The fundamental principle of a directed line’s motion in connection with a solid body is referred to as the R S concept in spatial kinematics. This notion holds great importance in conventional differential geometry and has been the subject of extensive research by numerous scholars, as demonstrated by [1–7]. From a geometric perspective, the properties of R S and their offset surfaces have been analyzed in both Euclidean and non-Euclidean spaces. Bertrand curves were examined in the field of line-geometry by Ravani and Ku, revealing that R S can possess an infinite number of B O , similar to how a plane curve can possess an infinite number of B mates [8]. Küçük and Gürsoy provided certain characterizations of B O related to the trajectory of R S by studying the relationships between the projection areas for the spherical curves of B O and their integral invariants [9]. Kasap and Kuruoğlu conducted an analysis of the integral invariants of the couple of R S in the Euclidean 3-space E 3 , as documented in [10]. By considering the orthonormal frame along striction curve of a ruled surface, Önder has defined slant ruled surfaces in the Euclidean 3-space [11]. Moreover, Kaya and Önder have studied the position vectors and some differential equation characterizations for slant ruled surfaces in the Euclidean 3-space E 3 [12–14]. They have also defined a new type of slant ruled surface as the Darboux slant ruled surface and characterized for this type of slant surfaces [15]. In [16], Önder introduced some characterizations for a non-null ruled surface to be a slant ruled surface in Minkowski 3-space E 1 3 . In their study, Kasap and Kuruoğlu investigated B O of R S in Minkowski 3-space E 1 3 , as documented in [17]. [18] demonstrated the involute–evolute offsets of R S . Orbay et al. began studying the Mannheim offsets of R S in [19]. Önder and Uğurlu conducted a study on the relationships between invariants of Mannheim offsets of T L R S . They also formulated many considerations for the development of these surface offsets [20, 21]. In view of the involute–evolute offsets of the ruled surface in [7], Şentürk and Yüce described the integral invariants of the involute–evolute offsets of R S s using the geodesic Frenet frame [22].

In recent times, Yoon has investigated the evolute offsets of R S in E _ 1 3 with a stationary Gaussian curvature and mean curvature [23]. A plethora of comprehensive treatises has been published on this subject, as demonstrated by the numerous written works, such as [24–27]. However, to the best of our knowledge, no prior work has focused on constructing B O of slant S L R S , utilizing the geometric attributes of the striction curve ( S C ) . Here, we intend to fill the gap in the existing literature.

In this paper, with the identification of slant curves, we treat the structure issue of the B O of a slant S L R S family in Minkowski 3-space E 1 3 . Therefore, we extend the parametrization of B O for any slant non-developable S L R S . Furthermore, we inquire into the ownerships of these S L R surfaces and grant their distribution. Meanwhile, we extend some interpretative paradigms to display S L R surfaces with their B O along mutual geodesic, line of curvature, and asymptotic curve. Our ramifications in this paper may be beneficial in any area that demands documentation around surfaces due to the descriptions supplying insights into surfaces theory.

Let E 1 3 indicate the Minkowski 3-space [28, 29]. For vectors a = a 1 , a 2 , a 3 and b = ( b 1 , b 2 , b 3 ) in E 1 3 , a , b = a 1 b 1 − a 2 b 2 + a 3 b 3 is named the Lorentzian inner product. We also explain a vector a × v = a 2 b 3 − a 3 b 2 , a 1 b 3 − a 3 b 1 , a 1 b 2 − a 2 b 1 . Since < , > is an indefinite metric, recall that a vector a ∈ E 1 3 can have one of three causal natures; it can be S L if a , a > 0 or a = 0 , timelike ( T L ) if a , a < 0 , and null or lightlike if a , a = 0 and a ≠ 0 . The norm of a ∈ E 1 3 is explained by a = a , a ; then, the hyperbolic and Lorentzian (de Sitter space) unit spheres are H + 2 = a ∈ E 1 3 ∣ a 2 ≔ a 1 2 − a 2 2 + a 3 2 = − 1 (1) and S 1 2 = a ∈ E 1 3 ∣ a 2 ≔ a 1 2 − a 2 2 + a 3 2 = 1 . (2)

In this section, we contemplate and analyze the B O for slant S L R S . Then, a theory hassling to the theory of the Bertrand curves can be broadened for such surfaces.

In comparable with [30], a point e 0 ( v ) ∈ S 1 2 will be heading an e k curvature axis of the curve b ( v ) ∈ S 1 2 ; for all v such that < e 0 , b ( v ) > = 0 , but < e 0 , b 1 t + 1 ( v ) > ≠ 0 . Here, b 1 t + 1 signalizes the tth derivative of b ( v ) with regard to v . For the first curvature axis e of b ( v ) , we find < e , b ′ > = ± < e , z > = 0 , and < e , b ′ ′ > = ± < e , b + γ g > ≠ 0 . So, e is at least an e 2 curvature axis of b ( v ) ∈ S 1 2 . We now sign a height function d : I × S 1 2 → R , by d ( v , e 0 ) = < e 0 , b > . We set the notation d ( v ) = d ( v , e 0 ) for any specified point e 0 ∈ S 1 2 .

