Yanlin LipersName, Zhizhi ChenpersName, Sahar H. NazrapersName, Rashad A. Abdel-BakypersName (2023date). “Singularities for Timelike Developable Surfaces in Minkowski 3-Space.” Symmetrytitle·j.
Object of study — the singularities and geometrical properties of timelike developable surfacesterm built with the Bishop frameterm in Minkowski 3-spaceterm
Research question — under what conditions the two timelike developable surfacesterm have singularities, and how to recognise the different singularityterm types
Contribution — a classification of the generic singularities of the two timelike developable surfacesterm as the cuspidal edgeterm and the swallowtailterm, obtained through singularityterm theory and the Minkowski Bishop frameterm
phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text| phenomenon | at their singular points the timelike developable surfaces are locally diffeomorphic to a cuspidal edge or a swallowtail, according to precise curvature conditions — “(1) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the cuspidal edge (CE) C(2,3) × 𝑅 at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , and 𝜇 𝑖 ′ ( 𝑠 0 ) ≠ 0 . (2) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the Swallowtail (SW) at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , 𝜇 𝑖 ′ ( 𝑠 0 ) = 0 , and 𝜇 𝑖 − 1 ( 𝑠 0 ) ″ ≠ 0 .” | |
| claim-type | existence — “(1) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the cuspidal edge (CE) C(2,3) × 𝑅 at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , and 𝜇 𝑖 ′ ( 𝑠 0 ) ≠ 0 . (2) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the Swallowtail (SW) at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , 𝜇 𝑖 ′ ( 𝑠 0 ) = 0 , and 𝜇 𝑖 − 1 ( 𝑠 0 ) ″ ≠ 0 .” | |
| method — evidence | — “Now we employ some public results on the singularity theory for families of function germs [27,28]. Let F: 𝑅 × 𝑅 𝑟 , ( 𝑠 0 , 𝑥 0 ) → 𝑅 be a smooth function, and 𝑓 ( 𝑠 ) = 𝐹 𝑥 0 ( 𝑠 , 𝑥 0 ) . Then F is called an r-parameter unfolding of 𝑓 ( 𝑠 ) . We say that 𝑓 ( 𝑠 ) has Ak-singularity at 𝑠 0 if 𝑓 ( 𝑝 ) ( 𝑠 0 ) = 0 for all 1 ≤ 𝑝 ≤ 𝑘 , and 𝑓 ( 𝑘 + 1 ) ( 𝑠 0 ) ≠ 0 . We also say that f has A ⩾ 𝑘 -singularity ( 𝑘 ⩾ 1 ) at s0. Let the 𝑘 − 1 -jet of the partial derivative ∂ 𝐹 ∂ 𝑥 𝑖 at 𝑠 0 be 𝑗 𝑘 − 1 ∂ 𝐹 ∂ 𝑥 𝑖 𝑠 , 𝑥 0 𝑠 0 = 𝛴 𝑗 = 0 𝑘 − 1 𝐿 𝑗 𝑖 𝑠 − 𝑠 0 𝑗 (without the constant term), for 𝑖 = 1 , . . . , 𝑟 . Then 𝐹 ( 𝑠 ) is called an p-versal unfolding if the 𝑘 × 𝑟 matrix of coefficients 𝐿 𝑗 𝑖 has rank k 𝑘 ≤ 𝑟 . Therefore, we write serious sets on the unfolding regarding to the above notations. The discriminant set of order t of F is the set 𝐷 𝐹 𝑡 = 𝑥 ∈ 𝑅 𝑟 ∣ ∃ 𝑠 ∈ 𝑅 , 𝐹 𝑠 , 𝑥 = , ∂ 𝐹 ∂ 𝑠 𝑠 , 𝑥 = . . . = ∂ 𝑡 𝐹 ∂ 𝑠 𝑡 𝑠 , 𝑥 = 0 at ( 𝑠 , 𝑥 ) . (8) Then 𝐷 𝐹 1 = 𝐷 𝐹 , and 𝐷 𝐹 2 is the set of singular points of 𝐷 𝐹 .” | |
| object of study — evidence | — “In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results.” | |
| research question — evidence | — “Let 𝑦 ( 𝑠 , 𝑡 ) be a parametric sweeping given by: 𝑦 ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑟 1 ( 𝑡 ) 𝜉 1 ( 𝑠 ) + 𝑟 2 ( 𝑡 ) 𝜉 2 ( 𝑠 ) . (7) Here 𝛾 ( 𝑠 ) is a timelike unit speed spine curve, 0 ≤ 𝑠 ≤ 𝑇 , s is the arc length parameter, and 𝑟 ( 𝑡 ) = ( 0 , 𝑟 1 ( 𝑡 ) , 𝑟 2 ( 𝑡 ) ) is the characteristic circle, with another parameter 𝑡 ∈ 𝑅 . It can be seen that 𝑦 ( 𝑠 , 𝑡 ) is a timelike surface. From Equation (7) it follows that the expression of the two timelike developable surfaces is 𝐷 𝑖 ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑡 𝜉 𝑖 ( 𝑠 ) , ( i = 1 , 2 ) . It is clear that 𝐷 2 ( 𝑠 , 0 ) = 𝛾 ( 𝑠 ) (resp. 𝐷 1 ( 𝑠 , 0 ) = 𝛾 ( 𝑠 ) ), 0 ≤ 𝑠 ≤ 𝐿 , that is, the surface 𝐷 2 (resp. 𝐷 1 ) interpolate the curve 𝛾 ( 𝑠 ) . Now, it seems natural to pose the following two questions. Under what condition does 𝐷 𝑖 have singularities, and how do we recognize their different types? The answer can be stated as: To discuss the singularities of 𝐷 𝑖 ( 𝑠 , 𝑡 ) , the cross product can be obtained as ∂ 𝐷 1 ∂ 𝑠 × ∂ 𝐷 1 ∂ 𝑡 = 1 − 𝑡 𝜇 1 𝜉 2 ( 𝑠 ) , and ∂ 𝐷 2 ∂ 𝑠 × ∂ 𝐷 2 ∂ 𝑡 = − 1 − 𝑡 𝜇 2 𝜉 1 ( 𝑠 ) . It follows that 𝐷 1 (resp. 𝐷 2 ) is non-singular at ( 𝑠 0 , 𝑡 0 ) if and only if 1 − 𝑡 0 𝜇 1 ( 𝑠 0 ) ≠ 0 (resp. ( 1 − 𝑡 0 𝜇 2 ( 𝑠 0 ) ≠ 0 ) . Hence, according to Theorem 3.2 in Ref. [10], we can give the following corollary:” | |
| contribution — evidence | — “In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results.” | |
| mechanism | rotation-minimizing frame The Bishop (rotation-minimizing) frame stays smoothly defined along a space curve even where the Serret-Frenet frame becomes undefined, such as at an inflection point where curvature vanishes. — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | mechanisms[0] |
| mechanism | height-function contact An A_k singularity of the Bishop height function corresponds to (k+1)-point contact between the curve and the Bishop rectifying bundle, which detects the surface singularity. — “First, we consider the affirmation (1). For 𝑥 0 = 𝐷 1 ( 𝑠 0 , 𝑡 0 ) , since BR 2 ( 𝑥 0 ) = 𝐻 2 − 1 ( 0 ) , where 0 is a regular value of 𝐻 2 ; ℎ 2 𝑥 0 ( 𝑠 ) has the A 𝑘 -singularity at s0 if and only if 𝛾 ( 𝑠 0 ) and BR 2 ( 𝑥 0 ) have (k+1)-point contact for 𝑠 0 . On the other hand, the discriminant set 𝐷 𝐻 2 of 𝐻 2 is 𝐷 𝐻 2 = 𝑥 = 𝛾 ( 𝑠 ) + 𝑡 𝜉 1 ( 𝑠 ) ∣ 𝑠 ∈ 𝐼 , 𝑡 ∈ 𝑅 . The affirmation (1) follows from Theorem 2 and Proposition 1. Since the trajectory of the singularities of CE is locally diffeomorphic to the line, the affirmation (b) holds. Since the trajectory of singularities of SW is C(2,3,4), the affirmation (c) holds.” | mechanisms[1] |
| mechanism | versal-unfolding classification When the height function is a versal unfolding, its discriminant set is locally diffeomorphic to the standard cuspidal-edge or swallowtail model, transferring the classification to the surface. — “Now we employ some public results on the singularity theory for families of function germs [27,28]. Let F: 𝑅 × 𝑅 𝑟 , ( 𝑠 0 , 𝑥 0 ) → 𝑅 be a smooth function, and 𝑓 ( 𝑠 ) = 𝐹 𝑥 0 ( 𝑠 , 𝑥 0 ) . Then F is called an r-parameter unfolding of 𝑓 ( 𝑠 ) . We say that 𝑓 ( 𝑠 ) has Ak-singularity at 𝑠 0 if 𝑓 ( 𝑝 ) ( 𝑠 0 ) = 0 for all 1 ≤ 𝑝 ≤ 𝑘 , and 𝑓 ( 𝑘 + 1 ) ( 𝑠 0 ) ≠ 0 . We also say that f has A ⩾ 𝑘 -singularity ( 𝑘 ⩾ 1 ) at s0. Let the 𝑘 − 1 -jet of the partial derivative ∂ 𝐹 ∂ 𝑥 𝑖 at 𝑠 0 be 𝑗 𝑘 − 1 ∂ 𝐹 ∂ 𝑥 𝑖 𝑠 , 𝑥 0 𝑠 0 = 𝛴 𝑗 = 0 𝑘 − 1 𝐿 𝑗 𝑖 𝑠 − 𝑠 0 𝑗 (without the constant term), for 𝑖 = 1 , . . . , 𝑟 . Then 𝐹 ( 𝑠 ) is called an p-versal unfolding if the 𝑘 × 𝑟 matrix of coefficients 𝐿 𝑗 𝑖 has rank k 𝑘 ≤ 𝑟 . Therefore, we write serious sets on the unfolding regarding to the above notations. The discriminant set of order t of F is the set 𝐷 𝐹 𝑡 = 𝑥 ∈ 𝑅 𝑟 ∣ ∃ 𝑠 ∈ 𝑅 , 𝐹 𝑠 , 𝑥 = , ∂ 𝐹 ∂ 𝑠 𝑠 , 𝑥 = . . . = ∂ 𝑡 𝐹 ∂ 𝑠 𝑡 𝑠 , 𝑥 = 0 at ( 𝑠 , 𝑥 ) . (8) Then 𝐷 𝐹 1 = 𝐷 𝐹 , and 𝐷 𝐹 2 is the set of singular points of 𝐷 𝐹 .” | mechanisms[2] |
| builds-on | Izumiya's singularity theory and Legendrian duality for submanifolds — “The present work studies the singularities and geometrical properties of timelike developable surfaces according to the Minkowski Bishop frame. The field, as far as we are aware, is not covered by any articles, though it has been carefully and ably worked by many researchers. Thus, the present paper hopes to help such a need, and it is inspired by the works of Izumiya, and Haiming et al. [3,23,24,25,26]. In this article, we construct the Bishop frame of speed timelike curve and take advantage of the singularity theory to study the singularities and geometrical properties of timelike developable.” | claim_relations[0] |
| builds-on | Bishop's rotation-minimizing frame — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | claim_relations[1] |
| extends | Euclidean Bishop-frame surface theory — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | claim_relations[2] |
| sense | singularity - differential-geometry sense: a point where the surface map fails to be an immersion, locally modelled by a cuspidal edge or swallowtail — “(1) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the cuspidal edge (CE) C(2,3) × 𝑅 at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , and 𝜇 𝑖 ′ ( 𝑠 0 ) ≠ 0 . (2) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the Swallowtail (SW) at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , 𝜇 𝑖 ′ ( 𝑠 0 ) = 0 , and 𝜇 𝑖 − 1 ( 𝑠 0 ) ″ ≠ 0 .” | senses[0] |
| sense | developable surface - sense: a surface that can be flattened onto a plane without distortion — “A channel or canal surface is a surface traced by a one-parameter family of spheres, whose center is on a smooth space curve, its directrix or spine. If the radii of the traced spheres are stationary then the canal surface is named a tubular or sweeping surface. There are several famous examples: circular cylinder, right circular cone, torus, and rotation surface. This visualization is a generalization of the classical registration of a partner of a planar curve [1,2,3,4,5,6]. One of the eminent facts about sweeping surface is that the sweeping surface can be a developable surface, that is, a smooth surface can be flattened onto a plane without distortion. Therefore, the developable surface can be made out of sheet metal, since such a surface must be attainable by modification from a plane, and every point on such a surface lies on at least one straight line. Therefore, sweeping surfaces and developable surfaces have been paid interest to in engineering, architecture, and design, etc. [7,8,9,10,11,12,13,14,15,16,17,18].” | senses[1] |
| sense | frame - Serret-Frenet versus Bishop (rotation-minimizing) sense: two moving frames of a curve, the latter defined where the former is not — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | senses[2] |
| sense | contact - k-point sense: the tangency order between a curve and a level set measured by vanishing derivatives — “Now, we should address the next concept regarding the contact of curves with some surfaces. Let F: 𝐸 1 3 → 𝑅 be a regular function, and 𝛾 : 𝑅 → 𝐸 1 3 be a smooth unit speed timelike Frenet–Serret curve. We say that 𝛾 ( 𝑠 ) and 𝐹 − 1 ( 0 ) have k-point contact for 𝑠 = 𝑠 0 if the function 𝑔 ( 𝑠 0 ) = 𝐹 ∘ 𝛾 satisfies 𝑔 ( 𝑠 0 ) = 𝑔 ′ ( 𝑠 0 ) = 𝑔 ″ ( 𝑠 0 ) = . . . = 𝑔 ( 𝑘 − 1 ) ( 𝑠 0 ) = 0 , and 𝑔 ( 𝑘 ) ( 𝑠 0 ) ≠ 0 . Further, for any fixed 𝑥 0 ∈ 𝐸 1 3 , we define the set BR 𝑗 ( 𝑥 0 ) = < 𝜉 𝑖 , 𝑥 0 − 𝛾 ( 𝑠 ) > = 0 , 𝑗 = 1 , 2 . We call it the first (resp. second) Bishop rectifying bundle of 𝛾 ( 𝑠 ) through 𝑥 0 when j = 1 (resp. j = 2).” | senses[3] |
