality for diverse geometric structures. The proposed theoretical framework is further supported by computational visualization, and all figures and numerical illustrations presented in this study were generated using MATLAB R2024a, ensuring a consistent implementation of the proposed superelliptic model. Full article

This paper investigates and characterizes Smarandache space curves, an important class of curves, using the Caputo fractional Frenet frame. The Frenet frame and fractional curvature functions have been calculated for these fractional Smarandache curves. To demonstrate the theoretical results obtained, an example of a fractional Smarandache curve derived from a helical curve is considered, and the curvatures of this curve are explicitly calculated. Finally, to show the effect of the fractional order parameter on the geometric behavior, a graphical analysis of the curvatures obtained for different fractional orders is presented. Full article

Fiber-optic shape sensing enables real-time monitoring of structural deformation across a wide range of applications. For large-scale structures, Brillouin-based distributed sensing, typically implemented through Brillouin Optical Time Domain Analysis (BOTDA), offers an extended range for quasi-static measurements, albeit its limited spatial resolution degrades reconstruction accuracy. This study addresses this fundamental limitation through the introduction of a novel error compensation algorithm, particularly suited for a Brillouin-based shape sensing system, yet agnostic with respect to the sensing technology. The method leverages both the initial and final points of the sensing path, performing both forward and backward reconstructions and fusing the two trajectories by testing several polynomial and exponential weighting strategies. The algorithm is experimentally validated on a 28.91 m four-core shape sensing fiber cable (length = L), interrogated through BOTDA operating at 50 cm spatial resolution, and reconstructed through the Frenet–Serret frame formulation. Calibration procedures include radial-offset tuning and segment alignment via a hotspot reference. A non-trivial S-shaped geometry is adopted as a case study, specifically addressing curvature discontinuities arising from mixed straight and curved segments. Reconstruction accuracy is quantified through a Euclidean-distance-based Figure of Merit (FOMs). The cubic weighting strategy demonstrates improvements exceeding 86% in all FOMs compared to classical methods without compensation. Specifically, it achieves an RMSE of 0.145 m (0.50% of L), a MAE of 0.109 m (0.38% of L), and a maximum error of 0.341 m (1.18% of L). Remarkably, these percentage errors are of the same order of magnitude as those reported in the literature for Fiber Bragg Grating (FBG) and Optical Frequency Domain Reflectometry (OFDR) systems, indicating that the proposed compensation strategy enables BOTDA-based shape sensing to achieve comparable reconstruction accuracy despite its lower spatial resolution. Full article

We propose a comprehensive rotation-minimizing (RM) Darboux framework for the study of curve theory and relativistic ruled surfaces in Minkowski three-space ${\mathbb{E}}_1^3$. The construction merges the adaptability of the classical Darboux frame to surface geometry with the reduced rotational behavior characteristic of RM frames, yielding a natural geometric description of curves in a Lorentzian environment. For unit speed non-null curves, the governing equations of the RM Darboux frame are derived, and precise connections between the RM curvature functions and the classical Frenet and Darboux invariants are obtained, thereby elucidating the geometric significance of RM curvatures in Lorentzian geometry. Within this setting, multiple classes of ruled surfaces are generated using RM Darboux frame vector fields. Necessary and sufficient conditions for developability, minimality, and flatness are formulated exclusively in terms of RM curvature quantities. The role of the causal character of the generating curve is analyzed in detail, revealing distinct geometric behaviors for space-like and time-like cases. These findings indicate that the RM Darboux framework constitutes a flexible and effective approach for modeling curve-induced surface geometries in Minkowski space, with potential relevance to relativistic kinematics, world sheet constructions, and geometric problems arising in mathematical physics. Full article

