Spacelike Sweeping Surfaces and Singularities in Minkowski 3-Space

Mofarreh, Fatemah; Abdel-Baky, Rashad A.; Alluhaibi, Nadia

doi:https://doi.org/10.1155/2021/5130941

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5130941 | https://doi.org/10.1155/2021/5130941

Spacelike Sweeping Surfaces and Singularities in Minkowski 3-Space

Fatemah Mofarreh,¹Rashad A. Abdel-Baky,²and Nadia Alluhaibi³

Academic Editor: Alessandro Gasparetto

Received31 May 2020

Revised23 Dec 2020

Accepted12 Jan 2021

Published30 Jan 2021

Abstract

We considered the spacelike sweeping surface with rotation minimizing frames at Minkowski 3-space . We presented the new geometric invariant to demonstrate geometric properties and local singularities for this surface. Then, we derived sufficient and necessary conditions of the surface to become developable ruled surfaces. Additionally, its singularities are studied. Finally, examples are illustrated to explain the applications of the theoretical results.

1. Introduction

The vision of the computer is the automatical analysis of image sequences in order to build the 3-dimensional surface form. Recently, various majors of mathematics are used for computer vision and medical imaging. Projective geometry, as an old mathematical subject, still used to characterize connections between both lines and points in several images to the same theme. In addition, differential geometry is used to characterize the shape of the curve and surface in engineering. Both Rene Thom, French mathematician, and Hassler Whitney introduced some developments in mathematical thinking and methods, especially the concept of singularity theory that contains catastrophes and bifurcations. Singularity theory now as the direct application of differential calculus is important to gain vital results in many subjects as computer vision and medical imaging (e.g., [1–7]).

A canal surface is the surface that can be generated by the one-parameter set of spheres determined by a radius function and center curves: in case a radius function is a constant function, the canal surface is the envelope of the moving sphere and named the sweeping surface. Some well-known examples of the sweeping surface are circular cylinder and circular cone (radius of spheres is not constant), surface of revolution, and Dupin cycloids. More specific, the sweeping surface named the tubular surface in case the radius of the generating spheres is constant. Sweeping surfaces are very important for descriptive geometry, especially for solid and surface modeling at computer-aided design, computer-aided manufacturing (CAD/CAM), and the design of trajectory movement for robots [8–14]. It is a fact that a sweeping surface can be a developable surface. The developable surface defines the surface that can become unfolded (or developed) to the plane with the absence of any stretch or tear. As known at differential geometry, with considering sufficient differentiability, the developable surface defines the plane, conical surface, cylindrical surface, or tangent surface of the curve or the structure of one of those kinds. Therefore, the developable surface is considered as the ruled surface, such that every point at the same ruling shares the common tangent plane. The rulings are principal curvature lines which vanish normal curvature and Gaussian curvature that is vanishing at every point. As a result, the developable surface is a significant surface in (CAD/CAM) and geometric modeling as it is used for motion analysis or designing cars and ships [15–19].

The essential tool to analyze the curve and surface at differential geometry is the Serret–Frenet frame that is the most used frame at Euclidean 3-space and Minkowski 3-space [16–20]. The main apparatus in the previous literatures are Serret–Frenet formulas and some linked functions on the curve as a distance-squared function in addition to the height function. Based on Serret–Frenet formulas, the singularity of those functions can be studied from the view of extrinsic differential geometry. Because the Serret–Frenet will not be defined everywhere, there is a need for a new mathematical tool to be used for analysis purposes. In [20], Bishop gave the alternative moving frame to points on the curve at Euclidean 3-space using parallel vector fields. It named rotation minimizing frame (RMF) or Bishop frame of the curves [19–21]. Analogous to Bishop frame in Euclidean 3-space, there is a similar Lorentzian frame which named Lorentzian Bishop frame, constructed along the curve at Lorentzian space, and it is the analog of the Bishop type frame as applied to Lorentzian geometry. At Lorentzian space, using Minkowski Bishop frame through the curve as a basic tool is preferred than using Serret–Frenet frame [21–23].

In fact, there is no more literature review regarding singularities of sweeping surfaces relating to the Minkowski Bishop frame. Therefore, this study aims to cover some needs, where it is inspired by the study of Izumiya et al. [7] and Bishop [19]. At this study, we establish the Lorentzian Bishop frame along the unit speed spacelike curve with timelike principal normal and develop the local differential geometry of spacelike sweeping surfaces at Minkowski 3-space. Using unfolding theory at singularity theory in this study, generic singularities of this sweeping surface are classified. A new invariant relating to singularities of this sweeping surface is presented. It is founded that generic singularities of this sweeping surface are cuspidal edge and swallowtail, and these kind of singularities can be characterized by this invariant, in the same order. Afterward, we solved this problem of requiring the surface which is the spacelike sweeping surface and, at the same time, the spacelike developable surface. Some examples are introduced in order to demonstrate theoretical results.