| note | Izumiya's singularity theory and Legendrian duality for submanifolds — The study of singularities of the developable surfaces is inspired by and follows Izumiya-type methods. — “The present work studies the singularities and geometrical properties of timelike developable surfaces according to the Minkowski Bishop frame. The field, as far as we are aware, is not covered by any articles, though it has been carefully and ably worked by many researchers. Thus, the present paper hopes to help such a need, and it is inspired by the works of Izumiya, and Haiming et al. [3,23,24,25,26]. In this article, we construct the Bishop frame of speed timelike curve and take advantage of the singularity theory to study the singularities and geometrical properties of timelike developable.” | claim_relations[0] |
| weight | Izumiya's singularity theory and Legendrian duality for submanifolds — 0.9 — “The present work studies the singularities and geometrical properties of timelike developable surfaces according to the Minkowski Bishop frame. The field, as far as we are aware, is not covered by any articles, though it has been carefully and ably worked by many researchers. Thus, the present paper hopes to help such a need, and it is inspired by the works of Izumiya, and Haiming et al. [3,23,24,25,26]. In this article, we construct the Bishop frame of speed timelike curve and take advantage of the singularity theory to study the singularities and geometrical properties of timelike developable.” | claim_relations[0] |
| note | Bishop's rotation-minimizing frame — The whole construction replaces the Serret-Frenet frame with Bishop's alternative frame, extended to Minkowski space. — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | claim_relations[1] |
| weight | Bishop's rotation-minimizing frame — 0.92 — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | claim_relations[1] |
| note | Euclidean Bishop-frame surface theory — The Euclidean Bishop frame idea is expanded to the Minkowski (Lorentzian) setting. — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | claim_relations[2] |
| weight | Euclidean Bishop-frame surface theory — 0.9 — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | claim_relations[2] |
| audit | developable surface —is-a→ ruled surface — Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the span says a ruled surface D is developable if and only if a determinant vanishes; is-a is loose. — “A ruled surface in 𝐸 1 3 is locally the map 𝐷 ( 𝛾 , 𝑥 ) : 𝐼 × 𝑅 → 𝐸 1 3 defined by 𝐷 ( 𝛾 , 𝑥 ) ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) , t ∈ 𝑅 , (6) where 𝛾 ( 𝑠 ) is named the base curve, and 𝑥 ( 𝑠 ) the director curve. The straight lines 𝑡 → 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) are named rulings. It is well known that 𝐷 ( 𝛾 , 𝑥 ) is a developable surface if and only if det( 𝛾 ′ ( 𝑠 ) , 𝑥 ( 𝑠 ) , 𝑥 ′ ( 𝑠 ) ) = 0 .” | concept_edges[0] |
| note | developable surface —is-a→ ruled surface — Definitional subset of ruled surfaces; flat categorical. — “A ruled surface in 𝐸 1 3 is locally the map 𝐷 ( 𝛾 , 𝑥 ) : 𝐼 × 𝑅 → 𝐸 1 3 defined by 𝐷 ( 𝛾 , 𝑥 ) ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) , t ∈ 𝑅 , (6) where 𝛾 ( 𝑠 ) is named the base curve, and 𝑥 ( 𝑠 ) the director curve. The straight lines 𝑡 → 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) are named rulings. It is well known that 𝐷 ( 𝛾 , 𝑥 ) is a developable surface if and only if det( 𝛾 ′ ( 𝑠 ) , 𝑥 ( 𝑠 ) , 𝑥 ′ ( 𝑠 ) ) = 0 .” | concept_edges[0] |
| relation | developable surface —is-a→ ruled surface — A developable surface is a ruled surface whose base-curve, director and director-derivative determinant vanishes. — “A ruled surface in 𝐸 1 3 is locally the map 𝐷 ( 𝛾 , 𝑥 ) : 𝐼 × 𝑅 → 𝐸 1 3 defined by 𝐷 ( 𝛾 , 𝑥 ) ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) , t ∈ 𝑅 , (6) where 𝛾 ( 𝑠 ) is named the base curve, and 𝑥 ( 𝑠 ) the director curve. The straight lines 𝑡 → 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) are named rulings. It is well known that 𝐷 ( 𝛾 , 𝑥 ) is a developable surface if and only if det( 𝛾 ′ ( 𝑠 ) , 𝑥 ( 𝑠 ) , 𝑥 ′ ( 𝑠 ) ) = 0 .” | concept_edges[0] |
| note | Bishop frame —contrasted-with→ Serret–Frenet frame — Differentiator stated: smooth definition at inflection points where curvature vanishes. — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | concept_edges[1] |
| relation | Bishop frame —contrasted-with→ Serret–Frenet frame — The Bishop frame stays smoothly defined where the Serret-Frenet frame becomes undefined, such as at an inflection point. — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | concept_edges[1] |
| weight | Bishop frame —contrasted-with→ Serret–Frenet frame — 0.9 — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | concept_edges[1] |
| relation | Bishop height function → singularity — Singularities of the surface are read off from singularities of the height function. — “First, we consider the affirmation (1). For 𝑥 0 = 𝐷 1 ( 𝑠 0 , 𝑡 0 ) , since BR 2 ( 𝑥 0 ) = 𝐻 2 − 1 ( 0 ) , where 0 is a regular value of 𝐻 2 ; ℎ 2 𝑥 0 ( 𝑠 ) has the A 𝑘 -singularity at s0 if and only if 𝛾 ( 𝑠 0 ) and BR 2 ( 𝑥 0 ) have (k+1)-point contact for 𝑠 0 . On the other hand, the discriminant set 𝐷 𝐻 2 of 𝐻 2 is 𝐷 𝐻 2 = 𝑥 = 𝛾 ( 𝑠 ) + 𝑡 𝜉 1 ( 𝑠 ) ∣ 𝑠 ∈ 𝐼 , 𝑡 ∈ 𝑅 . The affirmation (1) follows from Theorem 2 and Proposition 1. Since the trajectory of the singularities of CE is locally diffeomorphic to the line, the affirmation (b) holds. Since the trajectory of singularities of SW is C(2,3,4), the affirmation (c) holds.” | unhoused_relations[0] |
| weight | Bishop height function → singularity — 0.9 — “First, we consider the affirmation (1). For 𝑥 0 = 𝐷 1 ( 𝑠 0 , 𝑡 0 ) , since BR 2 ( 𝑥 0 ) = 𝐻 2 − 1 ( 0 ) , where 0 is a regular value of 𝐻 2 ; ℎ 2 𝑥 0 ( 𝑠 ) has the A 𝑘 -singularity at s0 if and only if 𝛾 ( 𝑠 0 ) and BR 2 ( 𝑥 0 ) have (k+1)-point contact for 𝑠 0 . On the other hand, the discriminant set 𝐷 𝐻 2 of 𝐻 2 is 𝐷 𝐻 2 = 𝑥 = 𝛾 ( 𝑠 ) + 𝑡 𝜉 1 ( 𝑠 ) ∣ 𝑠 ∈ 𝐼 , 𝑡 ∈ 𝑅 . The affirmation (1) follows from Theorem 2 and Proposition 1. Since the trajectory of the singularities of CE is locally diffeomorphic to the line, the affirmation (b) holds. Since the trajectory of singularities of SW is C(2,3,4), the affirmation (c) holds.” | unhoused_relations[0] |
| no type for this | Bishop height function → singularity — The A_k singularity of the Bishop height function detects and classifies the singularity of the developable surface. — “First, we consider the affirmation (1). For 𝑥 0 = 𝐷 1 ( 𝑠 0 , 𝑡 0 ) , since BR 2 ( 𝑥 0 ) = 𝐻 2 − 1 ( 0 ) , where 0 is a regular value of 𝐻 2 ; ℎ 2 𝑥 0 ( 𝑠 ) has the A 𝑘 -singularity at s0 if and only if 𝛾 ( 𝑠 0 ) and BR 2 ( 𝑥 0 ) have (k+1)-point contact for 𝑠 0 . On the other hand, the discriminant set 𝐷 𝐻 2 of 𝐻 2 is 𝐷 𝐻 2 = 𝑥 = 𝛾 ( 𝑠 ) + 𝑡 𝜉 1 ( 𝑠 ) ∣ 𝑠 ∈ 𝐼 , 𝑡 ∈ 𝑅 . The affirmation (1) follows from Theorem 2 and Proposition 1. Since the trajectory of the singularities of CE is locally diffeomorphic to the line, the affirmation (b) holds. Since the trajectory of singularities of SW is C(2,3,4), the affirmation (c) holds.” | unhoused_relations[0] |
| limitation | The results are local: each surface is only shown to be locally diffeomorphic to a cuspidal edge or swallowtail at a singular point, not classified globally. — “(1) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the cuspidal edge (CE) C(2,3) × 𝑅 at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , and 𝜇 𝑖 ′ ( 𝑠 0 ) ≠ 0 . (2) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the Swallowtail (SW) at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , 𝜇 𝑖 ′ ( 𝑠 0 ) = 0 , and 𝜇 𝑖 − 1 ( 𝑠 0 ) ″ ≠ 0 .” | limitations[0] |
| limitation | The theorems assume a unit-speed timelike curve with both Bishop functions nonzero, a genericity condition that excludes degenerate cases. — “Let γ: 𝐼 → 𝐸 1 3 be a unit speed timelike curve, with 𝜇 𝑖 ≠ 0 ; i = 1, 2. Then we: A- For 𝑥 0 = 𝐷 1 ( 𝑠 0 , 𝑡 0 ) and the second Bishop rectifying bundle BR 2 ( 𝑥 0 ) of 𝛾 ( 𝑠 ) . One has the following:” | limitations[1] |
| limitation | The worked example exhibits a point where the surface fails to be a swallowtail because the height function is not a versal unfolding there, showing the classification does not cover every point. — “(2) If 𝑟 1 ( 𝑡 ) = 𝑟 2 ( 𝑡 ) = 𝑡 , then 𝐷 1 ( 𝑠 , 𝑡 ) , and 𝐷 2 ( 𝑠 , 𝑡 ) , respectively, are: 𝐷 1 ( 𝑠 , 𝑡 ) = 3 sinh 𝑠 + 𝑡 𝜉 11 , 3 cosh 𝑠 + 𝑡 𝜉 13 , 2 𝑠 + 𝑡 𝜉 12 , and 𝐷 2 ( 𝑠 , 𝑡 ) = 3 sinh 𝑠 + 𝑡 𝜉 21 , 3 cosh 𝑠 + 𝑡 𝜉 23 , 2 𝑠 + 𝑡 𝜉 22 . In addition, the singular locus of 𝐷 1 ( 𝑠 , 𝑡 ) , and 𝐷 2 ( 𝑠 , 𝑡 ) , respectively, are: 𝐷 1 ( 𝑠 ) = 3 sinh 𝑠 + 1 3 cos 2 𝑠 𝜉 11 , 3 cosh 𝑠 + 1 3 cos 2 𝑠 𝜉 13 , 2 𝑠 + 1 3 cos 2 𝑠 𝜉 12 , and 𝐷 2 ( 𝑠 ) = 3 sinh 𝑠 + 1 3 sin 2 𝑠 𝜉 21 , 3 cosh 𝑠 + 1 3 sin 2 𝑠 𝜉 23 , 2 𝑠 + 1 3 sin 2 𝑠 𝜉 22 . We consider a local part of this curve when 𝜋 6 2 ≤ 𝑠 ≤ 𝜋 3 2 . We see that 𝜇 1 − 1 ( 𝑠 ) = 1 3 cos 2 𝑠 ≠ 0 , and 𝜇 1 ′ ( 𝑠 ) = − 6 sin 2 𝑠 ≠ 0 for 𝜋 6 2 ≤ 𝑠 ≤ 𝜋 3 2 . This means that 𝐷 1 ( 𝑠 , 𝑡 ) is locally diffeomorphic to a CE and its singular locus is locally diffeomorphic to a line (the red line), see Figure 4. For 𝐷 2 ( 𝑠 , 𝑡 ) , when 𝜋 6 2 ≤ 𝑠 ≤ 5 𝜋 6 2 . We see that 𝜇 2 − 1 ( 𝑠 ) = 1 3 cos 2 𝑠 ≠ 0 , 𝜇 2 ′ ( 𝑠 ) = 6 cos 2 𝑠 = 0 gives one real root 𝑠 = 𝜋 2 2 . We can also get that 𝜇 2 − 1 ( 𝜋 2 2 ) ″ = 0 , that is, 𝐷 2 ( 𝑠 , 𝑡 ) fails to be locally diffeomorphic to SW and its singular locus is not locally diffeomorphic to the C(2,3,4)-cusp at 𝑠 = 𝜋 2 2 . Hence 𝐻 2 ( 𝑠 , 𝑥 ) fails to be a versal unfolding of the ℎ 2 𝑥 (s) at 𝑠 = 𝜋 2 2 ; see Figure 5.” | limitations[2] |
finding · quantity · variable · definition · key-term — grounded in the full text| finding | A ruled surface D(gamma,x) is developable if and only if the determinant of the base-curve derivative, director curve, and its derivative vanishes. — “A ruled surface in 𝐸 1 3 is locally the map 𝐷 ( 𝛾 , 𝑥 ) : 𝐼 × 𝑅 → 𝐸 1 3 defined by 𝐷 ( 𝛾 , 𝑥 ) ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) , t ∈ 𝑅 , (6) where 𝛾 ( 𝑠 ) is named the base curve, and 𝑥 ( 𝑠 ) the director curve. The straight lines 𝑡 → 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) are named rulings. It is well known that 𝐷 ( 𝛾 , 𝑥 ) is a developable surface if and only if det( 𝛾 ′ ( 𝑠 ) , 𝑥 ( 𝑠 ) , 𝑥 ′ ( 𝑠 ) ) = 0 .” | content_tags.findings[0] |
