Evolutoids and pedaloids of frontals on timelike surfaces

1 Introduction

The research of special curves and surfaces has attracted much more attention of mathematicians and physicists. For example, in optics, caustics and orthotomics can be regarded as evolutes and involutes of curves [1,2]. Helices and helicoids are frequently used in mechanics [3–5]. Moreover, pedal models are useful for the study of astrophysics. Blaschke et al. [6] applied pedal coordinates to show a surprising connection between the mechanical system of a free double linkage and the orbits of particles inside a Schwarzschild Black Hole. In addition, a special solution to the Dark Kepler problem was provided by this mechanical system.

In differential geometry, the evolute is the locus of the centers of curvature of a plane curve. From the viewpoint of the envelope theory, an evolute is the envelope of normal lines of a plane curve, while the plane curve is the envelope of its own tangent lines. Giblin and Warder considered the question of what lies between the evolute and the original curve, and then defined the evolutoid of a plane curve as the envelope of a family of lines such that each line makes a constant angle with the tangent line of the original curve [7]. The evolutoid coincides with the evolute or the original curve if the angle equals π ∕ 2 or zero, respectively. Another important curve in terms of its geometric properties is the pedal curve, which is defined by the orthogonal projection of a fixed point on the tangent lines of a plane curve. The pedal curve is also the envelope of a family of circles. Analogous to the evolutoid, Izumiya and Takeuchi gave the definition of the pedaloid in [8]. Moreover, the connection between pedal curves and evolutes of plane curves was found by them when the angle changed.

Recently, there have been some new results about evolutoids and pedaloids of curves in Minkowski space [9,10]. Šekerci and Izumiya [9] defined the evolutoids and the pedaloids of curves in Minkowski plane by using the Lorentzian angle. They also showed the relationships among evolutoids, pedaloids, and contrapedaloids of curves in the Minkowski plane. Šekerci [10] investigated the geometric properties of evolutoids and pedaloids for curves on spacelike surfaces. The most interesting result is that the evolutoid is a ruled surface, not a curve. Due to the tangent space of each point on the spacelike surface being identified with the Euclidean plane, Šekerci defined the evolutoid of a curve on the spacelike surface by using the standpoint of the circle in the Euclidean plane. As a part of our research projects about the differential geometry of timelike submanifolds, we focus our attention on the study of the evolutoids and the pedaloids of non-lightlike frontals on timelike surfaces. The essential difference between a spacelike surface and a timelike surface in Minkowski 3-space is that the tangent space of the timelike surface is a Minkowski plane for any point. As mentioned in [9,11,12], the evolute and the pedal curve in the Minkowski plane present certain peculiarities when compared with the Euclidean case. For instance, the evolute of a circle is a quadrangular star in the Minkowski plane while it is a point in the Euclidean plane (Figure 1). Furthermore, the frontal we consider here may have singular points, which is a natural extension of the regular curve.

Figure 1

A circle and its evolute in Minkowski plane.

In this article, we define the evolutoids of non-lightlike frontals on timelike surfaces by taking into account the connection between the evolutoids of plane frontals in Minkowski 2-space and Minkowski 3-space. As a result, the evolutoids of non-lightlike frontals on timelike surfaces are ruled surfaces with lightlike direction. In physics, ruled surfaces with lightlike direction are of importance since they provide models of different horizon types, such as event horizons of isolated horizons, Kerr black holes, Kruskal horizons, Cauchy horizons, and Killing horizons [13–19]. In addition, singularity theory is a very useful tool for studying the geometric properties of submanifolds immersed in different ambient spaces. For instance, Chen [20] investigated the geometric properties of spacelike curves on a timelike surface from the contact viewpoint by applying the singularity theory. Regarding singularities, the evolutoids will inevitably have singularities. We study the singularities of evolutoids and classify these singularities by using the singularity theory. Furthermore, we define the pedaloids of non-lightlike frontals on timelike surfaces and show the relationship between the pedaloids and the pedal surfaces of evolutoids.

The structure of this article is as follows. In Section 2, we introduce some necessary definitions and show the local geometry of frontals in Minkowski space. In view of the evolutoids of curves in the Euclidean plane and the Minkowski plane, the evolutoids of plane frontals are defined in Minkowski 3-space in Section 3. We generalize this concept to non-lightlike frontals on timelike surfaces in Section 4. The evolutoids of non-lightlike frontals are ruled surfaces with lightlike direction, and we obtain the concrete expressions for their singularities. In Section 5, we define two kinds of functions on non-lightlike frontals and obtain two geometric invariants. By using the unfolding theory, we give the classification of singularities of evolutoids. The types of singularities are closely related to the new geometric invariants. In addition, we study the pedaloids of non-lightlike frontals in Section 6. Meanwhile, we give the equivalent condition for singular points of pedaloids and prove that the pedal surfaces of evolutoids could be represented by the pedaloids and some functions. Finally, we give some examples in Section 7.