2. Preliminaries

Some definitions and basic concepts are given which will be used (for instance [8, 24, 25]). Suppose is 3-dimensional Minkowski space, the 3-dimensional real vector space considers the metricwhere denotes the canonical coordinates in . Any vector of named spacelike in case or , timelike in case , and lightlike or null in case and . The timelike or lightlike vector at is named causal. Also, with the norm , the vector is the spacelike unit vector if and a timelike unit vector if . Therefore, we say that a smooth map is spacelike, timelike, or lightlike, if its velocity vector is spacelike, timelike, or lightlike, in the same order. Similarly, the surface is named spacelike, timelike, or lightlike if its tangent planes are spacelike, timelike, or lightlike, respectively. For any two vectors , , the inner product is a real number , and the vector product is given aswhere , , are the canonical bases of . For a fixed point and the positive number , hyperbolic and Lorentzian (de Sitter space) spheres, in the same order, are given as

We defineand it is called the (open) lightcone at the vertex . In case and , we define , , in addition to , in the same order.

Suppose is the unit speed spacelike curve with timelike principal normal in , and suppose and are the natural curvature and torsion of , in the same order. Suppose , , is a Serret–Frenet frame related to the curve , then the Serret–Frenet formulas readwhere is the Darboux vector of the Serret–Frenet frame. At this study, denotes the derivatives of respecting to arc length parameter. The Serret–Frenet vector fields satisfy the following relations:

Definition 1. The moving pseudoorthogonal frame , , , along the nonnull space curve , is rotation minimizing frame (RMF) respecting to in case its angular velocity insures or, similarly, the derivatives of and are both parallel to . Analogously, characterization holds when or is selected to be the reference direction.
Using Definition 1, it is observed that the Serret–Frenet frame is RMF respecting to the principal normal but not respecting to the tangent and the binormal . Even though the Serret–Frenet frame is not RMF respecting to , it is easy to derive similar RMF from it. New normal plane vectors () are determined along the rotation of () aswith a certain Lorentzian timelike angle . The set , , is called RMF or Bishop frame. The RMF vector insure the following relations:Therefore, the Bishop frame readswhere is the Bishop Darboux vector. Here, the Bishop curvatures are determined as and . Comparing equations (5) and (9), it is observed that the relative velocity isOne can show thatAs a consequence, both Serret–Frenet frame and RMF identical iff is the planar, which means .
The spacelike vector is defined asand we name it the modified Bishop Darboux vector through . A Bishop spherical Darboux image is defined byTherefore, we consider a new geometric invariant .

Definition 2. The sweeping surface through is the surface given aswhere named the (at least -continuous) spine curve, , and is the arc length parameter. is the planar profile (cross-section) curve defined as parametric presentation, , “T” refers to transposition, and . The semiorthogonal matrix , , specifies the RMF through .
Kinematically, the sweeping surface is generating by the moving of the profile curve through the spine curve with the orientation as introduced by . The profile curve is in the 2D or 3D space that passes through the spine curve during sweeping. Interestingly, RMF allows for a simple characterization of spine curve.

Definition 3. The surface at Minkowski 3-space named the timelike surface in case the induced metric at the surface is the Lorentz metric and also it named the spacelike surface in case the induced metric at the surface is a positive definite Riemannian metric, which means the normal vector on spacelike (timelike) surface is the timelike (spacelike) vector.

3. Spacelike Sweeping Surface and Its Singularities

We present the spacelike sweeping surface at Minkowski 3-space . Consider the planar profile (cross-section) that is defined as . Using equation (14), we obtain

Using equation (9) resulted in

The unit normal vector of is

The main aim of this study is given in the following theorem:

Theorem 1. Suppose : is the unit speed spacelike curve with a timelike principal normal, and . Then, for any fixed , one has the following: A.(1)The image of Bishop spherical Darboux is locally diffeomorphic to the line {} at iff (2)The image of Bishop spherical Darboux is locally diffeomorphic to a cusp at if , as well B.(1) is locally diffeomorphic to cuspidal edge at iff , as well (2) is locally diffeomorphic to swallowtail SW at iff , , as well Here, , , and . The pictures of , , and are shown in Figures 1–3.

3.1. Lorentzian Bishop Height Functions

We will introduce two different families of Lorentzian Bishop height functions that will be useful to study singularities of as follows [1–3]: , by . It is called Lorentzian Bishop height function. The notation will be used for all fixed . In addition, it is defined , using . It is called extended Lorentzian Bishop height function of . It is denoted that . From here, parameter will not be written.

We have the following proposition:

Proposition 1. Let : be the unit speed spacelike curve with a timelike principal normal, with . Then, A.(1) iff , and .(2) iff (3) iff , and (4) iff , and (5) iff , and B.(1) iff there is (2) if there are , , that is, , and (3) iff , , and (4) iff , , and (5) iff , and (6) iff , , and

Proof. Using equation (9) results in that iff . A.(1)Because and , , is RMF through , then there are , such that . Because of , we have . The opposite holds as well.(2)Because , we have that . It follows from the fact that and . Therefore, we obtain Therefore, iff .(3)Because , using conditions of (2), we obtain Thus, iff , and .(4)Since by the conditions of (3), we have that Thus, iff , and .(5)Since , we have By using the conditions of (4), we have Therefore, iff , as well . B Using similar computation as in proof of A, we obtain B (1).