| finding | The surface D_i is non-singular at a point if and only if 1 minus t0 times the Bishop function at s0 is nonzero. — “Let 𝑦 ( 𝑠 , 𝑡 ) be a parametric sweeping given by: 𝑦 ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑟 1 ( 𝑡 ) 𝜉 1 ( 𝑠 ) + 𝑟 2 ( 𝑡 ) 𝜉 2 ( 𝑠 ) . (7) Here 𝛾 ( 𝑠 ) is a timelike unit speed spine curve, 0 ≤ 𝑠 ≤ 𝑇 , s is the arc length parameter, and 𝑟 ( 𝑡 ) = ( 0 , 𝑟 1 ( 𝑡 ) , 𝑟 2 ( 𝑡 ) ) is the characteristic circle, with another parameter 𝑡 ∈ 𝑅 . It can be seen that 𝑦 ( 𝑠 , 𝑡 ) is a timelike surface. From Equation (7) it follows that the expression of the two timelike developable surfaces is 𝐷 𝑖 ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑡 𝜉 𝑖 ( 𝑠 ) , ( i = 1 , 2 ) . It is clear that 𝐷 2 ( 𝑠 , 0 ) = 𝛾 ( 𝑠 ) (resp. 𝐷 1 ( 𝑠 , 0 ) = 𝛾 ( 𝑠 ) ), 0 ≤ 𝑠 ≤ 𝐿 , that is, the surface 𝐷 2 (resp. 𝐷 1 ) interpolate the curve 𝛾 ( 𝑠 ) . Now, it seems natural to pose the following two questions. Under what condition does 𝐷 𝑖 have singularities, and how do we recognize their different types? The answer can be stated as: To discuss the singularities of 𝐷 𝑖 ( 𝑠 , 𝑡 ) , the cross product can be obtained as ∂ 𝐷 1 ∂ 𝑠 × ∂ 𝐷 1 ∂ 𝑡 = 1 − 𝑡 𝜇 1 𝜉 2 ( 𝑠 ) , and ∂ 𝐷 2 ∂ 𝑠 × ∂ 𝐷 2 ∂ 𝑡 = − 1 − 𝑡 𝜇 2 𝜉 1 ( 𝑠 ) . It follows that 𝐷 1 (resp. 𝐷 2 ) is non-singular at ( 𝑠 0 , 𝑡 0 ) if and only if 1 − 𝑡 0 𝜇 1 ( 𝑠 0 ) ≠ 0 (resp. ( 1 − 𝑡 0 𝜇 2 ( 𝑠 0 ) ≠ 0 ) . Hence, according to Theorem 3.2 in Ref. [10], we can give the following corollary:” | content_tags.findings[1] |
| finding | D_i is locally diffeomorphic to the cuspidal edge when t0 equals the inverse Bishop function and its derivative is nonzero, and to the swallowtail when in addition that derivative vanishes and the next one does not. — “(1) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the cuspidal edge (CE) C(2,3) × 𝑅 at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , and 𝜇 𝑖 ′ ( 𝑠 0 ) ≠ 0 . (2) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the Swallowtail (SW) at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , 𝜇 𝑖 ′ ( 𝑠 0 ) = 0 , and 𝜇 𝑖 − 1 ( 𝑠 0 ) ″ ≠ 0 .” | content_tags.findings[2] |
| finding | The curve and the Bishop rectifying bundle have at least 2-point contact, rising to 3-point contact for the cuspidal-edge case and 4-point contact for the swallowtail case under stated derivative conditions. — “(a) The curve 𝛾 ( 𝑠 ) and BR 2 ( 𝑥 0 ) have at least a 2-point contact at s0; (b) The curve 𝛾 ( 𝑠 ) and BR 2 ( 𝑥 0 ) have at least a 3-point contact at s0 if and only if” | content_tags.findings[3] |
| finding | In the worked timelike-helix example D_1 is locally a cuspidal edge while D_2 fails to be a swallowtail at a specific parameter value. — “(2) If 𝑟 1 ( 𝑡 ) = 𝑟 2 ( 𝑡 ) = 𝑡 , then 𝐷 1 ( 𝑠 , 𝑡 ) , and 𝐷 2 ( 𝑠 , 𝑡 ) , respectively, are: 𝐷 1 ( 𝑠 , 𝑡 ) = 3 sinh 𝑠 + 𝑡 𝜉 11 , 3 cosh 𝑠 + 𝑡 𝜉 13 , 2 𝑠 + 𝑡 𝜉 12 , and 𝐷 2 ( 𝑠 , 𝑡 ) = 3 sinh 𝑠 + 𝑡 𝜉 21 , 3 cosh 𝑠 + 𝑡 𝜉 23 , 2 𝑠 + 𝑡 𝜉 22 . In addition, the singular locus of 𝐷 1 ( 𝑠 , 𝑡 ) , and 𝐷 2 ( 𝑠 , 𝑡 ) , respectively, are: 𝐷 1 ( 𝑠 ) = 3 sinh 𝑠 + 1 3 cos 2 𝑠 𝜉 11 , 3 cosh 𝑠 + 1 3 cos 2 𝑠 𝜉 13 , 2 𝑠 + 1 3 cos 2 𝑠 𝜉 12 , and 𝐷 2 ( 𝑠 ) = 3 sinh 𝑠 + 1 3 sin 2 𝑠 𝜉 21 , 3 cosh 𝑠 + 1 3 sin 2 𝑠 𝜉 23 , 2 𝑠 + 1 3 sin 2 𝑠 𝜉 22 . We consider a local part of this curve when 𝜋 6 2 ≤ 𝑠 ≤ 𝜋 3 2 . We see that 𝜇 1 − 1 ( 𝑠 ) = 1 3 cos 2 𝑠 ≠ 0 , and 𝜇 1 ′ ( 𝑠 ) = − 6 sin 2 𝑠 ≠ 0 for 𝜋 6 2 ≤ 𝑠 ≤ 𝜋 3 2 . This means that 𝐷 1 ( 𝑠 , 𝑡 ) is locally diffeomorphic to a CE and its singular locus is locally diffeomorphic to a line (the red line), see Figure 4. For 𝐷 2 ( 𝑠 , 𝑡 ) , when 𝜋 6 2 ≤ 𝑠 ≤ 5 𝜋 6 2 . We see that 𝜇 2 − 1 ( 𝑠 ) = 1 3 cos 2 𝑠 ≠ 0 , 𝜇 2 ′ ( 𝑠 ) = 6 cos 2 𝑠 = 0 gives one real root 𝑠 = 𝜋 2 2 . We can also get that 𝜇 2 − 1 ( 𝜋 2 2 ) ″ = 0 , that is, 𝐷 2 ( 𝑠 , 𝑡 ) fails to be locally diffeomorphic to SW and its singular locus is not locally diffeomorphic to the C(2,3,4)-cusp at 𝑠 = 𝜋 2 2 . Hence 𝐻 2 ( 𝑠 , 𝑥 ) fails to be a versal unfolding of the ℎ 2 𝑥 (s) at 𝑠 = 𝜋 2 2 ; see Figure 5.” | content_tags.findings[4] |
| definition | developable surface — “A channel or canal surface is a surface traced by a one-parameter family of spheres, whose center is on a smooth space curve, its directrix or spine. If the radii of the traced spheres are stationary then the canal surface is named a tubular or sweeping surface. There are several famous examples: circular cylinder, right circular cone, torus, and rotation surface. This visualization is a generalization of the classical registration of a partner of a planar curve [1,2,3,4,5,6]. One of the eminent facts about sweeping surface is that the sweeping surface can be a developable surface, that is, a smooth surface can be flattened onto a plane without distortion. Therefore, the developable surface can be made out of sheet metal, since such a surface must be attainable by modification from a plane, and every point on such a surface lies on at least one straight line. Therefore, sweeping surfaces and developable surfaces have been paid interest to in engineering, architecture, and design, etc. [7,8,9,10,11,12,13,14,15,16,17,18].” | content_tags.definitions[0] |
| definition | Minkowski 3-space — “In this section, we present some definitions and employ notions that we will use in this paper (see for instance Refs. [1,24]). Let 𝑅 3 = { 𝑝 1 , 𝑝 2 , 𝑝 3 ∣ , 𝑝 𝑖 ∈ 𝑅 (i = 1, 2, 3)} be a 3-dimensional Cartesian space. For any 𝑞 = 𝑞 1 , 𝑞 2 , 𝑞 3 , and 𝑝 = 𝑝 1 , 𝑝 2 , 𝑝 3 ∈ 𝑅 3 , the pseudo scalar product of 𝑞 and 𝑝 is defined by < 𝑞 , 𝑝 > = − 𝑞 1 𝑝 1 + 𝑞 2 𝑝 2 + 𝑞 3 𝑝 3 . We name ( 𝑅 3 , < , > ) Minkowski 3-space. We write 𝐸 1 3 instead of ( 𝑅 3 , < , > ) . We say that a non-zero vector 𝑞 ∈ 𝐸 1 3 is spacelike, lightlike, or timelike if < 𝑞 , 𝑞 > > 0 , < 𝑞 , 𝑞 > = 0 or < 𝑞 , 𝑞 > < 0 , respectively. The norm of the vector 𝑞 ∈ 𝐸 1 3 is defined to be 𝑞 = < 𝑞 , 𝑞 > . For any two vectors 𝑞 , 𝑝 ∈ 𝐸 1 3 , we define the cross product 𝑞 × 𝑝 by 𝑞 × 𝑝 = − 𝑖 𝑗 𝑘 𝑞 1 𝑞 2 𝑞 3 𝑝 1 𝑝 2 𝑝 3 = − ( 𝑞 2 𝑝 3 − 𝑞 3 𝑝 2 ) , ( 𝑞 3 𝑝 1 − 𝑞 1 𝑝 3 ) , ( 𝑞 1 𝑝 2 − 𝑞 2 𝑝 1 ) , where 𝑖 , 𝑗 , 𝑘 is the canonical basis of 𝐸 1 3 . The hyperbolic ( 𝐻 + 2 ) and Lorentzian unit spheres ( 𝑆 1 2 ), respectively, are 𝐻 + 2 = 𝑞 ∈ 𝐸 1 3 ∣ 𝑞 2 = − 𝑞 1 2 + 𝑞 2 2 + 𝑞 3 2 = − 1 , and 𝑆 1 2 = 𝑞 ∈ 𝐸 1 3 ∣ 𝑞 2 = − 𝑞 1 2 + 𝑞 2 2 + 𝑞 3 2 = 1 . Let 𝛾 : 𝐼 → 𝐸 1 3 be a smooth curve in 𝐸 1 3 , where I is an interval of 𝑅 . We call 𝛾 is 𝑠 𝑝 𝑎 𝑐 𝑒 𝑙 𝑖 𝑘 𝑒 , 𝑡 𝑖 𝑚 𝑒 𝑙 𝑖 𝑘 𝑒 , 𝑙 𝑖 𝑔 ℎ 𝑡 𝑙 𝑖 𝑘 𝑒 if 𝛾 ′ ( 𝑡 ) = 𝑑 𝛾 𝑑 𝑡 ( 𝑡 ) is spacelike, timelike, lightlike for any 𝑡 ∈ 𝑅 , respectively. A surface in the Minkowski 3-space 𝐸 1 3 is named a timelike surface if the induced metric on the surface is a Lorentzian metric and is named a spacelike surface if the induced metric on the surface is a positive definite Riemannian metric, i.e., the normal vector on spacelike (timelike) surface is a timelike (spacelike) vector. Let 𝛾 = 𝛾 ( 𝑠 ) be a unit speed timelike curve in 𝐸 1 3 ; by 𝜅 ( 𝑠 ) and 𝜏 ( 𝑠 ) we indicate the natural curvature and torsion, respectively. 𝛾 ( 𝑠 ) is called the Serret–Frenet curve if 𝜅 > 0 , and 𝜏 ≠ 0 . Consider the Serret–Frenet frame { 𝜒 1 ( 𝑠 ) , 𝜒 2 ( 𝑠 ) , 𝜒 3 ( 𝑠 ) } correlating with curve 𝛾 ( 𝑠 ) , then the Serret–Frenet formulae reads: 𝜒 1 ′ ( 𝑠 ) 𝜒 2 ′ ( 𝑠 ) 𝜒 3 ′ ( 𝑠 ) = 0 𝜅 0 𝜅 0 𝜏 0 − 𝜏 0 𝜒 1 ( 𝑠 ) 𝜒 2 ( 𝑠 ) 𝜒 3 ( 𝑠 ) , (1) where 𝜒 1 ( 𝑠 ) = 𝛾 ′ ( 𝑠 ) , 𝜒 2 ( 𝑠 ) = 𝛾 ″ ( 𝑠 ) / 𝛾 ″ ( 𝑠 ) and 𝜒 3 ( 𝑠 ) = 𝜒 1 ( 𝑠 ) × 𝜒 2 ( 𝑠 ) are named the unit tangent vector, the principal normal vector and the binormal vector, respectively. Here “prime” indicates the derivative with respect to the parameter s. The Serret–Frenet vector fields satisfy the relations: 𝜒 1 × 𝜒 2 = 𝜒 3 , 𝜒 1 × 𝜒 3 = − 𝜒 2 , 𝜒 2 × 𝜒 3 = − 𝜒 1 . (2) The Bishop frame or rotation minimizing frame (RMF) of 𝛾 ( 𝑠 ) is defined by 𝜉 ′ 𝜉 1 ′ 𝜉 2 ′ = 0 𝜇 1 − 𝜇 2 − 𝜇 1 0 0 − 𝜇 2 0 0 𝜉 𝜉 1 𝜉 2 , (3) Here, Bishop functions of 𝛾 ( 𝑠 ) are defined by 𝜇 1 ( 𝑠 ) = 𝜅 cos 𝜑 , 𝜇 2 ( 𝑠 ) = − 𝜅 sin 𝜑 . The relation matrix can be expressed as 𝜉 𝜉 1 𝜉 2 = 1 0 0 0 cos 𝜑 sin 𝜙 0 − sin 𝜑 cos 𝜑 𝜒 1 𝜒 2 𝜒 3 , (4) where 𝜑 ( 𝑠 ) ≥ 0 is a spacelike angle. One can show that 𝜇 1 2 + 𝜇 2 2 = 𝜅 2 , and 𝜗 = − tan − 1 𝜇 2 𝜇 1 ; 𝜇 1 ≠ 0 , 𝜑 ( 𝑠 ) = − ∫ 𝑠 𝑠 0 𝜏 𝑑 𝑠 + 𝜑 0 , 𝜑 0 = 𝜑 ( 𝑠 0 ) . (5)” | content_tags.definitions[1] |
| definition | ruled surface — “A ruled surface in 𝐸 1 3 is locally the map 𝐷 ( 𝛾 , 𝑥 ) : 𝐼 × 𝑅 → 𝐸 1 3 defined by 𝐷 ( 𝛾 , 𝑥 ) ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) , t ∈ 𝑅 , (6) where 𝛾 ( 𝑠 ) is named the base curve, and 𝑥 ( 𝑠 ) the director curve. The straight lines 𝑡 → 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) are named rulings. It is well known that 𝐷 ( 𝛾 , 𝑥 ) is a developable surface if and only if det( 𝛾 ′ ( 𝑠 ) , 𝑥 ( 𝑠 ) , 𝑥 ′ ( 𝑠 ) ) = 0 .” | content_tags.definitions[2] |
| definition | Bishop frame — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | content_tags.definitions[3] |
| definition | k-point contact — “Now, we should address the next concept regarding the contact of curves with some surfaces. Let F: 𝐸 1 3 → 𝑅 be a regular function, and 𝛾 : 𝑅 → 𝐸 1 3 be a smooth unit speed timelike Frenet–Serret curve. We say that 𝛾 ( 𝑠 ) and 𝐹 − 1 ( 0 ) have k-point contact for 𝑠 = 𝑠 0 if the function 𝑔 ( 𝑠 0 ) = 𝐹 ∘ 𝛾 satisfies 𝑔 ( 𝑠 0 ) = 𝑔 ′ ( 𝑠 0 ) = 𝑔 ″ ( 𝑠 0 ) = . . . = 𝑔 ( 𝑘 − 1 ) ( 𝑠 0 ) = 0 , and 𝑔 ( 𝑘 ) ( 𝑠 0 ) ≠ 0 . Further, for any fixed 𝑥 0 ∈ 𝐸 1 3 , we define the set BR 𝑗 ( 𝑥 0 ) = < 𝜉 𝑖 , 𝑥 0 − 𝛾 ( 𝑠 ) > = 0 , 𝑗 = 1 , 2 . We call it the first (resp. second) Bishop rectifying bundle of 𝛾 ( 𝑠 ) through 𝑥 0 when j = 1 (resp. j = 2).” | content_tags.definitions[4] |
| definition | Bishop height function — “Now we will consider two various sets of Bishop height functions, which will be helpful for studying the singularities of 𝐷 𝑖 ( 𝑠 , 𝑡 ) as follows [27,28]: 𝐻 𝑖 : 𝐼 × 𝐸 1 3 → 𝑅 , by 𝐻 𝑖 ( 𝑠 , 𝑥 ) = < 𝜉 𝑖 , 𝑥 − 𝛾 > ; i = 1, 2. We name it the first (resp. second) Bishop height function for i = 1 (resp. 2). We use the notation ℎ ix ( 𝑠 ) = 𝐻 𝑖 ( 𝑠 , 𝑥 ) for any fixed 𝑥 ∈ 𝐸 1 3 . From now on, we shall not often write the parameter s. Then, we have the following proposition:” | content_tags.definitions[5] |
