Int. J. Anal. Appl. (2022), 20:47

Evolutes of Fronts in de Sitter and Hyperbolic Spheres

M. Khalifa Saad1,∗, H. S. Abdel-Aziz2, A. A. Abdel-Salam2

1Department of Mathematics, Faculty of Science, Islamic University of Madinah, KSA
2Department of Mathematics, Faculty of Science, Sohag University, 82524 Sohag, Egypt

∗Corresponding author: mohammed.khalifa@iu.edu.sa

Abstract. The evolute of a regular curve is a classical object from the viewpoint of differential geometry.

We study some types of curves such as framed curves, framed immersion curves, frontal curves and

front curves in 2-dimensional de Sitter and hyperbolic spaces. Also, we investigate the evolutes and

some of their properties of fronts at singular points under some conditions. Finally, some computational

examples in support of our main results are given and plotted.

1. Introduction

In 1915, Einstein formulated general relativity as a theory of space, time and gravitation in semi-

Euclidean space. However, this subject has remained dormant for much of its history because its

understanding requires advanced mathematics knowledge. Since the end of the twentieth century,

semi-Euclidean geometry has been an active area of mathematical research, and it has been applied

to a variety of subjects related to differential geometry and general relativity.

It is well known that many important results in the theory of curves in R3 were initiated by G.
Monge and G. Darboux pionnered the moving frame idea. Thereafter, Frenet defined his moving

frame and special equations which are playing an important role in mechanics and kinematics as well

as in differential geometry [1].

At the beginning of the twentieth century, A. Einstein’s theory opened a door to use of new geome-

tries. One of them, Minkowski space-time, which is simultaneously the geometry of special relativity.

It is worth mentioning that the importance of the theory of singularity as a developing area which is

Received: Aug. 8, 2022.

2010 Mathematics Subject Classification. 53A25, 53C50.

Key words and phrases. Frenet frames; evolute curves; front curves; de Sitter space; hyperbolic space.

https://doi.org/10.28924/2291-8639-20-2022-47
ISSN: 2291-8639

© 2022 the author(s).

https://doi.org/10.28924/2291-8639-20-2022-47


2 Int. J. Anal. Appl. (2022), 20:47

related to nonlinear sciences, it has been extensively applied in studying the classifications of singu-

larities associated with some objects in Euclidean and semi-Euclidean spaces [2–4]. Therefore, it has

been a field of research for many researchers.

In this paper, we focus on the evolutes of curves at singular points in de Sitter and hyperbolic

spheres. The evolute of a plane curve is defined to be the locus of the center of its osculating

circles [5]. In particular, the evolute of a regular curve is a classical object from the viewpoint of

differential geometry. The evolute of a regular curve without inflection points is given by, not only,

the locus of all its centers of curvature but also the envelope of its normal lines. The properties of

evolutes can be discussed by Frenet-Serret formulas, distance squared functions and the theories of

Lagrangian and Legendre singularities. In general, there exist singular points along the evolute of a

regular curve and the singular points corresponding to the vertices of a regular curve. There are at

least four vertices for a simple closed curve. One can not define the evolutes of curves at singular

points. However, we can define evolutes of fronts under some conditions. In [6,7], T. Fukunaga and

M. Takahashi defined Legendre curves in Euclidean plane and studied evolutes of Legendre curves.

Moreover, S. Izumiya, D. He pei, T.Sano and E. Torii defined the evolute curve in hyperbolic 2-space

and found its equation (see [7]).

The paper can be organized as follows: Section 3 presents a framed curve and gives its moving

frame in de Sitter sphere. Moreover, we define a pair of smooth functions of this curve as a geodesic

curvature for a regular curve. We define evolutes of fronts in de Sitter sphere. The evolute of a front is

a generalization of the notion of an evolute of a regular curve. Therefore, we discuss some properties

of evolutes without inflection points. By the representation, we give properties for an evolute of the

front. For example, the evolute of a front is also a front (see Theorem 3.1). In section 4, similar to

the way that considered in the study of the framed curves, fronts and the evolutes in de Sitter sphere,

we do it in the hyperbolic sphere, see Theorem (4.1). We shall assume throughout the whole paper

that all manifolds and maps are C∞ unless the contrary is explicitly stated.

2. Geometric meanings and basis concepts

In this section, we present some of classical differential geometric properties of de Sitter and

hyperbolic spaces of plane curves. We adopt S21 and H
2
0 as models of de Sitter and hyperbolic spheres

in Minkowski 3-space E31, respectively. Since S
2
1 and H

2
0 are a Riemannian manifolds, so the explicit

differential geometry of the curves in these spheres is analogous to the differential geometry of the

curves in the Euclidean plane (for more details see [5,8]).