2 Basic notions

In this section, we introduce some necessary notations and concepts. Let R n be an n -dimension real vector space. The pseudo-scalar product on R n is defined by ⟨ a 1 , a 2 ⟩ = − a 1 1 a 2 1 + a 1 2 a 2 2 + ⋯ + a 1 n a 2 n , where a 1 = ( a 1 1 , a 1 2 , … , a 1 n ) and a 2 = ( a 2 1 , a 2 2 , … , a 2 n ) . We call ( R n , ⟨ , ⟩ ) a Minkowski n-space and denote it by R 1 n .

A non-zero vector a ∈ R 1 n is called spacelike, lightlike, and timelike if ⟨ a , a ⟩ > 0 , = 0 , < 0 , respectively. The norm of a is defined by ‖ a ‖ = ∣ ⟨ a , a ⟩ ∣ . If a 1 , a 2 , … , a n − 1 ∈ R 1 n , we define the pseudo-vector product of them by

We introduce three types of pseudo-spheres with the center q ∈ R 1 n and radius r ≥ 0 in R 1 n as follows:

We can denote S 1 n − 1 ( 0 , 1 ) and H n − 1 ( 0 , 1 ) as S 1 n − 1 and H n − 1 .

Let U be an open set in R 2 . We consider an embedding X : U → R 1 3 and write M = X ( U ) . The unit normal vector field along M for p = X ( u ) is defined as follows:

where u = ( u 1 , u 2 ) ∈ U . We say the embedding X is spacelike, lightlike, or timelike if the normal vector is timelike, lightlike, or spacelike for any p ∈ M , respectively. Let γ ¯ : I → U be a plane curve, then we have a curve γ : I → M defined by γ ( t ) = X ( γ ¯ ( t ) ) . In this article, we loosen up the restriction of regularity for the curve γ lying in a timelike surface M , namely, γ may have singular points. However, it is impossible to construct the Darboux-type frame along a singular curve γ as we deal with a regular curve. Inspired by the work of Honda et al. [21], we can define a moving frame for a frontal in T M ( 1 ) = { ( x , v ) ∈ T M ∣ ‖ v ‖ = 1 } , where T M ( 1 ) is the unit tangent bundle over M equipped with the canonical contact structure. We say that ( γ , b ) : I → T M ( 1 ) is a Legendre curve if ⟨ γ ′ ( t ) , b ( t ) ⟩ = 0 for any t ∈ I . We also say that γ is a frontal. For a timelike surface M = X ( U ) , let γ ¯ : I → U be a plane curve and ( γ , b ) : I → T M ( 1 ) a Legendre curve, where γ ( t ) = X ( γ ¯ ( t ) ) . The spacelike normal vector n γ ( t ) along γ is defined by n γ ( t ) = n ∘ γ ( t ) . Set t ( t ) = b ( t ) ∧ n γ ( t ) , we say γ is a timelike frontal, a lightlike frontal, or a spacelike frontal on M if t is timelike, lightlike, or spacelike for any t ∈ I , respectively. Moreover, the Frenet-Serret formula along a non-lightlike frontal γ is as follows:

Here, ε ( t ) = ⟨ t ( t ) , t ( t ) ⟩ . m ( t ) = ⟨ t ′ ( t ) , n γ ( t ) ⟩ , n ( t ) = − ε ( t ) ⟨ t ′ ( t ) , b ( t ) ⟩ , and l ( t ) = − ε ( t ) ⟨ n γ ′ ( t ) , b ( t ) ⟩ are curvature functions. Furthermore, there exists a smooth function α : I → R such that γ ′ ( t ) = α ( t ) t ( t ) . Obviously, t 0 is a singular point of γ if α ( t 0 ) = 0 .

Next, we introduce the definitions of evolutes in a Minkowski plane. Let γ : I → R 1 2 be a non-lightlike curve without inflection points and singular points. The evolute of γ is defined as follows:

where N ( t ) is the unit normal vector and κ ( t ) is the curvature function (see [22]). For a non-lightlike frontal γ in the Minkowski plane, it is defined as follows.