Proposition 1

Under the overhead presumptions, we capture the following:

i) d will be specified in the first evaluation iff e 0 ∈ S p { b , g } , that is, d ′ = 0 ⇔ < b ′ , e 0 > = 0 ⇔ < z , e 0 > = 0 ⇔ e 0 = c 1 b + c 2 g ; for real numbers c 1 , c 2 ∈ R , and c 1 2 + c 2 2 = 1 .

Proof. For d ′ , we determine d ′ = < b ′ , e 0 > . (18) So, we realize d ′ = 0 ⇔ < z , e 0 > = 0 ⇔ e 0 = c 1 b + c 3 g ; (19) for real numbers c 1 , c 2 ∈ R , and c 1 2 + c 2 2 = 1 , the consequence is evident.2- Derivation of Equation 18 displays that d ′ ′ = < b ′ ′ , e 0 > = < b + γ g , e 0 > . (20) By Equations 18–20, we determine d ′ = d ′ ′ = 0 ⇔ < b ′ , e 0 > = < b ′ ′ , e 0 > = 0 ⇔ e 0 = ± b ′ × b ′ ′ b ′ × b ′ ′ = ± e . 3- Differentiation of Equation 20 displays that d ′ ′ ′ = < b ′ ′ ′ , e 0 > = 1 + γ 2 < z , e 0 > + γ ′ < g , e 0 > . Thus, we gain d ′ = d ′ ′ = d ′ ′ ′ = 0 ⇔ b 0 = ± b , and  γ ′ ≠ 0 . 4- By corresponding debates, we can also determine d ′ = d ′ ′ = d ′ ′ ′ = d i v = 0 ⇔ b 0 = ± b ,  γ ′ = 0 , and  γ ′ ′ ≠ 0 . The proof is finished.

In view of Proposition 1, we determine

(a) The osculating circle S ( ρ , e 0 ) of b ( v ) ∈ S 1 2 is displayed by < e 0 , b > = 1 + ρ 2 , < b ′ , e 0 > = 0 , < b ′ ′ , e 0 > = 0 , which are pointed via the situation that the osculating circle must have touch of at least third order at b ( v 0 ) iff γ ′ ≠ 0 .

Through this method, by catching into meditation the curvature axes of b ( v ) ∈ S 1 2 , we can attain a concatenation of curvature axes e 2 , e 3 ,…, e n . The ownerships and the joint links via these curvature axes are much pleasant troubles. For example, it is facile to catch that if e 0 = ± e and γ ′ = 0 , b ( v ) located at ψ is specified relative to e 0 . At this position, the curvature axis is fixed up to second order and R is a slant T L R S .

Theorem 1

A non-developable S L R S is a slant SLRS iff its geodesic curvature γ ( v ) is fixed.

Definition 3

Let R and R * be two non-developable ruled S L surfaces in E 1 3 . R is entitled a B O of R * if there exists a bijection via their rulings such that R and R * possess a reciprocal central normal at the conformable striction points.

Let R * be a B O of R and c * ( v * ) b * ( v * ) , z * ( v * ) , g * ( v * ) is the B F of R * , as shown in Equations 7–9. Then, the surface R * can be allocated by R * : y * v * , t = c * v * + t b * v * ,   t ∈ R , (21) where c * v * = c v + Γ * v z v . (22) Here, Γ * ( v ) is the distance through the proportional striction points of R and R * . Through the differentiation of Equation 21 via v and considering Equation 22, we assign z * v * ′ = Δ + Γ * b + Γ * ′ z + δ + γ Γ * g . (23) Since z * = z at the congruent striction points of R and R * , we gain Γ * ′ = 0 ⇒ Γ * is fixed. Furthermore, given that Γ is the angle among the rulings of R and R * , that is, < b * , b > = cos Γ . (24) By differentiation of Equation 23, we gain < z * , b > v * ′ + < b * , z >   = − Γ ′ ⁡ sin Γ . (25) Since z * = z , then we realize Γ ′ = 0 ⇒ Γ is fixed. Moreover, at the congruent striction points of R and R * , we observe that < g * , g > = cos Γ . Then, by Equation 24 b * z * g * = cos Γ 0 sin Γ 0 1 0 − sin Γ 0 cos Γ b z g . (26) If Γ = 0 (resp. π 2 ), then R and R * are parallel (resp. oriented) offsets.

Theorem 2

The couple ( Γ , Γ * ) is fixed at the corresponding striction points of R and R * .

It is apparent from Theorem 2 that a non-developable S L R S , frequently, has a binary infinity of B O . Every B O can be displayed by a fixed linear offset Γ * ∈ R and a fixed-angle offset Γ ≥ 0 . Therefore, if R * is a B O of R , then R is also a B O of R * .

Let u * ( v * , t ) be the S L unit normal of R * . Then, as shown in Equation 10, we locate u * v * , t = y v * × y t * y v * × y t * = t g * + δ * z * − t 2 + δ * 2 ,   t > δ * , (27) where δ * is the distribution parameter of R * .