| definition | versal unfolding — “Now we employ some public results on the singularity theory for families of function germs [27,28]. Let F: 𝑅 × 𝑅 𝑟 , ( 𝑠 0 , 𝑥 0 ) → 𝑅 be a smooth function, and 𝑓 ( 𝑠 ) = 𝐹 𝑥 0 ( 𝑠 , 𝑥 0 ) . Then F is called an r-parameter unfolding of 𝑓 ( 𝑠 ) . We say that 𝑓 ( 𝑠 ) has Ak-singularity at 𝑠 0 if 𝑓 ( 𝑝 ) ( 𝑠 0 ) = 0 for all 1 ≤ 𝑝 ≤ 𝑘 , and 𝑓 ( 𝑘 + 1 ) ( 𝑠 0 ) ≠ 0 . We also say that f has A ⩾ 𝑘 -singularity ( 𝑘 ⩾ 1 ) at s0. Let the 𝑘 − 1 -jet of the partial derivative ∂ 𝐹 ∂ 𝑥 𝑖 at 𝑠 0 be 𝑗 𝑘 − 1 ∂ 𝐹 ∂ 𝑥 𝑖 𝑠 , 𝑥 0 𝑠 0 = 𝛴 𝑗 = 0 𝑘 − 1 𝐿 𝑗 𝑖 𝑠 − 𝑠 0 𝑗 (without the constant term), for 𝑖 = 1 , . . . , 𝑟 . Then 𝐹 ( 𝑠 ) is called an p-versal unfolding if the 𝑘 × 𝑟 matrix of coefficients 𝐿 𝑗 𝑖 has rank k 𝑘 ≤ 𝑟 . Therefore, we write serious sets on the unfolding regarding to the above notations. The discriminant set of order t of F is the set 𝐷 𝐹 𝑡 = 𝑥 ∈ 𝑅 𝑟 ∣ ∃ 𝑠 ∈ 𝑅 , 𝐹 𝑠 , 𝑥 = , ∂ 𝐹 ∂ 𝑠 𝑠 , 𝑥 = . . . = ∂ 𝑡 𝐹 ∂ 𝑠 𝑡 𝑠 , 𝑥 = 0 at ( 𝑠 , 𝑥 ) . (8) Then 𝐷 𝐹 1 = 𝐷 𝐹 , and 𝐷 𝐹 2 is the set of singular points of 𝐷 𝐹 .” | content_tags.definitions[6] |
| definition | discriminant set — “Now we employ some public results on the singularity theory for families of function germs [27,28]. Let F: 𝑅 × 𝑅 𝑟 , ( 𝑠 0 , 𝑥 0 ) → 𝑅 be a smooth function, and 𝑓 ( 𝑠 ) = 𝐹 𝑥 0 ( 𝑠 , 𝑥 0 ) . Then F is called an r-parameter unfolding of 𝑓 ( 𝑠 ) . We say that 𝑓 ( 𝑠 ) has Ak-singularity at 𝑠 0 if 𝑓 ( 𝑝 ) ( 𝑠 0 ) = 0 for all 1 ≤ 𝑝 ≤ 𝑘 , and 𝑓 ( 𝑘 + 1 ) ( 𝑠 0 ) ≠ 0 . We also say that f has A ⩾ 𝑘 -singularity ( 𝑘 ⩾ 1 ) at s0. Let the 𝑘 − 1 -jet of the partial derivative ∂ 𝐹 ∂ 𝑥 𝑖 at 𝑠 0 be 𝑗 𝑘 − 1 ∂ 𝐹 ∂ 𝑥 𝑖 𝑠 , 𝑥 0 𝑠 0 = 𝛴 𝑗 = 0 𝑘 − 1 𝐿 𝑗 𝑖 𝑠 − 𝑠 0 𝑗 (without the constant term), for 𝑖 = 1 , . . . , 𝑟 . Then 𝐹 ( 𝑠 ) is called an p-versal unfolding if the 𝑘 × 𝑟 matrix of coefficients 𝐿 𝑗 𝑖 has rank k 𝑘 ≤ 𝑟 . Therefore, we write serious sets on the unfolding regarding to the above notations. The discriminant set of order t of F is the set 𝐷 𝐹 𝑡 = 𝑥 ∈ 𝑅 𝑟 ∣ ∃ 𝑠 ∈ 𝑅 , 𝐹 𝑠 , 𝑥 = , ∂ 𝐹 ∂ 𝑠 𝑠 , 𝑥 = . . . = ∂ 𝑡 𝐹 ∂ 𝑠 𝑡 𝑠 , 𝑥 = 0 at ( 𝑠 , 𝑥 ) . (8) Then 𝐷 𝐹 1 = 𝐷 𝐹 , and 𝐷 𝐹 2 is the set of singular points of 𝐷 𝐹 .” | content_tags.definitions[7] |
| concept | singularity — not defined in this paper's own words; a point where the surface map fails to be regular, modelled by a standard singularity germ — “In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results.” | concepts[0] |
| concept | developable surface — not defined in this paper's own words; a surface flattenable onto a plane without distortion — “A channel or canal surface is a surface traced by a one-parameter family of spheres, whose center is on a smooth space curve, its directrix or spine. If the radii of the traced spheres are stationary then the canal surface is named a tubular or sweeping surface. There are several famous examples: circular cylinder, right circular cone, torus, and rotation surface. This visualization is a generalization of the classical registration of a partner of a planar curve [1,2,3,4,5,6]. One of the eminent facts about sweeping surface is that the sweeping surface can be a developable surface, that is, a smooth surface can be flattened onto a plane without distortion. Therefore, the developable surface can be made out of sheet metal, since such a surface must be attainable by modification from a plane, and every point on such a surface lies on at least one straight line. Therefore, sweeping surfaces and developable surfaces have been paid interest to in engineering, architecture, and design, etc. [7,8,9,10,11,12,13,14,15,16,17,18].” | concepts[1] |
| concept | timelike developable surfaces — not defined in this paper's own words; developable ruled surfaces in Minkowski 3-space with a Lorentzian induced metric — “In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results.” | concepts[2] |
| concept | Minkowski 3-space — not defined in this paper's own words; real 3-space with a signature minus-plus-plus pseudo scalar product — “In this section, we present some definitions and employ notions that we will use in this paper (see for instance Refs. [1,24]). Let 𝑅 3 = { 𝑝 1 , 𝑝 2 , 𝑝 3 ∣ , 𝑝 𝑖 ∈ 𝑅 (i = 1, 2, 3)} be a 3-dimensional Cartesian space. For any 𝑞 = 𝑞 1 , 𝑞 2 , 𝑞 3 , and 𝑝 = 𝑝 1 , 𝑝 2 , 𝑝 3 ∈ 𝑅 3 , the pseudo scalar product of 𝑞 and 𝑝 is defined by < 𝑞 , 𝑝 > = − 𝑞 1 𝑝 1 + 𝑞 2 𝑝 2 + 𝑞 3 𝑝 3 . We name ( 𝑅 3 , < , > ) Minkowski 3-space. We write 𝐸 1 3 instead of ( 𝑅 3 , < , > ) . We say that a non-zero vector 𝑞 ∈ 𝐸 1 3 is spacelike, lightlike, or timelike if < 𝑞 , 𝑞 > > 0 , < 𝑞 , 𝑞 > = 0 or < 𝑞 , 𝑞 > < 0 , respectively. The norm of the vector 𝑞 ∈ 𝐸 1 3 is defined to be 𝑞 = < 𝑞 , 𝑞 > . For any two vectors 𝑞 , 𝑝 ∈ 𝐸 1 3 , we define the cross product 𝑞 × 𝑝 by 𝑞 × 𝑝 = − 𝑖 𝑗 𝑘 𝑞 1 𝑞 2 𝑞 3 𝑝 1 𝑝 2 𝑝 3 = − ( 𝑞 2 𝑝 3 − 𝑞 3 𝑝 2 ) , ( 𝑞 3 𝑝 1 − 𝑞 1 𝑝 3 ) , ( 𝑞 1 𝑝 2 − 𝑞 2 𝑝 1 ) , where 𝑖 , 𝑗 , 𝑘 is the canonical basis of 𝐸 1 3 . The hyperbolic ( 𝐻 + 2 ) and Lorentzian unit spheres ( 𝑆 1 2 ), respectively, are 𝐻 + 2 = 𝑞 ∈ 𝐸 1 3 ∣ 𝑞 2 = − 𝑞 1 2 + 𝑞 2 2 + 𝑞 3 2 = − 1 , and 𝑆 1 2 = 𝑞 ∈ 𝐸 1 3 ∣ 𝑞 2 = − 𝑞 1 2 + 𝑞 2 2 + 𝑞 3 2 = 1 . Let 𝛾 : 𝐼 → 𝐸 1 3 be a smooth curve in 𝐸 1 3 , where I is an interval of 𝑅 . We call 𝛾 is 𝑠 𝑝 𝑎 𝑐 𝑒 𝑙 𝑖 𝑘 𝑒 , 𝑡 𝑖 𝑚 𝑒 𝑙 𝑖 𝑘 𝑒 , 𝑙 𝑖 𝑔 ℎ 𝑡 𝑙 𝑖 𝑘 𝑒 if 𝛾 ′ ( 𝑡 ) = 𝑑 𝛾 𝑑 𝑡 ( 𝑡 ) is spacelike, timelike, lightlike for any 𝑡 ∈ 𝑅 , respectively. A surface in the Minkowski 3-space 𝐸 1 3 is named a timelike surface if the induced metric on the surface is a Lorentzian metric and is named a spacelike surface if the induced metric on the surface is a positive definite Riemannian metric, i.e., the normal vector on spacelike (timelike) surface is a timelike (spacelike) vector. Let 𝛾 = 𝛾 ( 𝑠 ) be a unit speed timelike curve in 𝐸 1 3 ; by 𝜅 ( 𝑠 ) and 𝜏 ( 𝑠 ) we indicate the natural curvature and torsion, respectively. 𝛾 ( 𝑠 ) is called the Serret–Frenet curve if 𝜅 > 0 , and 𝜏 ≠ 0 . Consider the Serret–Frenet frame { 𝜒 1 ( 𝑠 ) , 𝜒 2 ( 𝑠 ) , 𝜒 3 ( 𝑠 ) } correlating with curve 𝛾 ( 𝑠 ) , then the Serret–Frenet formulae reads: 𝜒 1 ′ ( 𝑠 ) 𝜒 2 ′ ( 𝑠 ) 𝜒 3 ′ ( 𝑠 ) = 0 𝜅 0 𝜅 0 𝜏 0 − 𝜏 0 𝜒 1 ( 𝑠 ) 𝜒 2 ( 𝑠 ) 𝜒 3 ( 𝑠 ) , (1) where 𝜒 1 ( 𝑠 ) = 𝛾 ′ ( 𝑠 ) , 𝜒 2 ( 𝑠 ) = 𝛾 ″ ( 𝑠 ) / 𝛾 ″ ( 𝑠 ) and 𝜒 3 ( 𝑠 ) = 𝜒 1 ( 𝑠 ) × 𝜒 2 ( 𝑠 ) are named the unit tangent vector, the principal normal vector and the binormal vector, respectively. Here “prime” indicates the derivative with respect to the parameter s. The Serret–Frenet vector fields satisfy the relations: 𝜒 1 × 𝜒 2 = 𝜒 3 , 𝜒 1 × 𝜒 3 = − 𝜒 2 , 𝜒 2 × 𝜒 3 = − 𝜒 1 . (2) The Bishop frame or rotation minimizing frame (RMF) of 𝛾 ( 𝑠 ) is defined by 𝜉 ′ 𝜉 1 ′ 𝜉 2 ′ = 0 𝜇 1 − 𝜇 2 − 𝜇 1 0 0 − 𝜇 2 0 0 𝜉 𝜉 1 𝜉 2 , (3) Here, Bishop functions of 𝛾 ( 𝑠 ) are defined by 𝜇 1 ( 𝑠 ) = 𝜅 cos 𝜑 , 𝜇 2 ( 𝑠 ) = − 𝜅 sin 𝜑 . The relation matrix can be expressed as 𝜉 𝜉 1 𝜉 2 = 1 0 0 0 cos 𝜑 sin 𝜙 0 − sin 𝜑 cos 𝜑 𝜒 1 𝜒 2 𝜒 3 , (4) where 𝜑 ( 𝑠 ) ≥ 0 is a spacelike angle. One can show that 𝜇 1 2 + 𝜇 2 2 = 𝜅 2 , and 𝜗 = − tan − 1 𝜇 2 𝜇 1 ; 𝜇 1 ≠ 0 , 𝜑 ( 𝑠 ) = − ∫ 𝑠 𝑠 0 𝜏 𝑑 𝑠 + 𝜑 0 , 𝜑 0 = 𝜑 ( 𝑠 0 ) . (5)” | concepts[3] |
| concept | Bishop frame — not defined in this paper's own words; the rotation-minimizing moving frame of a space curve — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | concepts[4] |
| concept | ruled surface — not defined in this paper's own words; a surface swept by a straight line moving along a base curve — “A ruled surface in 𝐸 1 3 is locally the map 𝐷 ( 𝛾 , 𝑥 ) : 𝐼 × 𝑅 → 𝐸 1 3 defined by 𝐷 ( 𝛾 , 𝑥 ) ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) , t ∈ 𝑅 , (6) where 𝛾 ( 𝑠 ) is named the base curve, and 𝑥 ( 𝑠 ) the director curve. The straight lines 𝑡 → 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) are named rulings. It is well known that 𝐷 ( 𝛾 , 𝑥 ) is a developable surface if and only if det( 𝛾 ′ ( 𝑠 ) , 𝑥 ( 𝑠 ) , 𝑥 ′ ( 𝑠 ) ) = 0 .” | concepts[5] |
| concept | cuspidal edge — not defined in this paper's own words; the singularity model C(2,3) x R the surface is locally diffeomorphic to — “(1) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the cuspidal edge (CE) C(2,3) × 𝑅 at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , and 𝜇 𝑖 ′ ( 𝑠 0 ) ≠ 0 . (2) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the Swallowtail (SW) at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , 𝜇 𝑖 ′ ( 𝑠 0 ) = 0 , and 𝜇 𝑖 − 1 ( 𝑠 0 ) ″ ≠ 0 .” | concepts[6] |
| concept | swallowtail — not defined in this paper's own words; the higher singularity model the surface is locally diffeomorphic to under stronger degeneracy — “(1) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the cuspidal edge (CE) C(2,3) × 𝑅 at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , and 𝜇 𝑖 ′ ( 𝑠 0 ) ≠ 0 . (2) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the Swallowtail (SW) at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , 𝜇 𝑖 ′ ( 𝑠 0 ) = 0 , and 𝜇 𝑖 − 1 ( 𝑠 0 ) ″ ≠ 0 .” | concepts[7] |
| concept | Bishop height function — not defined in this paper's own words; the function whose singularities encode the surface singularities — “Now we will consider two various sets of Bishop height functions, which will be helpful for studying the singularities of 𝐷 𝑖 ( 𝑠 , 𝑡 ) as follows [27,28]: 𝐻 𝑖 : 𝐼 × 𝐸 1 3 → 𝑅 , by 𝐻 𝑖 ( 𝑠 , 𝑥 ) = < 𝜉 𝑖 , 𝑥 − 𝛾 > ; i = 1, 2. We name it the first (resp. second) Bishop height function for i = 1 (resp. 2). We use the notation ℎ ix ( 𝑠 ) = 𝐻 𝑖 ( 𝑠 , 𝑥 ) for any fixed 𝑥 ∈ 𝐸 1 3 . From now on, we shall not often write the parameter s. Then, we have the following proposition:” | concepts[8] |
| concept | k-point contact — not defined in this paper's own words; the order of tangency between the curve and a level set — “Now, we should address the next concept regarding the contact of curves with some surfaces. Let F: 𝐸 1 3 → 𝑅 be a regular function, and 𝛾 : 𝑅 → 𝐸 1 3 be a smooth unit speed timelike Frenet–Serret curve. We say that 𝛾 ( 𝑠 ) and 𝐹 − 1 ( 0 ) have k-point contact for 𝑠 = 𝑠 0 if the function 𝑔 ( 𝑠 0 ) = 𝐹 ∘ 𝛾 satisfies 𝑔 ( 𝑠 0 ) = 𝑔 ′ ( 𝑠 0 ) = 𝑔 ″ ( 𝑠 0 ) = . . . = 𝑔 ( 𝑘 − 1 ) ( 𝑠 0 ) = 0 , and 𝑔 ( 𝑘 ) ( 𝑠 0 ) ≠ 0 . Further, for any fixed 𝑥 0 ∈ 𝐸 1 3 , we define the set BR 𝑗 ( 𝑥 0 ) = < 𝜉 𝑖 , 𝑥 0 − 𝛾 ( 𝑠 ) > = 0 , 𝑗 = 1 , 2 . We call it the first (resp. second) Bishop rectifying bundle of 𝛾 ( 𝑠 ) through 𝑥 0 when j = 1 (resp. j = 2).” | concepts[9] |