A curve ( γ , ν ) → R 1 2 × H 1 (or R 1 2 × S 1 1 ) is called a spacelike Legendrian curve (or timelike Legendrian curve) if ( γ ( t ) , ν ( t ) ) ∗ θ = 0 , where θ is a canonical contact structure on R 1 2 × H 1 (or R 1 2 × S 1 1 ). The condition ( γ ( t ) , ν ( t ) ) ∗ θ = 0 is equivalent to ⟨ γ ′ ( t ) , ν ( t ) ⟩ = 0 for any t ∈ I . We say a curve γ is a spacelike frontal (or timelike frontal) if there exists ν such that ( γ , ν ) is a spacelike Legendrian curve (or timelike Legendrian curve).

Let ( γ , ν ) → R 1 2 × H 1 (or R 1 2 × S 1 1 ) be a non-lightlike Legendrian curve. Then, we have a moving pseudo-orthonormal frame { ν , μ } along γ and a Frenet-type formula of γ as follows:

In this case, there exists a smooth function β ( t ) such that γ ′ ( t ) = β ( t ) μ ( t ) . The function pair ( ι ( t ) , β ( t ) ) is the curvature function of the non-lightlike Legendrian curve ( γ , ν ) . Moreover, t 0 is an inflection point of γ if ι ( t 0 ) = 0 . Assume that there exists a function ρ such that ρ ι = β . The evolute of γ is defined as follows:

For more details about evolutes of plane frontals, see [11,23].

3 Evolutoids of plane curves

It is well known that the envelope of the family of normal lines of a curve is the evolute of this curve. Caustic is an important research object in geometric optics, which is the envelope of light rays. So the evolute of a curve is a caustic. The evolutoid of a plane curve is defined by the envelope of a family of lines such that each line has a constant angle with the tangent line of the original curve. If we consider an evolutoid in Euclidean plane, the evolutoid is the evolute if the lines have the angle π ∕ 2 with the tangent lines, and the evolutoid is the original curve γ if the angle equals zero.

Let γ : I → R 2 be a regular curve with the unit tangent vector field T and the unit normal vector field N . The curvature function κ satisfies T ′ ( s ) = κ ( s ) N ( s ) and N ′ ( s ) = − κ ( s ) T ( s ) . We consider the family of lines defined by

The envelope of this family of lines is the θ -evolutoid of γ . Since { T , N } is an orthogonal basis, any vector has the form of λ T + μ N . Applying this to e − γ if e is the θ -evolutoid, we obtain the following two equations:

We have

Therefore,

It is obvious that e ( s ) is the evolute of γ ( s ) if θ = π 2 and e ( s ) is the original curve γ ( s ) if θ = 0 .

On the other hand, we find that cos θ N − sin θ T is perpendicular to sin θ N + cos θ T and consider a function F : I × R 2 × [ 0 , π 2 ] → R , which is defined as follows:

For a fixed θ , denote f s ( x ) = F ( s , x , θ ) , then f s − 1 ( 0 ) is a line family with the angle θ with T . Moreover, the θ -evolutoid satisfies the condition F ( s , x , θ ) = ∂ F ∂ s ( s , x , θ ) = 0 .

If γ is a non-lightlike curve in Minkowski plane, the situation is different since the causal characters of tangent vectors and normal vectors are opposite and the angle is not the conventional Euclidean angle anymore. By using the Lorentzian angle ϕ , Šekerci and Izumiya [9] defined two line families. One family of lines make an angle ϕ with tangent vectors of γ . The direction of these lines are cosh ϕ T + sinh ϕ N . Note that whatever the angle ϕ varies, the vector cosh ϕ T + sinh ϕ N could not rotate to the direction paralleling to N . Hence, they defined the other family of lines which make an angle ϕ with normal vectors N . Two kinds of evolutoids of γ ( ϕ T -evolutoid and ϕ N -evolutoid) were obtained as follows:

For the case of ϕ T -evolutoid, the direction of line family is cosh ϕ T + sinh ϕ N . Then, the vector filed cosh ϕ N + sinh ϕ T is pseudo-orthogonal to the family of lines, and the family of lines satisfies the following vector equation:

Since ϕ T -evolutoid is the envelope of these lines, then ϕ T -evolutoid satisfies the following condition:

Analog to the ϕ T -evolutoid, we can obtain the expression of ϕ N -evolutoid. Even if γ is a singular curve in Minkowski plane, the evolutoid of γ was also studied in [9]. Let ( γ , ν ) be a Legendrian curve with the moving pseudo-orthonormal frame { ν , μ } and the curvature function ( ι , β ) . Assume that there exists a function ρ such that ρ ι = β . The ϕ μ -evolutoid and ϕ ν -evolutoid of γ are defined as follows:

Note that E v μ [ ϕ ] ( t ) is the original curve γ ( t ) and E v ν [ ϕ ] ( t ) is the evolute of γ ( t ) if ϕ = 0 .

