Abstract
In this paper, we propose a method for constructing families of spacelike surfaces in Minkowski 3-space that share Bertrand curves as asymptotic curves. By using marching-scale functions, we derive the necessary conditions, provide flexible formulations, and establish a framework for constructing mutual spacelike Bertrand curves. Examples show how different functions generate surfaces interpolating the common asymptotic curves, offering new insights for geometric modeling and ruled surface theory.
1 Introduction
Among the various curves found on surfaces, asymptotic curves hold a special place due to their distinct geometric behavior. An asymptotic curve is one where the tangent vector continuously aligns with a direction that renders the normal curvature zero. In practical terms, this means the surface remains momentarily aligned with its tangent plane along the curve, a condition further reinforced by the persistent alignment of the curve’s binormal vector with the surface normal. Consequently, the Gaussian curvature along such a curve is non-positive—that is, it is either zero or negative [, ]. Recent investigations have emphasized the importance of these curves, particularly in applications such as astronomy. For example, Hartman and Wintner [] stressed that regularity conditions are critical for understanding asymptotic curves on surfaces with negative Gaussian curvature, whereas Kitagawa [] showed that a flat torus embedded in a unit 3-sphere features cyclic asymptotic curves. Further studies by Garcia and Sotomayor [] have shed light on the intrinsic properties of asymptotic curves in Euclidean spaces, and Garcia et al. [] demonstrated that networks of these curves can maintain topological stability under small perturbations. Moreover, asymptotic curves play a significant role in diverse fields such as architecture and computer-aided design. In astrophysics, for instance, the analysis of Lyapunov orbits—which is crucial for understanding stellar escape trajectories—relies heavily on the geometry of asymptotic curves. Contopoulos [] examined how unstable orbits can escape along these curves, whereas Efthymiopoulos et al. [] observed that chaotic trajectories within fractal-like sets often mirror the behavior of unstable asymptotic curves. In the realm of free-form architecture, Flöry and Pottmann [] developed a geometric modeling framework that uses strips of ruled surfaces, constructed by aligning rulings with asymptotic curves, and refined these initial designs to match specific target shapes.
In a related development, the concept of a surface family featuring a unique characteristic curve was first introduced by Wang et al. [] in Euclidean 3-space , where they constructed a family of surfaces sharing a common geodesic. This pioneering work has inspired numerous subsequent studies on surface families with shared key curves in both Euclidean and non-Euclidean contexts (e.g., [–32]). In curve theory, establishing a robust correspondence between curves remains a fundamental challenge. One well-known example is the Bertrand pair, which is a classical case where two curves are in bijective correspondence and share the same principal normals [, ]. These Bertrand curves also serve as models for offset curves, which are integral to computer-aided design and manufacturing processes (see [33–35]). Despite these advances, no prior research has focused on constructing spacelike surfaces in that incorporate Bertrand curves as asymptotic curves. In this study, we seek to bridge that gap by exploring how Bertrand curves can be utilized as asymptotic curves to generate families of spacelike surfaces in .
In this study, we introduce a method for constructing families of spacelike surfaces in Minkowski 3-space that share a common Bertrand curve as an asymptotic curve. By utilizing spacelike Bertrand curves and aligning the tangent planes of the surfaces with the osculating planes of the curves, parametric equations of the surfaces are developed using marching-scale functions. Necessary and sufficient conditions are derived to guarantee the asymptotic nature of the common curve on each surface. Several flexible formulations are proposed, and examples illustrate the generation of spacelike surfaces through different choices of marching-scale functions. The significance of this work lies in providing a systematic and versatile framework for generating geometrically meaningful surfaces, enhancing the theoretical understanding of Lorentzian geometry, and offering practical tools for applications in computer-aided geometric design, relativity, and broader differential geometry contexts.
2 Preliminaries
In this section, we provide a concise overview of the fundamentals of curves and surfaces in Minkowski 3-space [, ]. Consider vectors and in . Their Lorentzian inner product is defined by
We also define the cross product of and as
As the Lorentzian inner product is an indefinite metric, any vector can be classified by its causal character. Specifically, is called spacelike if or , timelike () if , and lightlike or null if while . The norm of is defined as . Accordingly, the hyperbolic unit sphere and the Lorentzian (de Sitter) unit sphere are given byand
Consider a unit-speed -curve in that possesses a -binormal vector. Its curvature and torsion are denoted by and , respectively. Let be the corresponding Serret–Frenet frame along , where , , and represent the unit tangent, principal normal, and binormal vectors, respectively. The frame vectors satisfy the following normalization conditions under the Lorentzian inner product:
The derivatives of the frame vectors with respect to the arc-length parameter are then expressed aswith the prime indicating differentiation with respect to . The subspaces spanned by , , and are referred to as the normal plane, osculating plane, and rectifying plane, respectively.