| concept | versal unfolding — not defined in this paper's own words; an unfolding rich enough to carry the standard singularity model on its discriminant — “Now we employ some public results on the singularity theory for families of function germs [27,28]. Let F: 𝑅 × 𝑅 𝑟 , ( 𝑠 0 , 𝑥 0 ) → 𝑅 be a smooth function, and 𝑓 ( 𝑠 ) = 𝐹 𝑥 0 ( 𝑠 , 𝑥 0 ) . Then F is called an r-parameter unfolding of 𝑓 ( 𝑠 ) . We say that 𝑓 ( 𝑠 ) has Ak-singularity at 𝑠 0 if 𝑓 ( 𝑝 ) ( 𝑠 0 ) = 0 for all 1 ≤ 𝑝 ≤ 𝑘 , and 𝑓 ( 𝑘 + 1 ) ( 𝑠 0 ) ≠ 0 . We also say that f has A ⩾ 𝑘 -singularity ( 𝑘 ⩾ 1 ) at s0. Let the 𝑘 − 1 -jet of the partial derivative ∂ 𝐹 ∂ 𝑥 𝑖 at 𝑠 0 be 𝑗 𝑘 − 1 ∂ 𝐹 ∂ 𝑥 𝑖 𝑠 , 𝑥 0 𝑠 0 = 𝛴 𝑗 = 0 𝑘 − 1 𝐿 𝑗 𝑖 𝑠 − 𝑠 0 𝑗 (without the constant term), for 𝑖 = 1 , . . . , 𝑟 . Then 𝐹 ( 𝑠 ) is called an p-versal unfolding if the 𝑘 × 𝑟 matrix of coefficients 𝐿 𝑗 𝑖 has rank k 𝑘 ≤ 𝑟 . Therefore, we write serious sets on the unfolding regarding to the above notations. The discriminant set of order t of F is the set 𝐷 𝐹 𝑡 = 𝑥 ∈ 𝑅 𝑟 ∣ ∃ 𝑠 ∈ 𝑅 , 𝐹 𝑠 , 𝑥 = , ∂ 𝐹 ∂ 𝑠 𝑠 , 𝑥 = . . . = ∂ 𝑡 𝐹 ∂ 𝑠 𝑡 𝑠 , 𝑥 = 0 at ( 𝑠 , 𝑥 ) . (8) Then 𝐷 𝐹 1 = 𝐷 𝐹 , and 𝐷 𝐹 2 is the set of singular points of 𝐷 𝐹 .” | concepts[10] |
| concept | discriminant set — not defined in this paper's own words; the locus where the unfolding and its s-derivatives vanish — “Now we employ some public results on the singularity theory for families of function germs [27,28]. Let F: 𝑅 × 𝑅 𝑟 , ( 𝑠 0 , 𝑥 0 ) → 𝑅 be a smooth function, and 𝑓 ( 𝑠 ) = 𝐹 𝑥 0 ( 𝑠 , 𝑥 0 ) . Then F is called an r-parameter unfolding of 𝑓 ( 𝑠 ) . We say that 𝑓 ( 𝑠 ) has Ak-singularity at 𝑠 0 if 𝑓 ( 𝑝 ) ( 𝑠 0 ) = 0 for all 1 ≤ 𝑝 ≤ 𝑘 , and 𝑓 ( 𝑘 + 1 ) ( 𝑠 0 ) ≠ 0 . We also say that f has A ⩾ 𝑘 -singularity ( 𝑘 ⩾ 1 ) at s0. Let the 𝑘 − 1 -jet of the partial derivative ∂ 𝐹 ∂ 𝑥 𝑖 at 𝑠 0 be 𝑗 𝑘 − 1 ∂ 𝐹 ∂ 𝑥 𝑖 𝑠 , 𝑥 0 𝑠 0 = 𝛴 𝑗 = 0 𝑘 − 1 𝐿 𝑗 𝑖 𝑠 − 𝑠 0 𝑗 (without the constant term), for 𝑖 = 1 , . . . , 𝑟 . Then 𝐹 ( 𝑠 ) is called an p-versal unfolding if the 𝑘 × 𝑟 matrix of coefficients 𝐿 𝑗 𝑖 has rank k 𝑘 ≤ 𝑟 . Therefore, we write serious sets on the unfolding regarding to the above notations. The discriminant set of order t of F is the set 𝐷 𝐹 𝑡 = 𝑥 ∈ 𝑅 𝑟 ∣ ∃ 𝑠 ∈ 𝑅 , 𝐹 𝑠 , 𝑥 = , ∂ 𝐹 ∂ 𝑠 𝑠 , 𝑥 = . . . = ∂ 𝑡 𝐹 ∂ 𝑠 𝑡 𝑠 , 𝑥 = 0 at ( 𝑠 , 𝑥 ) . (8) Then 𝐷 𝐹 1 = 𝐷 𝐹 , and 𝐷 𝐹 2 is the set of singular points of 𝐷 𝐹 .” | concepts[11] |
| concept | Legendrian duality — not defined in this paper's own words; a duality between spherical indicatrices used to relate frame vectors — “In this section, we will investigate the singularities of two timelike developable ruled surfaces according to Bishop frame of regular unit speed timelike curve in Minkowski 3-space 𝐸 1 3 . Next we deduce Legendrian dualities between spherical indicatrixes of curves by using the theory of Legendrian duality as follows: By the theory of Legendrian duality we define one-forms < 𝑑 𝑣 , 𝜔 > = − 𝜔 1 𝑑 𝑣 1 + 𝜔 2 𝑑 𝑣 2 + 𝜔 3 𝑑 𝑣 3 , and < 𝑑 𝜔 , 𝑣 > = − 𝑣 1 𝑑 𝜔 1 + 𝑣 2 𝑑 𝜔 2 + 𝑣 3 𝑑 𝜔 3 . Let 𝑆 1 2 be the Lorentzian (de Sitter space) unit sphere in 𝐸 1 3 . Then we have the following double fibration: ( 𝑎 ) 𝑆 1 2 × 𝑆 1 2 ⊇ △ = ( 𝑣 , 𝜔 ) ∣ < 𝜔 , 𝑣 > = 0 , ( 𝑏 ) 𝜋 11 : △ → 𝑆 1 2 , 𝜋 12 : △ → 𝑆 1 2 , ( 𝑐 ) 𝜃 11 = < 𝑑 𝑣 , 𝜔 > ∣ △ , 𝜃 12 = < 𝑣 , 𝑑 𝜔 > ∣ △ . Here 𝜋 11 ( 𝑣 , 𝜔 ) = 𝑣 , 𝜋 12 ( 𝑣 , 𝜔 ) = 𝜔 . 𝜃 11 − 1 ( 0 ) , and 𝜃 12 − 1 ( 0 ) define the same tangent plane on △, which is denoted by K, and indicates that (△,K) is a contact manifold and each of 𝜋 1 𝑗 (j = 1, 2) is Legendrian fibration. If there exists an isotropic mapping i: 𝐿 → △ , which means that 𝑖 × 𝜃 11 = 0 , we say that 𝜋 11 ( 𝑖 ( 𝐿 ) ) and 𝜋 12 ( 𝑖 ( 𝐿 ) ) are △-dual to each other. It is uncomplicated to see that the condition 𝑖 × 𝜃 11 = 0 is equivalent to 𝑖 × 𝜃 12 = 0 . Then we have:” | concepts[12] |
| concept | timelike curve — not defined in this paper's own words; a curve whose velocity vector is timelike in the Lorentzian metric — “In this section, we present some definitions and employ notions that we will use in this paper (see for instance Refs. [1,24]). Let 𝑅 3 = { 𝑝 1 , 𝑝 2 , 𝑝 3 ∣ , 𝑝 𝑖 ∈ 𝑅 (i = 1, 2, 3)} be a 3-dimensional Cartesian space. For any 𝑞 = 𝑞 1 , 𝑞 2 , 𝑞 3 , and 𝑝 = 𝑝 1 , 𝑝 2 , 𝑝 3 ∈ 𝑅 3 , the pseudo scalar product of 𝑞 and 𝑝 is defined by < 𝑞 , 𝑝 > = − 𝑞 1 𝑝 1 + 𝑞 2 𝑝 2 + 𝑞 3 𝑝 3 . We name ( 𝑅 3 , < , > ) Minkowski 3-space. We write 𝐸 1 3 instead of ( 𝑅 3 , < , > ) . We say that a non-zero vector 𝑞 ∈ 𝐸 1 3 is spacelike, lightlike, or timelike if < 𝑞 , 𝑞 > > 0 , < 𝑞 , 𝑞 > = 0 or < 𝑞 , 𝑞 > < 0 , respectively. The norm of the vector 𝑞 ∈ 𝐸 1 3 is defined to be 𝑞 = < 𝑞 , 𝑞 > . For any two vectors 𝑞 , 𝑝 ∈ 𝐸 1 3 , we define the cross product 𝑞 × 𝑝 by 𝑞 × 𝑝 = − 𝑖 𝑗 𝑘 𝑞 1 𝑞 2 𝑞 3 𝑝 1 𝑝 2 𝑝 3 = − ( 𝑞 2 𝑝 3 − 𝑞 3 𝑝 2 ) , ( 𝑞 3 𝑝 1 − 𝑞 1 𝑝 3 ) , ( 𝑞 1 𝑝 2 − 𝑞 2 𝑝 1 ) , where 𝑖 , 𝑗 , 𝑘 is the canonical basis of 𝐸 1 3 . The hyperbolic ( 𝐻 + 2 ) and Lorentzian unit spheres ( 𝑆 1 2 ), respectively, are 𝐻 + 2 = 𝑞 ∈ 𝐸 1 3 ∣ 𝑞 2 = − 𝑞 1 2 + 𝑞 2 2 + 𝑞 3 2 = − 1 , and 𝑆 1 2 = 𝑞 ∈ 𝐸 1 3 ∣ 𝑞 2 = − 𝑞 1 2 + 𝑞 2 2 + 𝑞 3 2 = 1 . Let 𝛾 : 𝐼 → 𝐸 1 3 be a smooth curve in 𝐸 1 3 , where I is an interval of 𝑅 . We call 𝛾 is 𝑠 𝑝 𝑎 𝑐 𝑒 𝑙 𝑖 𝑘 𝑒 , 𝑡 𝑖 𝑚 𝑒 𝑙 𝑖 𝑘 𝑒 , 𝑙 𝑖 𝑔 ℎ 𝑡 𝑙 𝑖 𝑘 𝑒 if 𝛾 ′ ( 𝑡 ) = 𝑑 𝛾 𝑑 𝑡 ( 𝑡 ) is spacelike, timelike, lightlike for any 𝑡 ∈ 𝑅 , respectively. A surface in the Minkowski 3-space 𝐸 1 3 is named a timelike surface if the induced metric on the surface is a Lorentzian metric and is named a spacelike surface if the induced metric on the surface is a positive definite Riemannian metric, i.e., the normal vector on spacelike (timelike) surface is a timelike (spacelike) vector. Let 𝛾 = 𝛾 ( 𝑠 ) be a unit speed timelike curve in 𝐸 1 3 ; by 𝜅 ( 𝑠 ) and 𝜏 ( 𝑠 ) we indicate the natural curvature and torsion, respectively. 𝛾 ( 𝑠 ) is called the Serret–Frenet curve if 𝜅 > 0 , and 𝜏 ≠ 0 . Consider the Serret–Frenet frame { 𝜒 1 ( 𝑠 ) , 𝜒 2 ( 𝑠 ) , 𝜒 3 ( 𝑠 ) } correlating with curve 𝛾 ( 𝑠 ) , then the Serret–Frenet formulae reads: 𝜒 1 ′ ( 𝑠 ) 𝜒 2 ′ ( 𝑠 ) 𝜒 3 ′ ( 𝑠 ) = 0 𝜅 0 𝜅 0 𝜏 0 − 𝜏 0 𝜒 1 ( 𝑠 ) 𝜒 2 ( 𝑠 ) 𝜒 3 ( 𝑠 ) , (1) where 𝜒 1 ( 𝑠 ) = 𝛾 ′ ( 𝑠 ) , 𝜒 2 ( 𝑠 ) = 𝛾 ″ ( 𝑠 ) / 𝛾 ″ ( 𝑠 ) and 𝜒 3 ( 𝑠 ) = 𝜒 1 ( 𝑠 ) × 𝜒 2 ( 𝑠 ) are named the unit tangent vector, the principal normal vector and the binormal vector, respectively. Here “prime” indicates the derivative with respect to the parameter s. The Serret–Frenet vector fields satisfy the relations: 𝜒 1 × 𝜒 2 = 𝜒 3 , 𝜒 1 × 𝜒 3 = − 𝜒 2 , 𝜒 2 × 𝜒 3 = − 𝜒 1 . (2) The Bishop frame or rotation minimizing frame (RMF) of 𝛾 ( 𝑠 ) is defined by 𝜉 ′ 𝜉 1 ′ 𝜉 2 ′ = 0 𝜇 1 − 𝜇 2 − 𝜇 1 0 0 − 𝜇 2 0 0 𝜉 𝜉 1 𝜉 2 , (3) Here, Bishop functions of 𝛾 ( 𝑠 ) are defined by 𝜇 1 ( 𝑠 ) = 𝜅 cos 𝜑 , 𝜇 2 ( 𝑠 ) = − 𝜅 sin 𝜑 . The relation matrix can be expressed as 𝜉 𝜉 1 𝜉 2 = 1 0 0 0 cos 𝜑 sin 𝜙 0 − sin 𝜑 cos 𝜑 𝜒 1 𝜒 2 𝜒 3 , (4) where 𝜑 ( 𝑠 ) ≥ 0 is a spacelike angle. One can show that 𝜇 1 2 + 𝜇 2 2 = 𝜅 2 , and 𝜗 = − tan − 1 𝜇 2 𝜇 1 ; 𝜇 1 ≠ 0 , 𝜑 ( 𝑠 ) = − ∫ 𝑠 𝑠 0 𝜏 𝑑 𝑠 + 𝜑 0 , 𝜑 0 = 𝜑 ( 𝑠 0 ) . (5)” | concepts[13] |
| concept | Serret–Frenet frame — not defined in this paper's own words; the classical moving frame of a curve, undefined at inflection points — “In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frame can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frame becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frame, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frame in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frame can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frame. We can find many articles related with the Minkowski Bishop frames, for example, Refs. [20,21,22,23].” | concepts[14] |
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Singularities for Timelike Developable Surfacesterm in Minkowski 3-Spaceterm Abstract Abstract In this paper, we consider the singularities and geometrical properties of timelike developable surfacesterm with Bishop frameterm in Minkowski 3-spaceterm. Taking advantage of the singularityterm theory, we give the classification of generic singularities of these developable surfacesterm. Furthermore, an example of application is given to illustrate the applications of the results. Keywords: timelike developable surfacesterm; singularities; developable surfacesterm 1. Introduction A channel or canal surface is a surface traced by a one-parameter family of spheres, whose center is on a smooth space curve, its directrix or spine. If the radii of the traced spheres are stationary then the canal surface is named a tubular or sweeping surface. There are several famous examples: circular cylinder, right circular cone, torus, and rotation surface. This visualization is a generalization of the classical registration of a partner of a planar curve [1,2,3,4,5,6]. One of the eminent facts about sweeping surface is that the sweeping surface can be a developable surfaceterm, that is, a smooth surface can be flattened onto a plane without distortion. Therefore, the developable surfaceterm can be made out of sheet metal, since such a surface must be attainable by modification from a plane, and every point on such a surface lies on at least one straight line. Therefore, sweeping surfaces and developable surfacesterm have been paid interest to in engineering, architecture, and design, etc. [7,8,9,10,11,12,13,14,15,16,17,18].
In Euclidean 3-space 𝐸 3 , in spite of the fact that the Serret–Frenet frameterm can simply be calculated, it is not continuously defined for a 𝐶 1 -continuous space curve, and even for a 𝐶 2 -continuous space curve the Serret–Frenet frameterm becomes undefined at an inflection point (i.e., curvature κ = 0), thus causing unacceptable discontinuity when utilized for surface modeling. Thus, in Ref. [19], Bishop defined a novel frame named the alternative frame or Bishop frameterm, which could yield the desired means to move smoothly on a space curve. We know that the Bishop frameterm in Euclidean 3-space is a sharp tool used to investigate the topological and geometrical properties of curves in Euclidean 3-space. Even though the second derivative of the curves are vanishing, the Bishop frameterm can still work for them. The idea can be expanded to the Minkowski space. Therefore, we give the Minkowski version moving frame, which is the Minkowski Bishop frameterm. We can find many articles related with the Minkowski Bishop framesterm, for example, Refs. [20,21,22,23].
The present work studies the singularities and geometrical properties of timelike developable surfacesterm according to the Minkowski Bishop frameterm. The field, as far as we are aware, is not covered by any articles, though it has been carefully and ably worked by many researchers. Thus, the present paper hopes to help such a need, and it is inspired by the works of Izumiya, and Haiming et al. [3,23,24,25,26]. In this article, we construct the Bishop frameterm of speed timelike curveterm and take advantage of the singularityterm theory to study the singularities and geometrical properties of timelike developable.