In physics, a caustic characterizes a family of rays since every ray of the family is tangent to the caustic. In Minkowski 3-space, the caustic can be regarded as an envelope of a family of lines with lightlike directors. The reason is that if the direction vectors of lines are timelike or spacelike, their speed will be lower than or exceed the speed of rays, respectively. Thus, rays in Minkowski 3-space are lightlike lines. Consider a unit timelike vector w in R 1 2 . We have a lightlike vector corresponds to w in R 1 3 defined by w + e 3 , where e 3 = ( 0 , 0 , 1 ) . Therefore, for a timelike line γ ˜ 0 ( t ) = x 0 + t w , which passes through x 0 , the corresponding lightlike line is γ 0 ( t ) = x 0 + t ( w + e 3 ) . It follows that a ruled surface with lightlike direction is formed by these lightlike lines if x 0 moves in R 1 2 . Hence, in the sense of caustics, we deal with ruled surfaces with lightlike direction in order to investigate the evolutoids of non-lightlike frontals on timelike surfaces.

We first apply this thought to non-lightlike frontals in R 1 2 . Let γ be a non-lightlike frontal in R 1 2 = { ( x 1 , x 2 , 0 ) } ⊂ R 1 3 with the moving pseudo-orthonormal frame { ν , μ } . Assume that there exists a function ρ such that ρ ι = β , we distinguish two cases as follows:

Case 1: γ is a spacelike frontal.

Consider a ruled surface with lightlike direction defined by

We define the ϕ -evolutoid of the spacelike frontal γ in R 1 3 as the envelope of the above family. Since { μ , ν , e 3 } is an orthogonal basis, for this, any vector has the form of λ μ + ξ ν + u e 3 . Applying this to e − γ if e is the ϕ -evolutoid, we obtain two equations:

where β and ι are the curvatures of γ . By calculation, we have

Thus, by e = γ + λ μ + ξ ν + u e 3 , we obtain the ϕ -evolutoid as follows.

We can observe that E v [ ϕ ] γ ( s , u ) is the ϕ μ -evolutoid of γ if u = 0 .

Furthermore, since μ ± ν are also two lightlike vector fields along γ , we obtain two families of lightlike lines with the direction μ ± ν . However, there is no envelope for these two families. Next, we consider the other case.

Case 2: γ is a timelike frontal.

A ruled surface with lightlike direction X [ ϕ ] γ along γ is defined as follows:

The ϕ -evolutoid of the timelike curve γ in R 1 3 is defined as the envelope of the above family. Since { μ , ν , e 3 } is an orthogonal basis, for this, any vector has the form of λ μ + ξ ν + u e 3 . Applying this to e − γ if e is the ϕ -evolutoid, we obtain two equations:

By calculation, we have

Thus, by the form e = γ + λ μ + ξ ν + u e 3 , we obtain the ϕ -evolutoid as follows:

We can see that E v [ ϕ ] γ ( t , u ) is the ϕ ν -evolutoid of γ if u = 0 .

By the above method, we define the evolutoids of frontals on Minkowski plane in R 1 3 . Since the Minkowski plane is a flat timelike surface in R 1 3 such that t ( t ) = μ ( t ) , b ( t ) = ν ( t ) and n γ ( t ) = e 3 . Naturally, we extend the definition of evolutoids of non-lightlike frontals on timelike surfaces in R 1 3 .

4 Evolutoids of non-lightlike frontals on timelike surfaces

Let γ be a spacelike frontal on a timelike surface. We consider a ruled surface with lightlike rulings along γ , which is defined as follows:

We define the envelope of the above family as the ϕ -evolutoid of γ . According to the definition of the envelope, we obtain the ϕ -evolutoid under the condition

as follows:

In addition, if γ is a timelike frontal on a timelike surface, we consider a ruled surface with lightlike direction as follows:

With the condition

the ϕ -evolutoid of γ is obtained in the following form:

We can find that the ϕ -evolutoids of the spacelike frontal and the timelike frontal are both ruled surfaces with lightlike direction. A ruled surface with lightlike direction X ( t , u ) = B ( t ) + u D ( t ) is degenerate if ⟨ B ′ ( t ) , D ( t ) ⟩ = 0 . The definition of degenerate ruled surface with lightlike direction was provided by Liu [24].