Let R3 = {(x1,x2,x3) | x1,x2,x3 ∈ R} be a 3-dimensional vector space, and x = (x1,x2,x3) and
y = (y1,y2,y3) be two vectors in R3. The pseudo scalar product of x and y is defined by 〈x,y〉 =
−x1y1+x2y2+x3y3. We call (R3,〈,〉) a 3-dimensional pseudo Euclidean space, or Minkowski 3-space.
We write E31 instead of (R

3,〈,〉). We say that a vector x in E31 is spacelike, lightlike or timelike if



Int. J. Anal. Appl. (2022), 20:47 3

〈x,x〉 > 0, 〈x,x〉=0 or 〈x,x〉 < 0, respectively. Now, we define two spheres in E31 as follows:

Q2� =


H

2
0 = {x ∈E

3
1 | −x

2
1 +x

2
2 +x

2
3 =−1}, if � =−

S21 = {x ∈E
3
1 | −x

2
1 +x

2
2 +x

2
3 =1}, if � =+,

and we take

H20 =


H

2
+ = {x ∈E31 | −x

2
1 +x

2
2 +x

2
3 =−1, x1 ≥ 1}

H2− = {x ∈E31 | −x
2
1 +x

2
2 +x

2
3 =−1, x1 ≤−1},

where H20 =H
2
+ ∪H2−. We call H20 a hyperbolic sphere and S

2
1 a de Sitter sphere.

Let γ : I −→Q2� ⊂E31; γ(t)= (x1(t),x2(t),x3(t)) be a smooth regular curve in Q
2
� (i.e.,γ

′(t) 6=0)
for any t ∈ I, where I is an open interval. It is easy to show that 〈γ′(t),γ′(t)〉 > 0. Throughout the
remainder in this paper, we denote the parameter s of γ as the arc-length parameter. Let us denote

T(s)= γ̇(s), and we call T(s) a unit tangent vector of γ at s.

For any x = (x1,x2,x3),y = (y1,y2,y3) ∈ R31, the pseudo vector product of x and y is defined as
follows:

x ∧y =

∣∣∣∣∣∣∣∣
−e1 e2 e3
x1 x2 x3

y1 y2 y3

∣∣∣∣∣∣∣∣ =(−(x2y3 −x3y2),x3y1 −x1y3,x1y2 −x2y1).
Here, we set a vector E(s) = γ(s)∧T(s). By definition, we can calculate that 〈E(s),E(s)〉 = 1
and 〈γ(s),γ(s)〉 = −1. Also, we can show that T(s)∧E(s) = −γ(s) and γ(s)∧E(s) = −T(s).
Therefore, we have a pseudo-orthonormal frame {γ(s),T(s),E(s)} along γ(s). The de Sitter Frenet-
Serret formula of plane curve are:


γ̇(s)

Ṫ(s)

Ė(s)


 =



0 1 0

−1 0 κg
0 κg 0





γ(s)

T(s)

E(s)


 , (2.1)

where κg is the geodesic curvature of γ in S21, which is given by

κg(s)= det(γ(s) T(s) Ṫ(s)). (2.2)

For the general parameter t, we get T(t) =
γ′(t)

‖γ′(t)‖
and E(t) = γ(t) ∧ T(t). Then, de Sitter

Frenet-Serret formula of γ(t) is expressed as:

γ′(t)

T′(t)

E′(t)


 =




0 ‖γ′(t)‖ 0
−‖γ′(t)‖ 0 ‖γ′(t)‖κg
0 ‖γ′(t)‖κg 0





γ(t)

T(t)

E(t)


 , (2.3)

where

κg(t)=
det (γ(t) γ′(t) γ′′(t))

‖γ′(t)‖3
. (2.4)



4 Int. J. Anal. Appl. (2022), 20:47

Also, the hyperbolic Frenet-Serret formula is given by:

γ̇(s)

Ṫ(s)

Ė(s)


 =



0 1 0

1 0 κg

0 −κg 0





γ(s)

T(s)

E(s)


 , (2.5)

where κg is the geodesic curvature of the curve γ in H20, which is defined as

κg(s)= det(γ(s) T(s) Ṫ(s)). (2.6)

We have the following hyperbolic Frenet-Serret formula of γ(t)

γ′(t)

T′(t)

E′(t)


 =




0 ‖γ′(t)‖ 0
‖γ′(t)‖ 0 ‖γ′(t)‖κg
0 −‖γ′(t)‖κg 0





γ(t)

T(t)

E(t)


 , (2.7)

Definition 2.1. Under the assumption κ2g 6= ±1, the evolute of a regular curve γ is defined as
Eγ : I −→Q2�;

Eγ(t)=
1√

|κ2g(t)−1|
(κg(t)γ(t)+ �E(t)); (2.8)

Eγ is called hyperbolic evolute or de Sitter evolute of γ when � =1 or � =−1, respectively (see [5,8]).

Remark 2.1. Eγ(t) is located in H20 with κ
2
g(t) > 1, and it is in S

2
1 with κ

2
g(t) < 1.