Definition 2.1([, ]) Let and be two curves in , with the respective principal normal vectors and. The curves form a Bertrand pair if, at corresponding points, their principal normals are linearly dependent. Equivalently, there exists a constant such thatHere, is a fixed scalar.
We denote a surface by its parametrization
Here, if we let , for and , the surface’s normal vector is defined aswith the property that .
Definition 2.2([, ]) A curve on a surface is called asymptotic if, at every point on the curve, the surface normal is aligned with the curve’s binormal vector.
A curve on a surface is called an iso-parametric curve if one of its parameters remains constant. In other words, there exists a fixed value such that , or similarly, a constant with . When a given curve serves both as an asymptotic curve and as an iso-parametric curve on , we refer to it as an iso-asymptotic of the surface.
Definition 2.3A surface in Minkowski 3-space is termed a surface if its induced metric is Lorentzian, and it is called an surface if the induced metric is a positive-definite Riemannian metric. Equivalently, a surface has a timelike normal vector, whereas a surface possesses a -normal vector.
3 Main results
In this section, we present a novel method for constructing families of surfaces in that share Bertrand curves as common asymptotic curves. Our approach leverages -Bertrand curves in such a way that the tangent planes of the resulting surfaces align with the osculating planes of these -Bertrand curves. To achieve this, we assume that the curves and are -Bertrand curves with -binormal vectors, as described in Equations 2.1, 2.2. Based on this assumption, one can define a surface family bywhich constitutes an -surface family sharing the curve as mutual curve in common. Similarly, the surfacealso forms an -surface family with as a common curve. In these constructions, the functions and (which belong to ) are termed marching-scale functions, with the condition for some . To ensure that is an asymptotic curve on , we must determine the appropriate conditions for the marching-scale functions. Computing the partial derivatives of giveswhere subscripts denote partial differentiation with respect to and , respectively. The surface normal is then defined by
As is an iso-parametric curve on , there exists a parameter value such that . This implies that
Evaluating the normal vector at then simplifies to
This result shows that the surface normal along is parallel to the binormal vector, ensuring that is indeed an -asymptotic curve on . From the relations in Equations 3.1–3.6, we consequently derive the following theorem.
A curve is an iso-asymptotic (i.e., an asymptotic) curve on the -surface family if and only iffor some and for all .
For simplification and practical evaluation, we assume that the marching-scale functions and can be separated as follows:where , , and are functions that are not identically zero. From Theorem 3.1, we then obtain the following:
Corollary 3.1The curve is an asymptotic curve on the -surface family if and only ifwith and .
To construct surfaces
and
that interpolate
and
as common asymptotic
-Bertrand curves, we first define the marching-scale functions as in
Equation 3.7and then substitute these into
Equations 3.1,
3.2to obtain the parametrization. Moreover, for practical flexibility, the functions
and
can be expressed in alternative forms while still allowing sufficient freedom to ensure that the surfaces
and
interpolate
and
as mutual asymptotic curves. Therefore, we assume that
and
may also be given in two additional formats:
- 1.
We assume that the marching-scale functions are defined by
where
,
,
, and
belong to
; the coefficients
and
are real numbers for
; and neither
nor
is identically zero. In this setting, the sufficient conditions to ensure that the pair of curves {
,
} are common asymptotic curves on the surfaces {
,
} are given by
- 2.
Alternatively, we assume the functions take a compositional form:
then the corresponding conditions become
where
,
, and
, with
,
for
, and
,
, and
are not identically zero. As there are no additional restrictions imposed by the given curve in
Equations 3.8,
3.9or
Equation 3.10, one can always construct the pair of surfaces {
,
} with {
,
} as their mutual asymptotic curves by suitably choosing these flexible marching-scale functions.
Example 3.1We consider the curveThen one may define the associated frame vectors asWith these, the -surface family is given by where .Now, if we set in Equation 2.1, then the curve is modified asFor this modified curve, the corresponding frame vectors becomeThus, the -surface family is now parameterized by
In each case, the functions
and
(the marching-scale functions) provide the necessary degrees of freedom to generate the family of
surfaces, with
serving as the common asymptotic curve.
1. In the case, where , , , and , the conditions in Equation 3.8 are satisfied. Consequently, the constructed pair of surfaces {, } (Figure 1) features on the surface , whereas the green curve represents on the surface .
2. If we define
for
and
, with
, and coefficients
,
, then the conditions given in
Equation 3.9are fulfilled. For example, if we choose
, the corresponding pair of surfaces {
,
} is illustrated in
Figure 2. In this figure, the blue curve represents
on
, whereas the green curve represents
on
.