2. Preliminaries In this section, we present some definitions and employ notions that we will use in this paper (see for instance Refs. [1,24]). Let 𝑅 3 = { 𝑝 1 , 𝑝 2 , 𝑝 3 ∣ , 𝑝 𝑖 ∈ 𝑅 (i = 1, 2, 3)} be a 3-dimensional Cartesian space. For any 𝑞 = 𝑞 1 , 𝑞 2 , 𝑞 3 , and 𝑝 = 𝑝 1 , 𝑝 2 , 𝑝 3 ∈ 𝑅 3 , the pseudo scalar product of 𝑞 and 𝑝 is defined by < 𝑞 , 𝑝 > = − 𝑞 1 𝑝 1 + 𝑞 2 𝑝 2 + 𝑞 3 𝑝 3 . We name ( 𝑅 3 , < , > ) Minkowski 3-spaceterm. We write 𝐸 1 3 instead of ( 𝑅 3 , < , > ) . We say that a non-zero vector 𝑞 ∈ 𝐸 1 3 is spacelike, lightlike, or timelike if < 𝑞 , 𝑞 > > 0 , < 𝑞 , 𝑞 > = 0 or < 𝑞 , 𝑞 > < 0 , respectively. The norm of the vector 𝑞 ∈ 𝐸 1 3 is defined to be 𝑞 = < 𝑞 , 𝑞 > . For any two vectors 𝑞 , 𝑝 ∈ 𝐸 1 3 , we define the cross product 𝑞 × 𝑝 by 𝑞 × 𝑝 = − 𝑖 𝑗 𝑘 𝑞 1 𝑞 2 𝑞 3 𝑝 1 𝑝 2 𝑝 3 = − ( 𝑞 2 𝑝 3 − 𝑞 3 𝑝 2 ) , ( 𝑞 3 𝑝 1 − 𝑞 1 𝑝 3 ) , ( 𝑞 1 𝑝 2 − 𝑞 2 𝑝 1 ) , where 𝑖 , 𝑗 , 𝑘 is the canonical basis of 𝐸 1 3 . The hyperbolic ( 𝐻 + 2 ) and Lorentzian unit spheres ( 𝑆 1 2 ), respectively, are 𝐻 + 2 = 𝑞 ∈ 𝐸 1 3 ∣ 𝑞 2 = − 𝑞 1 2 + 𝑞 2 2 + 𝑞 3 2 = − 1 , and 𝑆 1 2 = 𝑞 ∈ 𝐸 1 3 ∣ 𝑞 2 = − 𝑞 1 2 + 𝑞 2 2 + 𝑞 3 2 = 1 . Let 𝛾 : 𝐼 → 𝐸 1 3 be a smooth curve in 𝐸 1 3 , where I is an interval of 𝑅 . We call 𝛾 is 𝑠 𝑝 𝑎 𝑐 𝑒 𝑙 𝑖 𝑘 𝑒 , 𝑡 𝑖 𝑚 𝑒 𝑙 𝑖 𝑘 𝑒 , 𝑙 𝑖 𝑔 ℎ 𝑡 𝑙 𝑖 𝑘 𝑒 if 𝛾 ′ ( 𝑡 ) = 𝑑 𝛾 𝑑 𝑡 ( 𝑡 ) is spacelike, timelike, lightlike for any 𝑡 ∈ 𝑅 , respectively. A surface in the Minkowski 3-spaceterm 𝐸 1 3 is named a timelike surface if the induced metric on the surface is a Lorentzian metric and is named a spacelike surface if the induced metric on the surface is a positive definite Riemannian metric, i.e., the normal vector on spacelike (timelike) surface is a timelike (spacelike) vector. Let 𝛾 = 𝛾 ( 𝑠 ) be a unit speed timelike curveterm in 𝐸 1 3 ; by 𝜅 ( 𝑠 ) and 𝜏 ( 𝑠 ) we indicate the natural curvature and torsion, respectively. 𝛾 ( 𝑠 ) is called the Serret–Frenet curve if 𝜅 > 0 , and 𝜏 ≠ 0 . Consider the Serret–Frenet frameterm { 𝜒 1 ( 𝑠 ) , 𝜒 2 ( 𝑠 ) , 𝜒 3 ( 𝑠 ) } correlating with curve 𝛾 ( 𝑠 ) , then the Serret–Frenet formulae reads: 𝜒 1 ′ ( 𝑠 ) 𝜒 2 ′ ( 𝑠 ) 𝜒 3 ′ ( 𝑠 ) = 0 𝜅 0 𝜅 0 𝜏 0 − 𝜏 0 𝜒 1 ( 𝑠 ) 𝜒 2 ( 𝑠 ) 𝜒 3 ( 𝑠 ) , (1) where 𝜒 1 ( 𝑠 ) = 𝛾 ′ ( 𝑠 ) , 𝜒 2 ( 𝑠 ) = 𝛾 ″ ( 𝑠 ) / 𝛾 ″ ( 𝑠 ) and 𝜒 3 ( 𝑠 ) = 𝜒 1 ( 𝑠 ) × 𝜒 2 ( 𝑠 ) are named the unit tangent vector, the principal normal vector and the binormal vector, respectively. Here “prime” indicates the derivative with respect to the parameter s. The Serret–Frenet vector fields satisfy the relations: 𝜒 1 × 𝜒 2 = 𝜒 3 , 𝜒 1 × 𝜒 3 = − 𝜒 2 , 𝜒 2 × 𝜒 3 = − 𝜒 1 . (2) The Bishop frameterm or rotation minimizing frame (RMF) of 𝛾 ( 𝑠 ) is defined by 𝜉 ′ 𝜉 1 ′ 𝜉 2 ′ = 0 𝜇 1 − 𝜇 2 − 𝜇 1 0 0 − 𝜇 2 0 0 𝜉 𝜉 1 𝜉 2 , (3) Here, Bishop functions of 𝛾 ( 𝑠 ) are defined by 𝜇 1 ( 𝑠 ) = 𝜅 cos 𝜑 , 𝜇 2 ( 𝑠 ) = − 𝜅 sin 𝜑 . The relation matrix can be expressed as 𝜉 𝜉 1 𝜉 2 = 1 0 0 0 cos 𝜑 sin 𝜙 0 − sin 𝜑 cos 𝜑 𝜒 1 𝜒 2 𝜒 3 , (4) where 𝜑 ( 𝑠 ) ≥ 0 is a spacelike angle. One can show that 𝜇 1 2 + 𝜇 2 2 = 𝜅 2 , and 𝜗 = − tan − 1 𝜇 2 𝜇 1 ; 𝜇 1 ≠ 0 , 𝜑 ( 𝑠 ) = − ∫ 𝑠 𝑠 0 𝜏 𝑑 𝑠 + 𝜑 0 , 𝜑 0 = 𝜑 ( 𝑠 0 ) . (5) A ruled surfaceterm in 𝐸 1 3 is locally the map 𝐷 ( 𝛾 , 𝑥 ) : 𝐼 × 𝑅 → 𝐸 1 3 defined by 𝐷 ( 𝛾 , 𝑥 ) ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) , t ∈ 𝑅 , (6) where 𝛾 ( 𝑠 ) is named the base curve, and 𝑥 ( 𝑠 ) the director curve. The straight lines 𝑡 → 𝛾 ( 𝑠 ) + 𝑡 𝑥 ( 𝑠 ) are named rulings. It is well known that 𝐷 ( 𝛾 , 𝑥 ) is a developable surfaceterm if and only if det( 𝛾 ′ ( 𝑠 ) , 𝑥 ( 𝑠 ) , 𝑥 ′ ( 𝑠 ) ) = 0 .
3. Singularities for Timelike Developable Surfacesterm In this section, we will investigate the singularities of two timelike developable ruled surfacesterm according to Bishop frameterm of regular unit speed timelike curveterm in Minkowski 3-spaceterm 𝐸 1 3 . Next we deduce Legendrian dualities between spherical indicatrixes of curves by using the theory of Legendrian dualityterm as follows: By the theory of Legendrian dualityterm we define one-forms < 𝑑 𝑣 , 𝜔 > = − 𝜔 1 𝑑 𝑣 1 + 𝜔 2 𝑑 𝑣 2 + 𝜔 3 𝑑 𝑣 3 , and < 𝑑 𝜔 , 𝑣 > = − 𝑣 1 𝑑 𝜔 1 + 𝑣 2 𝑑 𝜔 2 + 𝑣 3 𝑑 𝜔 3 . Let 𝑆 1 2 be the Lorentzian (de Sitter space) unit sphere in 𝐸 1 3 . Then we have the following double fibration: ( 𝑎 ) 𝑆 1 2 × 𝑆 1 2 ⊇ △ = ( 𝑣 , 𝜔 ) ∣ < 𝜔 , 𝑣 > = 0 , ( 𝑏 ) 𝜋 11 : △ → 𝑆 1 2 , 𝜋 12 : △ → 𝑆 1 2 , ( 𝑐 ) 𝜃 11 = < 𝑑 𝑣 , 𝜔 > ∣ △ , 𝜃 12 = < 𝑣 , 𝑑 𝜔 > ∣ △ . Here 𝜋 11 ( 𝑣 , 𝜔 ) = 𝑣 , 𝜋 12 ( 𝑣 , 𝜔 ) = 𝜔 . 𝜃 11 − 1 ( 0 ) , and 𝜃 12 − 1 ( 0 ) define the same tangent plane on △, which is denoted by K, and indicates that (△,K) is a contact manifold and each of 𝜋 1 𝑗 (j = 1, 2) is Legendrian fibration. If there exists an isotropic mapping i: 𝐿 → △ , which means that 𝑖 × 𝜃 11 = 0 , we say that 𝜋 11 ( 𝑖 ( 𝐿 ) ) and 𝜋 12 ( 𝑖 ( 𝐿 ) ) are △-dual to each other. It is uncomplicated to see that the condition 𝑖 × 𝜃 11 = 0 is equivalent to 𝑖 × 𝜃 12 = 0 . Then we have: Let γ: 𝐼 → 𝐸 1 3 be a unit speed timelike curveterm, with 𝜇 𝑖 ≠ 0 (i = 1, 2), we have: (1) 𝜒 1 ( 𝑠 ) and 𝜒 3 ( 𝑠 ) is Δ-dual to each other; (2) 𝜉 1 ( 𝑠 ) and 𝜉 2 is Δ-dual to each other.
(1) Consider the curve 𝐿 ( 𝑠 ) = ( 𝜒 1 ( 𝑠 ) , 𝜒 3 ( 𝑠 ) ) ⊂ 𝐻 + 2 × 𝑆 1 2 . Then we have < 𝜒 1 ( 𝑠 ) , 𝜒 3 ( 𝑠 ) > = 0 , and < 𝜒 1 ′ ( 𝑠 ) , 𝜒 3 ( 𝑠 ) > = < 𝜅 𝜒 2 , 𝜒 3 ( 𝑠 ) > = 0 . The affirmation (1) holds.
(2) Consider the curve 𝐿 ( 𝑠 ) = ( 𝜉 1 ( 𝑠 ) , 𝜉 2 ( 𝑠 ) ) ⊂ 𝑆 1 2 × 𝑆 1 2 . Then, we have < 𝜉 1 ( 𝑠 ) , 𝜉 2 ( 𝑠 ) > = 0 , and < 𝜉 1 ′ ( 𝑠 ) , 𝜁 2 ( 𝑠 ) > = < 𝜇 1 𝜉 , 𝜁 2 ( 𝑠 ) > = 0 . The affirmation (2) holds. □ Let 𝑦 ( 𝑠 , 𝑡 ) be a parametric sweeping given by: 𝑦 ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑟 1 ( 𝑡 ) 𝜉 1 ( 𝑠 ) + 𝑟 2 ( 𝑡 ) 𝜉 2 ( 𝑠 ) . (7) Here 𝛾 ( 𝑠 ) is a timelike unit speed spine curve, 0 ≤ 𝑠 ≤ 𝑇 , s is the arc length parameter, and 𝑟 ( 𝑡 ) = ( 0 , 𝑟 1 ( 𝑡 ) , 𝑟 2 ( 𝑡 ) ) is the characteristic circle, with another parameter 𝑡 ∈ 𝑅 . It can be seen that 𝑦 ( 𝑠 , 𝑡 ) is a timelike surface. From Equation (7) it follows that the expression of the two timelike developable surfacesterm is 𝐷 𝑖 ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑡 𝜉 𝑖 ( 𝑠 ) , ( i = 1 , 2 ) . It is clear that 𝐷 2 ( 𝑠 , 0 ) = 𝛾 ( 𝑠 ) (resp. 𝐷 1 ( 𝑠 , 0 ) = 𝛾 ( 𝑠 ) ), 0 ≤ 𝑠 ≤ 𝐿 , that is, the surface 𝐷 2 (resp. 𝐷 1 ) interpolate the curve 𝛾 ( 𝑠 ) . Now, it seems natural to pose the following two questions. Under what condition does 𝐷 𝑖 have singularities, and how do we recognize their different types? The answer can be stated as: To discuss the singularities of 𝐷 𝑖 ( 𝑠 , 𝑡 ) , the cross product can be obtained as ∂ 𝐷 1 ∂ 𝑠 × ∂ 𝐷 1 ∂ 𝑡 = 1 − 𝑡 𝜇 1 𝜉 2 ( 𝑠 ) , and ∂ 𝐷 2 ∂ 𝑠 × ∂ 𝐷 2 ∂ 𝑡 = − 1 − 𝑡 𝜇 2 𝜉 1 ( 𝑠 ) . It follows that 𝐷 1 (resp. 𝐷 2 ) is non-singular at ( 𝑠 0 , 𝑡 0 ) if and only if 1 − 𝑡 0 𝜇 1 ( 𝑠 0 ) ≠ 0 (resp. ( 1 − 𝑡 0 𝜇 2 ( 𝑠 0 ) ≠ 0 ) . Hence, according to Theorem 3.2 in Ref. [10], we can give the following corollary: For the developable ruled surfacesterm 𝐷 𝑖 ( 𝑠 , 𝑡 ) = 𝛾 ( 𝑠 ) + 𝑡 𝜉 𝑖 ( 𝑠 ) , 𝑠 ∈ 𝐼 , 𝑡 ∈ 𝑅 , we have: (1) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the cuspidal edgeterm (CE) C(2,3) × 𝑅 at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , and 𝜇 𝑖 ′ ( 𝑠 0 ) ≠ 0 . (2) 𝐷 𝑖 ( 𝑠 , 𝑡 ) is locally diffeomorphic to the Swallowtailterm (SW) at ( 𝑠 0 , 𝑡 0 ) if and only if 𝑡 0 = 𝜇 𝑖 − 1 ( 𝑠 0 ) ≠ 0 , 𝜇 𝑖 ′ ( 𝑠 0 ) = 0 , and 𝜇 𝑖 − 1 ( 𝑠 0 ) ″ ≠ 0 .
Now, we should address the next concept regarding the contact of curves with some surfaces. Let F: 𝐸 1 3 → 𝑅 be a regular function, and 𝛾 : 𝑅 → 𝐸 1 3 be a smooth unit speed timelike Frenet–Serret curve. We say that 𝛾 ( 𝑠 ) and 𝐹 − 1 ( 0 ) have k-point contactterm for 𝑠 = 𝑠 0 if the function 𝑔 ( 𝑠 0 ) = 𝐹 ∘ 𝛾 satisfies 𝑔 ( 𝑠 0 ) = 𝑔 ′ ( 𝑠 0 ) = 𝑔 ″ ( 𝑠 0 ) = . . . = 𝑔 ( 𝑘 − 1 ) ( 𝑠 0 ) = 0 , and 𝑔 ( 𝑘 ) ( 𝑠 0 ) ≠ 0 . Further, for any fixed 𝑥 0 ∈ 𝐸 1 3 , we define the set BR 𝑗 ( 𝑥 0 ) = < 𝜉 𝑖 , 𝑥 0 − 𝛾 ( 𝑠 ) > = 0 , 𝑗 = 1 , 2 . We call it the first (resp. second) Bishop rectifying bundle of 𝛾 ( 𝑠 ) through 𝑥 0 when j = 1 (resp. j = 2).
The major aim of this work is in the following theorem: Let γ: 𝐼 → 𝐸 1 3 be a unit speed timelike curveterm, with 𝜇 𝑖 ≠ 0 ; i = 1, 2. Then we: A- For 𝑥 0 = 𝐷 1 ( 𝑠 0 , 𝑡 0 ) and the second Bishop rectifying bundle BR 2 ( 𝑥 0 ) of 𝛾 ( 𝑠 ) . One has the following: (a) The curve 𝛾 ( 𝑠 ) and BR 2 ( 𝑥 0 ) have at least a 2-point contact at s0; (b) The curve 𝛾 ( 𝑠 ) and BR 2 ( 𝑥 0 ) have at least a 3-point contact at s0 if and only if 𝑥 0 = 𝛾 ( 𝑠 0 ) − 1 𝜇 1 ( 𝑠 0 ) 𝜉 1 ( 𝑠 0 ) , 𝜇 1 ′ ( 𝑠 0 ) ≠ 0 . Under this situation 𝐷 1 ( 𝑠 , 𝑡 ) at ( 𝑠 0 , 𝑡 0 ) is locally diffeomorphic to CE C(2,3) × 𝑅 , and 𝐷 1 ( 𝑠 0 , 1 𝜇 1 ( 𝑠 0 ) ) is locally diffeomorphic to a line; (c) The curve 𝛾 ( 𝑠 ) and BR 2 ( 𝑥 0 ) has at least a 4-point contact at s0 if and only if 𝑥 0 = 𝛾 ( 𝑠 0 ) − 1 𝜇 1 ( 𝑠 0 ) 𝜉 1 ( 𝑠 0 ) , 𝜇 1 ′ ( 𝑠 0 ) = 0 , 𝜇 1 ″ ( 𝑠 0 ) ≠ 0 . Under this situation 𝐷 1 ( 𝑠 , 𝑡 ) at ( 𝑠 0 , 𝑡 0 ) is locally diffeomorphic to SW, and 𝐷 1 ( 𝑠 0 , 1 𝜇 1 ( 𝑠 0 ) ) is locally diffeomorphic to the (2,3,4)-cusp.