The dissimilarity between the normal to a R S and its B O is apparent from Equations 10, 26. This demonstrates that the B O of a R S is often not a parallel offset. Therefore, the parallel circumstances through R * in view of R can be exhibited by the following:

Theorem 3

R and R * are parallel offsets iff ( a ) δ = δ * , with ( b ) , their Blaschke frames, being conformable.

Proof. Let u * ( v * , t ) × u ( v , t ) = 0 , that is, R and R * are parallel offsets. Then, by Equations 10, 26, we acquire t δ * − δ ⁡ cos Γ b − t 2 ⁡ sin Γ z − t δ ⁡ sin Γ g = 0 , which is assumed true for any value t ≠ 0 , that is, δ = δ * , and Γ = 0 .

Let the two events hold true, that is, δ = δ * and Γ = 0 . Then, substituting them into u * ( v * , t ) × u ( v , t ) using Equation 27, we acquire u * v * , t × u v , t = t g * + δ * z * − t 2 + δ * 2 × t g + δ z − t 2 + δ 2 , which indicates that R and R * are parallel offsets since the previous u * ( v * , t ) × u ( v , t ) is a zero vector.

Using the same approach, but with a developable surface δ = 0 , we encounter the following:

Corollary 3

A developable S L R S and its developable B O are parallel offsets iff their Blaschke frames are identical.

Corollary 4

A developable S L R S and its non-developable B O cannot be parallel offsets.

Furthermore, we also detect d d v * b * z * g * = 0 1 0 1 0 γ * 0 γ * 0 b * z * g * , (28) where d v * = cos Γ + γ ⁡ sin Γ d v ,    γ * d v * = γ ⁡ cos Γ − sin Γ d v . (29) By takeoff d v * / d v , we locate using Equations 28, 29 γ − γ * cos Γ + γ * γ − 1 sin Γ = 0 . (30) This presents a new perspective of B O of S L R surfaces, specifically focusing on their geodesic curvatures.

Theorem 4

R and R * are BOSLR surfaces iff Equation 30 is fulfilled.

Corollary 5

R and R * are parallel offsets iff γ * − γ = 0 .

Corollary 6

R and R * are oriented offsets iff γ * γ − 1 = 0 .

For γ ( v ) being fixed, from Equations 7, 12, we have the ODE, b ′ ′ ′ − κ 2 b ′ = 0 . In accordance with several algebraic manipulations, the solution is b ϰ = cos ⁡ ψ , sin ⁡ ψ ⁡ sinh ϰ , sin ⁡ ψ ⁡ cosh ϰ , (31) where ψ is fixed and ϰ = 1 + γ 2 v . Then, z ϰ = d b d ϰ d b d ϰ − 1 = 0 , cosh ϰ , sinh ϰ , g ϰ = b × z = − sin ⁡ ψ , cos ⁡ ψ ⁡ sinh ϰ , cos ⁡ ψ ⁡ cosh ϰ . (32) Therefore, from Equations 8, 30, 31, S C c ( ϰ ) is expressed as c ϰ ≔ c 1 c 2 c 3 = ∫ 0 ϰ Δ d ϰ cos ⁡ ψ − ∫ 0 ϰ δ d ϰ sin ⁡ ψ ∫ 0 ϰ Δ ⁡ sinh ϰ d ϰ sin ⁡ ψ + ∫ 0 ϰ δ ⁡ sinh ϰ d ϰ cos ⁡ ψ ∫ 0 ϰ Δ ⁡ cosh ϰ d ϰ sin ⁡ ψ + ∫ 0 ϰ δ ⁡ cosh ϰ d ϰ cos ⁡ ψ . (33) Hence, from Equations 9, 30–33, the slant S L R S is expressed as R : r ϰ , t = c 1 , c 2 , c 3 + t cos ⁡ ψ , sin ⁡ ψ ⁡ sinh ϰ , sin ⁡ ψ ⁡ cosh ϰ . (34) Furthermore, by Equations 20, 25, 33, B O R * is expressed as R * : r * ϰ , t = c 1 + Γ * ⁡ sinh ϰ + t ⁡ cos ⁡ ϖ c 2 + Γ * ⁡ cosh ϰ + t ⁡ sin ⁡ ϖ ⁡ sinh ϰ c 3 + t ⁡ sin ⁡ ϖ ⁡ cosh ϰ ,   t ∈ R , (35) where ϖ = ψ + Γ and Γ * can control the shape of R * ; here, we will set Γ * = − 0.5 , ψ = π 4 , − 4 ≤ t ≤ 4 , and − 3 ≤ ϰ ≤ 3 .

This work explores the features of slant curves and develops and classifies slant S L R surfaces and their B O in Minkowski 3-space E 1 3 using the Blaschke domain. Next, we construct contemporary S L R surfaces in Lorentzian line space and determine their B O . In addition, we also obtain various groupings by a slant S L R S and its striction curve. These advancements are expected to enhance the usefulness of model-based manufacturing in mechanical outputs and geometric patterning. The authors intend to correlate this study across several domains and examine the classification of singularities, as identified in [31, 32].