3. If we define
where
,
, and set
, then the criteria outlined in
Equation 3.10are satisfied. Specifically, when choosing
for all
, the corresponding surfaces {
,
} are illustrated in
Figure 3. In this visualization, the blue curve denotes
on
, whereas the green curve represents
on the corresponding surface
.
FIGURE 1
FIGURE 2
FIGURE 3
3.1 ruled surfaces with -Bertrand curves
In this subsection, we analyze the structure of -ruled surfaces that incorporate -Bertrand curves as asymptotic curves. For the ease of interpretation, let be a unit-speed -curve with a -binormal vector in . We consider as an -ruled surface whose base curve is , and we assume that is also an iso-parametric curve of . Then, there exists a specific value , such that . This leads to the following relation:where is a -unit vector along the rulings. Using Equation 3.2, we obtainwhich represents a system of two equations involving the unknown functions and . To express these explicitly, we use
Equation 3.11 gives the necessary and sufficient conditions for to be a -ruled surface. According to Theorem 3.1, if the curve is an asymptotic curve on , then . Consequently, at any point along , the ruling direction belongs to the span of . Furthermore, and must not be collinear, leading to the expressionfor some real functions and . As a result, the family of iso-parametric -ruled surfaces sharing the common -asymptotic curve can be expressed asfor certain real-valued functions and . The unit normal to the surface is given by
Evaluating at , which corresponds to the curve , we obtain
Thus, remains an asymptotic curve on .
The pair of surfaces{, } interpolates the curves {, } as mutual asymptotic -Bertrand curves if and only if there exists and functions and such that and are represented by Equation 3.12 andfor some real-valued functions and .
It is important to highlight in Equation 3.12 (respectively, Equation 3.13) that a asymptotic curve passes through every point on the curve (respectively, . One of these curves is (rersp. itself, whereas the other corresponds to a -line aligned with the direction ((respectively, , as described in Equation 3.12 (respectively, (Equation 3.13)).
Example 3.2
Building upon
Example 3.1, we consider the following cases:
1. For and , the surfaces {, } interpolating {, } as mutual asymptotic -Bertrand curves are given as follows (Figure 4):
and
where
and
. In
Figure 4, the blue curve represents
on
, whereas the green curve illustrates
on
.
2. If , then the surfaces {, } interpolating {, } as mutual asymptotic -Bertrand curves are given by (Figure 5):
and
where
and
. In
Figure 5, the blue curve represents
on
, whereas the green curve corresponds to
on
.
3. If and , then the surfaces {, } interpolating {, } as mutual asymptotic -Bertrand curves are given by (Figure 6):
and
where
and
. In
Figure 6, the blue curve represents
on
, whereas the green curve corresponds to
on
.
FIGURE 4
FIGURE 5
FIGURE 6
4 Conclusion
In this study, we propose a method for constructing families of spacelike surfaces in Minkowski 3-space that share a common Bertrand curve as an asymptotic curve. By aligning surface tangent planes with the osculating planes of spacelike Bertrand curves and employing marching-scale functions, we establish a flexible parametrization framework and derive the necessary and sufficient conditions for asymptoticity. Examples highlight the versatility of the method, which enriches surface modeling in Lorentzian geometry and offers applications in differential geometry, relativity, and computer-aided design. Potential extensions include timelike or null surfaces, higher dimensions, and dynamic surface evolution.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.
Author contributions
AA: Data curation, Software, Resources, Conceptualization, Funding acquisition, Writing – review and editing, Writing original draft, Formal analysis, Investigation. RA-B: Supervision, Writing – review and editing, Conceptualization, Software, Investigation, Methodology, Funding acquisition, Resources, Formal analysis, Project administration, Writing – original draft, Validation, Data curation, Visualization.
Funding
The author(s) declare that financial support was received for the research and/or publication of this article. This study was financially supported by the Princess Nourah bint Abdulrahman University Researchers Supporting Project (PNURSP2025R337).
Acknowledgments
The authors would like to acknowledge the Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R337), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declare that no Generative AI was used in the creation of this manuscript.
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Summary
Keywords
Bertrand couple, tangent planes, iso-asymptotic curve, asymptotic, spacelike
Citation
Almoneef AA and Abdel-Baky RA (2025) Spacelike surface families interpolating common asymptotic curves in Minkowski 3-space. Front. Phys. 13:1650000. doi: 10.3389/fphy.2025.1650000
Received
19 June 2025
Revised
09 October 2025
Accepted
13 October 2025
Published
26 November 2025
Volume
13 - 2025
Edited by
Janos Polonyi, Université de Strasbourg, France
Reviewed by
Andreas Gustavsson, University of Seoul, Republic of Korea
Kemal Eren, Sakarya University, Türkiye
Updates
Copyright
© 2025 Almoneef and Abdel-Baky.
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*Correspondence: Areej A. Almoneef, aaalmoneef@pnu.edu.sa
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All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.