For 𝑥 0 = 𝐷 2 ( 𝑠 0 , 𝑡 0 ) and the first Bishop rectifying bundle BR 1 ( 𝑥 0 ) of 𝛾 ( 𝑠 ) . One has: (a) The curve 𝛾 ( 𝑠 ) and BR 1 ( 𝑥 0 ) have at least a 2-point contact at s0; (b) The curve 𝛾 ( 𝑠 ) and BR 1 ( 𝑥 0 ) have at least a 3-point contact at s0 if and only if 𝑥 0 = 𝛾 ( 𝑠 0 ) + 1 𝜇 2 ( 𝑠 0 ) 𝜉 2 ( 𝑠 0 ) , 𝜇 2 ′ ( 𝑠 0 ) ≠ 0 . Under this situation 𝐷 2 ( 𝑠 , 𝑡 ) at ( 𝑠 0 , 𝑡 0 ) is locally diffeomorphic to CE C(2,3) × 𝑅 , and 𝐷 2 ( 𝑠 0 , 1 𝜇 2 ( 𝑠 0 ) ) is locally diffeomorphic to a line; (c) The curve 𝛾 ( 𝑠 ) and BR 1 ( 𝑥 0 ) have at least a 4-point contact at s0 if and only if 𝑥 0 = 𝛾 ( 𝑠 0 ) + 1 𝜇 2 ( 𝑠 0 ) 𝜉 2 ( 𝑠 0 ) , 𝜇 2 ′ ( 𝑠 0 ) = 0 , 𝜇 2 ″ ( 𝑠 0 ) ≠ 0 . Under this situation 𝐷 2 ( 𝑠 , 𝑡 ) at ( 𝑠 0 , 𝑡 0 ) is locally diffeomorphic to SW, and 𝐷 2 ( 𝑠 0 , 1 𝜇 2 ( 𝑠 0 ) ) is locally diffeomorphic to the (2,3,4)-cusp.
Here, C(2,3) × 𝑅 = ( 𝑥 , 𝑦 ) ∣ 𝑥 2 = 𝑦 3 × 𝑅 , C(2, 3, 4) = ( 𝑥 , 𝑦 , 𝑧 ) ∣ 𝑥 = 𝑡 2 , 𝑦 = 𝑡 3 , 𝑧 = 𝑡 4 , and 𝑆 𝑊 = ( 𝑥 , 𝑦 , 𝑧 ) ∣ 𝑥 = 3 𝑢 4 + 𝑢 2 𝑣 , 𝑦 = 4 𝑢 3 + 2 𝑢 𝑣 , 𝑧 = 𝑣 . The graphs of cuspidal edgeterm and swallowtailterm can be seen in Figure 1 and Figure 2.
3.1. Bishop Height Functionsterm Now we will consider two various sets of Bishop height functionsterm, which will be helpful for studying the singularities of 𝐷 𝑖 ( 𝑠 , 𝑡 ) as follows [27,28]: 𝐻 𝑖 : 𝐼 × 𝐸 1 3 → 𝑅 , by 𝐻 𝑖 ( 𝑠 , 𝑥 ) = < 𝜉 𝑖 , 𝑥 − 𝛾 > ; i = 1, 2. We name it the first (resp. second) Bishop height functionterm for i = 1 (resp. 2). We use the notation ℎ ix ( 𝑠 ) = 𝐻 𝑖 ( 𝑠 , 𝑥 ) for any fixed 𝑥 ∈ 𝐸 1 3 . From now on, we shall not often write the parameter s. Then, we have the following proposition: Let γ: 𝐼 → 𝐸 1 3 be a unit speed timelike curveterm, with 𝜇 1 ≠ 0 , and 𝜇 2 ≠ 0 . Then: (A) (1) ℎ 1 𝑥 ( 𝑠 ) = 0 if and only if there exist a, 𝑏 ∈ 𝑅 such that 𝑥 − 𝛾 = 𝑎 𝜉 + 𝑏 𝜉 2 , (2) ℎ 1 𝑥 ( 𝑠 ) = ℎ 1 𝑥 ′ ( 𝑠 ) = 0 if and only if 𝑥 = 𝛾 ( 𝑠 ) + 𝑡 𝜉 2 ( 𝑠 ) . (3) ℎ 1 𝑥 ( 𝑠 ) = ℎ 1 𝑥 ′ ( 𝑠 ) = ℎ 1 𝑥 ″ ( 𝑠 ) = 0 if and only if 𝑥 = 𝛾 ( 𝑠 ) − 1 𝜇 2 ( 𝑠 ) 𝜉 2 ( 𝑠 ) . (4) ℎ 1 𝑥 ( 𝑠 ) = ℎ 1 𝑥 ′ ( 𝑠 ) = ℎ 1 𝑥 ″ ( 𝑠 ) = ℎ 1 𝑥 ‴ ( 𝑠 ) = 0 if and only if 𝑥 = 𝛾 ( 𝑠 ) − 1 𝜇 2 ( 𝑠 ) 𝜉 2 ( 𝑠 ) , and 𝜇 2 ′ ( 𝑠 ) = 0 . (5) ℎ 1 𝑥 ( 𝑠 ) = ℎ 1 𝑥 ′ ( 𝑠 ) = ℎ 1 𝑥 ″ ( 𝑠 ) = ℎ 1 𝑥 ‴ ( 𝑠 ) = ℎ 1 𝑥 ( 4 ) ( 𝑠 ) = 0 if and only if 𝑥 = 𝛾 ( 𝑠 ) − 1 𝜇 2 ( 𝑠 ) 𝜉 2 ( 𝑠 ) , and 𝜇 2 ′ ( 𝑠 ) = 𝜇 2 ″ ( 𝑠 ) = 0 . (B) (1) ℎ 2 𝑥 ( 𝑠 ) = 0 if and only if there exist a, 𝑏 ∈ 𝑅 such that 𝑥 − 𝛾 = 𝑎 𝜉 + 𝑏 𝜉 1 , (2) ℎ 2 𝑥 ( 𝑠 ) = ℎ 2 𝑥 ′ ( 𝑠 ) = 0 if and only if 𝑥 = 𝛾 ( 𝑠 ) + 𝑡 𝜉 1 ( 𝑠 ) . (3) ℎ 2 𝑥 ( 𝑠 ) = ℎ 2 𝑥 ′ ( 𝑠 ) = ℎ 2 𝑥 ″ ( 𝑠 ) = 0 if and only if 𝑥 = 𝛾 ( 𝑠 ) + 1 𝜇 1 ( 𝑠 ) 𝜉 1 ( 𝑠 ) . (4) ℎ 2 𝑥 ( 𝑠 ) = ℎ 2 𝑥 ′ ( 𝑠 ) = ℎ 2 𝑥 ″ ( 𝑠 ) = ℎ 2 𝑥 ‴ ( 𝑠 ) = 0 if and only if 𝑥 = 𝛾 ( 𝑠 ) + 1 𝜇 1 ( 𝑠 ) 𝜉 2 ( 𝑠 ) , and 𝜇 1 ′ ( 𝑠 ) = 0 . (5) ℎ 2 𝑥 ( 𝑠 ) = ℎ 2 𝑥 ′ ( 𝑠 ) = ℎ 2 𝑥 ″ ( 𝑠 ) = ℎ 2 𝑥 ‴ ( 𝑠 ) = ℎ 2 𝑥 ( 4 ) ( 𝑠 ) = 0 if and only if 𝑥 = 𝛾 ( 𝑠 ) + 1 𝜇 1 ( 𝑠 ) 𝜉 2 ( 𝑠 ) , and 𝜇 1 ′ ( 𝑠 ) = 𝜇 1 ″ ( 𝑠 ) = 0 .
(A). (1) Since ℎ 1 𝑥 ( 𝑠 ) = < 𝑥 − 𝛾 , 𝜉 1 > = 0 , and { 𝜉 , 𝜉 1 , 𝜉 2 } is RMF along 𝛾 ( 𝑠 ) , then there exists a, 𝑏 ∈ 𝑅 such that 𝑥 − 𝛾 = 𝑎 𝜉 + 𝑏 𝜉 2 .
(2) When ℎ 1 𝑥 ( 𝑠 ) = 0 , the affirmation (2) follows from the fact that ℎ 1 𝑥 ′ ( 𝑠 ) = < 𝜉 1 ′ , 𝑥 − 𝛾 > = < 𝜇 1 𝜉 , 𝑥 − 𝛾 > = 0 . Thus, we have that < 𝜇 1 𝜉 , 𝑥 − 𝛾 > = 0 . It follows from the fact 𝜇 1 ≠ 0 that < 𝜉 , 𝑥 − 𝛾 > , and 𝑥 − 𝛾 = 𝑏 𝜉 2 . Thus, we get that ℎ 1 𝑥 ( 𝑠 ) = ℎ 1 𝑥 ′ ( 𝑠 ) = 0 if and only if 𝑥 − 𝛾 = 𝑏 𝜉 2 .
(3) When ℎ 1 𝑥 ( 𝑠 ) = ℎ 1 𝑥 ′ ( 𝑠 ) = 0 , the affirmation (3) follows from the fact that ℎ 1 𝑥 ″ ( 𝑠 ) = − 𝜇 1 + < 𝑥 − 𝛾 , 𝜇 1 ′ 𝜉 + 𝜇 1 2 𝜉 1 − 𝜇 1 𝜇 2 𝜉 2 > = 0 = − 𝜇 1 ( 1 + 𝑏 𝜇 2 ) , Since 𝜇 1 ≠ 0 , we get that ℎ 1 𝑥 ( 𝑠 ) = ℎ 1 𝑥 ′ ( 𝑠 ) = ℎ 𝑥 ″ ( 𝑠 ) = 0 if and only if 𝑥 = 𝛾 ( 𝑠 ) − 1 𝜇 2 ( 𝑠 ) 𝜉 2 ( 𝑠 ) .
(4) Under the hypothesis that ℎ 1 𝑥 ( 𝑠 ) = ℎ 1 𝑥 ′ ( 𝑠 ) = ℎ 1 𝑥 ″ ( 𝑠 ) = 0 , this derivative is calculated as follows: ℎ 1 𝑥 ( 3 ) ( 𝑠 ) = − 2 𝜇 1 ′ + < 𝜇 1 ″ − 𝜇 1 3 + 𝜇 1 𝜇 2 2 ) 𝜉 + 3 𝜇 1 𝜇 1 ′ 𝜉 1 − 𝜇 1 𝜇 2 ′ + 2 𝜇 2 𝜇 1 ′ 𝜉 2 , 𝑥 − 𝛾 > = 𝜇 1 ( 𝑠 ) 𝜇 2 ′ ( 𝑠 ) 𝜇 2 ( 𝑠 ) = 0 . Since 𝜇 2 ( 𝑠 ) ≠ 0 , we get ℎ 1 𝑥 ( 3 ) ( 𝑠 ) = 0 , which is equivalent to the assumption 𝜇 2 ′ ( 𝑠 ) = 0 . The affirmation (4) follows.
(5) Under the hypothesis that ℎ 1 𝑥 ( 𝑠 ) = ℎ 1 𝑥 ′ ( 𝑠 ) = ℎ 1 𝑥 ″ ( 𝑠 ) = ℎ 1 𝑥 ( 3 ) ( 𝑠 ) = 0 , this derivative is calculated as: ℎ 1 𝑥 ( 4 ) ( 𝑠 ) = − 3 𝜇 1 ″ + 𝜇 1 3 − 𝜇 1 𝜇 2 2 + < 𝑎 𝜉 + 𝑎 1 𝜉 1 + 𝑎 2 𝜉 2 , , 𝑥 − 𝛾 > , where 𝑎 ( 𝑠 ) = 𝜇 1 ″ − 6 𝜇 1 ′ 𝜇 1 2 + 3 𝜇 1 ′ 𝜇 2 2 + 3 𝜇 1 ′ 𝜇 1 𝜇 2 ′ , 𝑎 1 ( 𝑠 ) = 3 𝜇 1 ′ 2 + 3 𝜇 1 𝜇 1 ‴ + 𝜇 1 4 + 𝜇 1 2 𝜇 1 2 , 𝑎 2 ( 𝑠 ) = − 3 𝜇 1 ″ 𝜇 2 − 3 𝜇 1 ′ 𝜇 2 ′ + 𝜇 1 3 𝜇 2 − 𝜇 1 𝜇 2 3 − 𝜇 1 𝜇 2 ″ . By 𝑥 = 𝛾 ( 𝑠 ) − 1 𝜇 2 ( 𝑠 ) 𝜉 2 ( 𝑠 ) in the above, we have that: ℎ 1 𝑥 ( 4 ) ( 𝑠 ) = − 3 𝜇 1 ″ + 𝜇 1 3 − 𝜇 1 𝜇 2 2 − 1 𝜇 2 − 3 𝜇 1 ‴ 𝜇 2 − 3 𝜇 1 ′ 𝜇 2 ′ + 𝜇 1 3 𝜇 2 − 𝜇 1 𝜇 2 3 − 𝜇 1 𝜇 2 ″ = 3 𝜇 1 ′ 𝜇 2 ′ + 𝜇 1 𝜇 2 ″ 𝜇 2 . Since 𝜇 1 ( 𝑠 ) ≠ 0 , 𝜇 2 ′ ( 𝑠 ) = 0 , we get that ℎ 1 𝑥 ( 4 ) ( 𝑠 ) = 0 is synonymous to the assumption 𝜇 2 ″ ( 𝑠 ) = 0 . The affirmation (5) follows. Using the identical computation as the proof of (A), we can get (B). □ By simple calculations, we have: Let γ: 𝐼 → 𝐸 1 3 be a unit speed timelike curveterm, with 𝜇 1 ≠ 0 , and 𝜇 2 ≠ 0 . Then: (1) 𝜇 2 ′ = 0 if and only if 𝑥 = 𝛾 ( 𝑠 ) + 1 𝜇 2 ( 𝑠 ) 𝜉 2 ( 𝑠 ) is a constant vector. (2) 𝜇 1 ′ = 0 if and only if 𝑥 = 𝛾 ( 𝑠 ) − 1 𝜇 1 ( 𝑠 ) 𝜉 1 ( 𝑠 ) is a constant vector.
3.2. Unfolding of Functions by One-Variable Now we employ some public results on the singularityterm theory for families of function germs [27,28]. Let F: 𝑅 × 𝑅 𝑟 , ( 𝑠 0 , 𝑥 0 ) → 𝑅 be a smooth function, and 𝑓 ( 𝑠 ) = 𝐹 𝑥 0 ( 𝑠 , 𝑥 0 ) . Then F is called an r-parameter unfolding of 𝑓 ( 𝑠 ) . We say that 𝑓 ( 𝑠 ) has Ak-singularity at 𝑠 0 if 𝑓 ( 𝑝 ) ( 𝑠 0 ) = 0 for all 1 ≤ 𝑝 ≤ 𝑘 , and 𝑓 ( 𝑘 + 1 ) ( 𝑠 0 ) ≠ 0 . We also say that f has A ⩾ 𝑘 -singularity ( 𝑘 ⩾ 1 ) at s0. Let the 𝑘 − 1 -jet of the partial derivative ∂ 𝐹 ∂ 𝑥 𝑖 at 𝑠 0 be 𝑗 𝑘 − 1 ∂ 𝐹 ∂ 𝑥 𝑖 𝑠 , 𝑥 0 𝑠 0 = 𝛴 𝑗 = 0 𝑘 − 1 𝐿 𝑗 𝑖 𝑠 − 𝑠 0 𝑗 (without the constant term), for 𝑖 = 1 , . . . , 𝑟 . Then 𝐹 ( 𝑠 ) is called an p-versal unfolding if the 𝑘 × 𝑟 matrix of coefficients 𝐿 𝑗 𝑖 has rank k 𝑘 ≤ 𝑟 . Therefore, we write serious sets on the unfolding regarding to the above notations. The discriminant setterm of order t of F is the set 𝐷 𝐹 𝑡 = 𝑥 ∈ 𝑅 𝑟 ∣ ∃ 𝑠 ∈ 𝑅 , 𝐹 𝑠 , 𝑥 = , ∂ 𝐹 ∂ 𝑠 𝑠 , 𝑥 = . . . = ∂ 𝑡 𝐹 ∂ 𝑠 𝑡 𝑠 , 𝑥 = 0 at ( 𝑠 , 𝑥 ) . (8) Then 𝐷 𝐹 1 = 𝐷 𝐹 , and 𝐷 𝐹 2 is the set of singular points of 𝐷 𝐹 .
We state the following theorem: Let F: 𝑅 × 𝑅 𝑟 , ( 𝑠 0 , 𝑥 0 ) → 𝑅 be an r-parameter unfolding of 𝑓 ( 𝑠 ) , which has the 𝐴 𝑘 singularityterm at 𝑠 0 .
Suppose that F is a p-versal unfolding.