Bertrand curve pairs share the same principal normals, creating a geometric symmetry useful in design and modeling. The geometry of a surface can be characterized and studied through three types of characteristic curves: geodesics, curvature lines, and asymptotic curves. We introduce a method to construct corresponding surface family pairs from a Bertrand curve pair, ensuring that both curves serve as the same type of characteristic curve on each surface family, thereby extending curve symmetry to surface symmetry. We build surface pairs by linearly combining the Frenet frame of a Bertrand curve, with coefficients acting as shape functions. We establish necessary and sufficient conditions that these functions must satisfy to guarantee that the Bertrand curves become the same characteristic type on both surface families. This provides flexible control over surface geometry and curve type. We further derive conditions for developable surface pairs, proving that no developable pair can contain a twisted Bertrand curve as a curvature line or asymptotic curve. To illustrate this, we construct surface pairs from a circular helix, and the resulting surfaces exhibit an aesthetically pleasing symmetry, demonstrating the flexibility and interactivity of our framework. Full article

This paper provides several new characterizations of the infinitesimal bending of dual curves, which is defined as an infinitesimal deformation preserving dual arc length (with appropriate precision). The main goal is to consider the infinitesimal deformations of ruled surfaces through the corresponding deformations of dual curves. Some useful properties of the infinitesimal bending of dual curves are obtained, and dual bending fields are determined. The Vekua-type characterization of the infinitesimal bending of dual curves is formulated in terms of the derivative of the dual arc length. Explicit formulas for dual infinitesimal bending fields of dual spherical curves are obtained using the Blaschke frame, considering both an arbitrary real parameter and the dual arc length. A necessary and sufficient condition for the infinitesimal bending of the dual curve to lie on the dual unit sphere is presented in terms of Blaschke and Frenet invariants. Several examples are illustrated graphically using our own software tool. Full article

In Minkowski 3-space, we establish a geometric framework to osculate Type-II ruled surfaces by utilizing the Type-II Bishop frame in $\left({\mathbb{E}}_1^3\right)$. Our analysis extends to higher-order singularities such as butterflies and pyramids, including explicit singularity loci. We also compare Type-II Bishop frames with rotation-minimizing frames using timelike base curves and spacelike normals. With RK4 integration, we develop a robust computational model for Weingarten surfaces and subclasses with constant curvature. The theoretical foundation for Type-II Bishop frames is extended to higher-dimensional Minkowski spaces ${\mathbb{E}}_1^n$ for $n>3$ through generalized Frenet-type equations and curvature functions. We determine exact stability conditions under perturbations of Bishop curvature using advanced singularity theory. The numerical implementations of our methods, including geometric modeling and relativistic geometry, demonstrate their effectiveness in both theoretical and applied contexts. Full article

In this study, equidistant ruled surfaces generated by the Darboux vector, which has significant kinematic importance and characterizes the instantaneous rotation of a moving frame, are investigated specifically for the Frenet frame. By establishing a structural relationship between a surface and its equidistant ruled surface, transition formulas are provided for shape operators, Gaussian and mean curvatures, and fundamental forms, revealing that the equidistant surface is a scaled transformation of the original one. The obtained results demonstrate that both surfaces are developable and that the geometric properties of the equidistant ruled surfaces can be expressed dependently on each other. Furthermore, it is shown that the geometric character of the equidistant surface, including the invariance of asymptotic lines and the preservation of umbilical points under constant angle conditions, is determined by the rotational dynamics of the base curve. These findings constitute a theoretical foundation for cases involving the use of Darboux axes of different frames in higher dimensions or the investigation of similar structures in different geometric spaces. The geometric interpretation of this theoretical framework is elucidated through the fundamental properties of the surfaces. Finally, a concrete example is presented, where the symmetry of the central planes of the equidistant ruled surfaces at appropriate points is visualized using Maple 2017 software. Full article

In this study, we investigate the focal surfaces associated with translation surfaces in Euclidean 3-space from the viewpoint of differential geometry. We begin by defining the translation surface generated by two planar curves and derive the corresponding focal surfaces using the framework of the Frenet frame. Analytical conditions are obtained under which the focal surfaces exhibit minimality or flatness. Several theorems are proven to classify the focal images, supported by illustrative examples. The results provide insights into the curvature structure of translation surfaces and contribute to the broader understanding of their geometric behavior. Full article