Proposition 2. Suppose : is the unit speed spacelike curve with the timelike principal normal, and . Then, we have iffis a constant vector.

Proof. Let . Using simple calculations, we have thatThus, iff .

Proposition 3. Suppose : is the unit speed spacelike curve with the timelike principal normal, and . Then, it is stated that(a) is the B-slant helix iff is a constant(b) is the part of circle at , and its center is the spacelike constant vector .

Proof. (a) Lets have . Hence, we can write This means that = constant, such that, is the B-slant helix.(b)Suppose that . SinceIn other words, is the part of circle at , and its center is the constant spacelike vector .

3.2. Unfolding of Functions by One-Variable

We use some general results at the singularity theory for families of function germs [1–3]. Suppose is the smooth function, and . Therefore, named the r-parameter unfolding of . It is said that has singularity at in case for every , and . Additionally, f has singularity () at . Suppose the jet of the partial derivative at is (without the constant term), for . Therefore, named the versal unfolding in case of the matrix of coefficients from the rank . So, we write an important set about the unfolding relative to the previous notations. The discriminant set of is

The bifurcation set of is

Similar to [1–3], we state the following theorem:

Theorem 2. Suppose is the r-parameter unfolding of , with singularity at .
Considering is the versal unfolding,(a)In case , is locally diffeomorphic to , and ;(b)In case , is locally diffeomorphic to , and is locally diffeomorphic to ;(c)In case , is locally diffeomorphic to , and is locally diffeomorphic to .Hence, we have the following proposition:

Proposition 4. Suppose : is the unit speed spacelike curve with the timelike principal normal, and .(1)In case has the singularity ( at , and is the p versal unfolding of (2)In case has the -singularity (, 3) at , and is the p–versal unfolding of

Proof. (1)Since , , and cannot be all zero. Without the loss of generality, suppose . Then, by , we have Thus, we have that Therefore, the 2 jets of at (i = 0, 1) are as follows: let , and assume , then(i)In case has the singularity at , then . So, the matrix of coefficients is If the matrix has rank equal zero, then Since , we have , so that we have the contradiction as follows: Then, rank , as well is the p versal unfolding of at .(ii)In case has the singularity at , then , and using Proposition 1, where , , and . So the matrix of the coefficients is It is required that the matrix is a nonsingular matrix. Clearly, this matrix determinate at is Since , we have . Substituting these relations to the previous equality, we obtain which resulted in rank .(2)Using similar notations as in (1), we getWe require the matrix,to have the maximal rank. Using case (1) in equation (38), the second raw of G will not be zero; then, rank .

Proof of Theorem 1. (1) Using Proposition 1, the bifurcation set of isThe assertion (1) of Theorem 1 follows from Propositions 1 and 4 and Theorem 2. The discriminant set of is written asThe assertion (1) of Theorem 1 follows from Propositions 1 and 4 and Theorem 2.

Example 1. Consider this spacelike helix:It is easy to show thatTaking , we obtain . By the use of equation (9), we haveHence, the geometric invariant isTherefore, the transformation matrix is written asFrom this, we haveHence, the Bishop spherical Darboux image (Figure 4)The spacelike sweeping surface (Figure 5) is

3.3. Singularities of Developable Surfaces

It is known that the developable surface is a special case of ruled surface. This kind of surfaces is used extensively in manufacture of automobile body parts, airplane wings, as well some ship parts. As a result, we analyze the case that the profile curve degenerates to the line. Therefore, we have the next spacelike developable surface:

Moreover, we have the following timelike developable surface:

Clearly, (respectively, ), , such that, the surface (respectively, ) interpolates the curve . We also haveas well

Therefore, we obtain (respectively, ) that is nonsingular at if and only if (respectively ). Hence, we can classify the singularities of developable surface by using .

Theorem 3. Suppose is the spacelike developable given as in equation (53). Therefore, we have(1) is locally diffeomorphic to cuspidal edge at iff and (2) is locally diffeomorphic to swallowtail at iff and

Proof. If there is a parameter , that is, , and (), is locally diffeomorphic to the cuspidal edge at . As a result, assertion (1) holds. In addition, in case there is a parameter , that is, , , and , and is locally diffeomorphic to swallowtail at , assertion (2) holds.

Example 2. By making using of Example 1, we have(1)If , then and . The timelike developable surface where is locally diffeomorphic to the cuspidal edge (Figure 6).(2)In case , and . The spacelike developable surface iswhere is locally diffeomorphic to swallowtail (Figure 7).

4. Conclusion

In this study, we introduced the notion of spacelike sweeping surfaces with rotation minimizing frames at Minkowski 3-space . Using the singularity, we classified the generic properties and present a new geometric invariant related to the singularities of this spacelike sweeping surface. It leads to that the generic singularities of this sweeping surface are cuspidal edge and swallowtail, and the types of these singularities can be characterized by this geometric invariant, in the same order. Subsequently, we derived the sufficient and necessary conditions for the surface to be developable ruled surfaces. At the end, some examples are introduced in order to clarify the theoretical results.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.

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Copyright © 2021 Fatemah Mofarreh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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