(a) If 𝑘 = 1 , then 𝐷 𝐹 is locally diffeomorphic to { 0 } × 𝑅 𝑟 − 1 , and 𝐷 𝐹 2 = ⌀ ; (b) If 𝑘 = 2 , then 𝐷 𝐹 is locally diffeomorphic to 𝐶 ( 2 , 3 ) × 𝑅 𝑟 − 2 , 𝐷 𝐹 2 is locally diffeomorphic to { 0 } × 𝑅 𝑟 − 2 , and 𝐷 𝐹 3 = ⌀ .
(c) If 𝑘 = 3 , then 𝐷 𝐹 is locally diffeomorphic to 𝑆 𝑊 × 𝑅 𝑟 − 3 , 𝐷 𝐹 3 is locally diffeomorphic to 𝐶 ( 2 , 3 , 4 ) × 𝑅 𝑟 − 3 , and 𝐷 𝐹 3 is locally diffeomorphic to { 0 } × 𝑅 𝑟 − 3 , and 𝐷 𝐹 4 = ⌀ .
Then, the following proposition can be obtained.
Under the situations of Proposition 1, if ℎ 1 𝑥 0 ( 𝑠 ) has 𝐴 𝑘 -singularity ( 𝑘 = 2 , 3 ) at 𝑠 0 ∈ 𝑅 , then 𝐻 1 ( 𝑠 , 𝑥 ) is a 𝑝 − versal unfoldingterm of ℎ 1 𝑥 0 ( 𝑠 0 ) . (2). If ℎ 2 𝑥 0 ( 𝑠 ) has 𝐴 𝑘 -singularity ( 𝑘 = 2 , 3 ) at 𝑠 0 ∈ 𝑅 , then 𝐻 2 ( 𝑠 , 𝑥 ) is a 𝑝 − versal unfoldingterm of ℎ 2 𝑥 0 ( 𝑠 0 ) .
(1) Let 𝛾 𝑠 = 𝛾 1 𝑠 , 𝛾 2 𝑠 , 𝛾 3 𝑠 , 𝑥 = 𝑥 1 , 𝑥 2 , 𝑥 3 in 𝐸 1 3 , and 𝜉 1 𝑠 = 𝜉 11 𝑠 , 𝜉 12 𝑠 , 𝜉 13 𝑠 ∈ 𝑆 1 2 . Then, 𝐻 1 ( 𝑠 , 𝑥 ) = − 𝛾 1 𝑠 − 𝑥 1 𝜉 11 ( 𝑠 ) + 𝛾 2 𝑠 − 𝑥 2 𝜉 12 ( 𝑠 ) + 𝛾 3 𝑠 − 𝑥 3 𝜉 13 ( 𝑠 ) . Thus, we have ∂ 𝐻 1 ∂ 𝑥 1 = 𝜉 11 ( 𝑠 ) , ∂ 𝐻 1 ∂ 𝑥 2 = − 𝜉 12 ( 𝑠 ) , ∂ 𝐻 1 ∂ 𝑥 3 = − 𝜉 13 ( 𝑠 ) ∂ 2 𝐻 1 ∂ 𝑠 ∂ 𝑥 1 = 𝜉 11 ′ ( 𝑠 ) , ∂ 2 𝐻 1 ∂ 𝑠 ∂ 𝑥 2 = − 𝜉 12 ′ ( 𝑠 ) , ∂ 2 𝐻 1 ∂ 𝑠 ∂ 𝑥 3 = − 𝜉 13 ′ ( 𝑠 ) . Therefore, the 2-jets of ∂ 𝐻 ∂ 𝑥 𝑖 at 𝑠 0 (i=0, 1) are: 𝑗 1 ∂ 𝐻 1 ∂ 𝑥 1 ( 𝑠 , 𝑥 0 ) = 𝜉 11 ′ ( 𝑠 − 𝑠 0 ) , 𝑗 1 ∂ 𝐻 1 ∂ 𝑥 2 ( 𝑠 , 𝑥 0 ) = − 𝜉 12 ′ ( 𝑠 ) ( 𝑠 − 𝑠 0 ) , 𝑗 1 ∂ 𝐻 1 ∂ 𝑥 3 ( 𝑠 , 𝑥 0 ) = − 𝜉 13 ′ ( 𝑠 − 𝑠 0 ) , and 𝑗 2 ∂ 𝐻 1 ∂ 𝑥 1 ( 𝑠 , 𝑥 0 ) = 𝜉 11 ′ ( 𝑠 − 𝑠 0 ) + 1 2 𝜉 11 ″ 𝑠 − 𝑠 0 2 , 𝑗 2 ∂ 𝐻 1 ∂ 𝑥 2 ( 𝑠 , 𝑥 0 ) = − 𝜉 12 ′ ( 𝑠 ) ( 𝑠 − 𝑠 0 ) − 1 2 𝜉 12 ″ 𝑠 − 𝑠 0 2 , 𝑗 2 ∂ 𝐻 1 ∂ 𝑥 2 ( 𝑠 , 𝑥 0 ) = − 𝜉 13 ′ ( 𝑠 ) ( 𝑠 − 𝑠 0 ) − 1 2 𝜉 13 ″ 𝑠 − 𝑠 0 2 .
(i) If ℎ 1 𝑥 0 ( 𝑠 0 ) has the 𝐴 2 -singularity at 𝑠 0 , then ℎ 1 𝑥 0 ′ ( 𝑠 0 ) = 0 . So the 1 × 3 matrix of coefficients 𝐿 𝑗 𝑖 is: 𝐴 = 𝜉 11 ′ − 𝜉 12 ′ − 𝜉 13 ′ . Suppose that the rank of the matrix A is zero, then we have: 𝜉 11 ′ = 𝜉 12 ′ = 𝜉 13 ′ = 0 .
Since 𝛾 ′ ( 𝑠 0 ) = 𝜉 ( 𝑠 0 ) 2 = − 1 , we have − 𝜉 11 ′ 2 + 𝜉 11 ′ 2 + 𝜉 11 ′ 2 = − 𝜇 1 2 ≠ 0 , so that we have a contradiction. Therefore rank( 𝐴 ) = 1 , and H is the (p) versal unfoldingterm of ℎ 1 𝑥 0 at 𝑠 0 .
(ii) If ℎ 1 𝑥 0 ( 𝑠 0 ) has the 𝐴 3 -singularity at 𝑠 0 ∈ 𝑅 , then ℎ 1 𝑥 0 ′ ( 𝑠 0 ) = ℎ 1 𝑥 0 ″ ( 𝑠 0 ) = 0 . Therefore, the 3 × 3 matrix of the coefficients 𝐿 𝑗 𝑖 is 𝐵 ( 𝑠 0 ) = 𝜉 11 − 𝜉 12 − 𝜉 13 𝜉 11 ′ − 𝜉 12 ′ − 𝜉 13 ′ 1 2 𝜉 11 ″ − 1 2 𝜉 12 ″ − 1 2 𝜉 13 ″ . For the intention, we also require the 3 × 3 matrix B to be non-singular, which it constantly is. Furthermore, the determinate of this matrix at 𝑠 0 is det 𝐵 = 1 2 𝜉 11 − 𝜉 12 − 𝜉 13 𝜉 11 ′ − 𝜉 12 ′ − 𝜉 13 ′ 𝜉 11 ″ − 𝜉 12 ″ − 𝜉 13 ″ = 1 2 < 𝜉 1 × 𝜉 1 ′ , 𝜉 1 ″ > . Since 𝜉 1 ′ = − 𝜇 1 𝜉 , we have 𝜉 1 ″ = − 𝜇 1 ′ 𝜉 − 𝜇 1 2 𝜉 1 + 𝜇 1 𝜇 2 𝜉 2 . Substituting these equations to the above equality, we obtain det 𝐵 = 1 2 𝜇 1 2 ( 𝑠 0 ) 𝜇 2 ( 𝑠 0 ) ≠ 0 . Thus rank ( 𝐵 ) = 3 , if we consider the matrix which consists of the first and the second row of the matrix B, so that rank ( 𝐵 ) = 2 . (2) Using the same computation as the proof of (1), we can get (2). □ (1) Let 𝛾 : 𝐼 → 𝐸 1 3 be a unit speed timelike curveterm, with 𝜇 1 ≠ 0 , and 𝜇 2 ≠ 0 . For 𝑥 0 = 𝛾 ( 𝑠 0 ) + 𝑡 0 𝜉 𝑖 ( 𝑠 0 ) , we define a function 𝐻 𝑖 ( 𝑝 ) = < 𝜉 𝑖 , 𝑥 0 − 𝑝 > (i = 1, 2), then we have ℎ 𝑖 𝑥 0 ( 𝑠 ) = 𝑁 𝑖 ( 𝛾 ( 𝑠 ) ) .
First, we consider the affirmation (1). For 𝑥 0 = 𝐷 1 ( 𝑠 0 , 𝑡 0 ) , since BR 2 ( 𝑥 0 ) = 𝐻 2 − 1 ( 0 ) , where 0 is a regular value of 𝐻 2 ; ℎ 2 𝑥 0 ( 𝑠 ) has the A 𝑘 -singularity at s0 if and only if 𝛾 ( 𝑠 0 ) and BR 2 ( 𝑥 0 ) have (k+1)-point contact for 𝑠 0 . On the other hand, the discriminant setterm 𝐷 𝐻 2 of 𝐻 2 is 𝐷 𝐻 2 = 𝑥 = 𝛾 ( 𝑠 ) + 𝑡 𝜉 1 ( 𝑠 ) ∣ 𝑠 ∈ 𝐼 , 𝑡 ∈ 𝑅 . The affirmation (1) follows from Theorem 2 and Proposition 1. Since the trajectory of the singularities of CE is locally diffeomorphic to the line, the affirmation (b) holds. Since the trajectory of singularities of SW is C(2,3,4), the affirmation (c) holds.
For the affirmation (2), the discriminant setterm of 𝐷 𝐻 1 of 𝐻 1 is 𝐷 𝐻 1 = 𝑥 = 𝛾 ( 𝑠 ) + 𝑡 𝜉 2 ( 𝑠 ) ∣ 𝑠 ∈ 𝐼 , 𝑡 ∈ 𝑅 . By a similar argument, we can also prove the affirmation (2). This completes the proof. □ Given the timelike helix: 𝛾 ( 𝑠 ) = ( 3 sinh 𝑠 , 3 cosh 𝑠 , 2 𝑠 ) , − 1 ≤ 𝑠 ≤ 1 , It is simple to have that 𝜒 1 ( 𝑠 ) = ( 3 cosh 𝑠 , 3 sinh 𝑠 , 2 ) , 𝜒 2 ( 𝑠 ) = ( sinh 𝑠 , cosh 𝑠 , 0 ) , 𝜒 3 ( 𝑠 ) = ( 2 cosh 𝑠 , 2 sinh 𝑠 , 3 ) , 𝜅 ( 𝑠 ) = 3 , 𝑎 𝑛 𝑑 𝜏 ( 𝑠 ) = 2 . Taking 𝜑 0 = 0 we have 𝜑 ( 𝑠 ) = − 2 𝑠 . Using Equations (3)–(5), we obtain 𝜇 1 ( 𝑠 ) = 3 cos 2 𝑠 , and 𝜇 2 ( 𝑠 ) = − 3 sin 2 𝑠 . Hence: 𝜉 𝜉 1 𝜉 2 = 1 0 0 0 cos 2 𝑠 𝑠 − sin 2 𝑠 0 sin 2 𝑠 cos 2 𝑠 𝜒 1 𝜒 2 𝜒 3 . Thus, we get 𝜉 1 = 𝜉 11 𝜉 12 𝜉 13 = sinh 𝑠 cos 2 𝑠 − 2 cosh 𝑠 sin 2 𝑠 cosh 𝑠 cos 2 𝑠 − 2 sinh 𝑠 sin 2 𝑠 − 3 sin 2 𝑠 , 𝜉 2 = 𝜉 21 𝜉 22 𝜉 23 = sinh 𝑠 sin 2 𝑠 + 2 cosh 𝑠 cos 2 𝑠 sin 2 𝑠 cosh 𝑠 + 2 cos 2 𝑠 sinh 𝑠 3 cos 2 𝑠 . So, the timelike developable surfaceterm is 𝑦 ( 𝑠 , 𝑡 ) = 3 sinh 𝑠 , 3 cosh 𝑠 , 2 𝑠 + 𝑟 1 ( 𝑡 ) 𝜉 11 𝜉 12 𝜉 13 + 𝑟 2 ( 𝑡 ) 𝜉 21 𝜉 22 𝜉 23 .
(1) If we take 𝑟 1 ( 𝑡 ) = cos 𝑡 , and 𝑟 2 ( 𝑡 ) = sin 𝑡 , then we instantly obtain the surface (see Figure 3).
(2) If 𝑟 1 ( 𝑡 ) = 𝑟 2 ( 𝑡 ) = 𝑡 , then 𝐷 1 ( 𝑠 , 𝑡 ) , and 𝐷 2 ( 𝑠 , 𝑡 ) , respectively, are: 𝐷 1 ( 𝑠 , 𝑡 ) = 3 sinh 𝑠 + 𝑡 𝜉 11 , 3 cosh 𝑠 + 𝑡 𝜉 13 , 2 𝑠 + 𝑡 𝜉 12 , and 𝐷 2 ( 𝑠 , 𝑡 ) = 3 sinh 𝑠 + 𝑡 𝜉 21 , 3 cosh 𝑠 + 𝑡 𝜉 23 , 2 𝑠 + 𝑡 𝜉 22 . In addition, the singular locus of 𝐷 1 ( 𝑠 , 𝑡 ) , and 𝐷 2 ( 𝑠 , 𝑡 ) , respectively, are: 𝐷 1 ( 𝑠 ) = 3 sinh 𝑠 + 1 3 cos 2 𝑠 𝜉 11 , 3 cosh 𝑠 + 1 3 cos 2 𝑠 𝜉 13 , 2 𝑠 + 1 3 cos 2 𝑠 𝜉 12 , and 𝐷 2 ( 𝑠 ) = 3 sinh 𝑠 + 1 3 sin 2 𝑠 𝜉 21 , 3 cosh 𝑠 + 1 3 sin 2 𝑠 𝜉 23 , 2 𝑠 + 1 3 sin 2 𝑠 𝜉 22 . We consider a local part of this curve when 𝜋 6 2 ≤ 𝑠 ≤ 𝜋 3 2 . We see that 𝜇 1 − 1 ( 𝑠 ) = 1 3 cos 2 𝑠 ≠ 0 , and 𝜇 1 ′ ( 𝑠 ) = − 6 sin 2 𝑠 ≠ 0 for 𝜋 6 2 ≤ 𝑠 ≤ 𝜋 3 2 . This means that 𝐷 1 ( 𝑠 , 𝑡 ) is locally diffeomorphic to a CE and its singular locus is locally diffeomorphic to a line (the red line), see Figure 4. For 𝐷 2 ( 𝑠 , 𝑡 ) , when 𝜋 6 2 ≤ 𝑠 ≤ 5 𝜋 6 2 . We see that 𝜇 2 − 1 ( 𝑠 ) = 1 3 cos 2 𝑠 ≠ 0 , 𝜇 2 ′ ( 𝑠 ) = 6 cos 2 𝑠 = 0 gives one real root 𝑠 = 𝜋 2 2 . We can also get that 𝜇 2 − 1 ( 𝜋 2 2 ) ″ = 0 , that is, 𝐷 2 ( 𝑠 , 𝑡 ) fails to be locally diffeomorphic to SW and its singular locus is not locally diffeomorphic to the C(2,3,4)-cusp at 𝑠 = 𝜋 2 2 . Hence 𝐻 2 ( 𝑠 , 𝑥 ) fails to be a versal unfoldingterm of the ℎ 2 𝑥 (s) at 𝑠 = 𝜋 2 2 ; see Figure 5.
4. Conclusions In this paper, we considered the notion of timelike developable surfacesterm with rotation minimizing frames in Minkowski 3–Spaceterm 𝐸 1 3 . By applying singularities theory, we classified the generic properties of the cuspidal edgeterm and swallowtailterm. Finally, an example of application is offered to demonstrate the theoretical results. Furthermore, recently the application of singularityterm theory and submanifolds theory and so forth, presented in Refs. [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57], has attracted great interest. In the following work, we will connect the results of this paper with the techniques and methods in Refs. [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57] to explore more results and theorems related to symmetric properties about this topic.