3. Evolutes of fronts in de Sitter sphere S21

If γ has a singular point, we can not construct a moving frame of γ in a traditional way. However,

we could define a moving frame of a front curve. For the case of Euclidean plane, there are some

creative works (see [6,7,9] ). Now, we give the following definitions.

Definition 3.1. We say that (γ,ν) : I −→ S21 × S
2
1 is a framed curve, if 〈γ(t),ν(t)〉 = 0 and

〈γ′(t),ν(t)〉=0 for all t ∈ I. Moreover, if (γ,ν) is an immersion, namely, (γ′(t),ν′(t)) 6=(0,0), we
call (γ,ν) a framed immersion curve.

Definition 3.2. We say that γ : I −→ S21 is a frontal curve if there exists a smooth mapping ν : I −→ S
2
1

such that (γ,ν) is a framed curve. We also say that γ : I −→ S21 is a front curve if there exists a
smooth mapping ν : I −→ S21 such that (γ,ν) is a framed immersion curve.

Throughout this paper, we assume that the pair (γ,ν) is co-oriented and the singular points of γ

are finite.

Let (γ,ν) : I −→ S21×S
2
1 be a framed curve. If γ is singular at t0, then we can’t define a frame in

a traditional way. However, ν always exists even if t is a singular point of γ. We take µ = ν ∧γ and



Int. J. Anal. Appl. (2022), 20:47 5

call the pair {γ,ν,µ} a moving frame of γ and then, de Sitter Frenet-Serret formula is given by:

γ′(t)

ν′(t)

µ′(t)


 =




0 0 m(t)

0 0 n(t)

−m(t) n(t) 0





γ(t)

ν(t)

µ(t)


 ,

where n(t) = 〈ν′(t),µ(t)〉, ν(t) and µ(t) are both unit spacelike vectors. We declare that (γ,−ν)
is also a framed curve. In this case, m(t) dose not change, but n(t) changes to −n(t). If (γ,ν) is
a framed immersion, we have (m(t),n(t)) 6= (0,0) for each t ∈ I. The pair (m,n) is an important
pair of functions of the framed curves as the geodesic curvature of a regular curve. We call the pair

(m,n) a geodesic curvature of the framed curve. Also, we have ν ∧µ = γ and γ ∧µ = ν (for more
details see [8,10]).

In what follows, some properties of the meant curves are introduced.

Proposition 3.1. If γ be a regular curve and (γ,ν) : I −→ S21 ×S
2
1 be its framed curve, then the

relationship between their geodesic curvatures is expressed as:

κg(t)=−
n(t)

|m(t)|
. (3.1)

Proof. By direct calculations, we have

γ′(t)= m(t)µ(t),

γ′′(t)= m′(t)µ(t)+m(t)µ′(t)

= m′(t)µ(t)+m(t)(−m(t)γ(t)+n(t)ν(t)),

where

κg(t)=
det(γ(t),γ′(t),γ′′(t))

‖γ′(t)‖3

=
det
(
γ(t),mµ(t),−m2γ(t)+mnν(t)+m′µ(t)

)
|m3|

,

so, we get

κg(t)=−
n(t)

|m(t)|
.

For a framed immersion (γ,ν), we say that t0 is an inflection point of the front γ if n(t0) = 0. And

then the condition m(t0) 6=0 and n(t0)=0, is equivalent to the condition κg(t0)=0. �

If γ is not a regular curve, then we can not define the evolute as in equation (2.8), since the geodesic

curvature may be divergence at a singular point. However, we can give the definition of the evolute of

a front and then focus on its properties. Hence, the notion of a parallel curve of γ can be presented

as follows:



6 Int. J. Anal. Appl. (2022), 20:47

Let (γ,ν) : I −→ S21 ×S
2
1 be a framed curve which has the geodesic curvature (m,n). So, we can

define a parallel curve γυ : I −→ S21 of γ as follows:

γυ(t)=
1√
|υ2 −1|

(γ(t)+υν(t)) , (3.2)

where υ ∈R and υ 6=±1.

Proposition 3.2. If γυ is a regular curve, then

κgυ(t)=
−n(t)−υm(t)
|m(t)+υn(t)|

. (3.3)

Proof. From Eq.(3.2), we have

γ′υ(t)=
m+υn√
|υ2 −1|

µ(t),

γ′′υ(t)=
−m2 −υmn√
|υ2 −1|

γ(t)+
mn+υn2√
|υ2 −1|

ν(t)+
m′ +υn′√
|υ2 −1|

µ(t),

then, we find

γυ ∧γ′υ =
1

(υ2 −1)

∣∣∣∣∣∣∣∣
−γ(t) ν(t) µ(t)
1 υ 0

0 0 m+υn

∣∣∣∣∣∣∣∣
=
−υ(m+υn)γ(t)− (m+υn)ν(t)

(υ2 −1)
.

Since

κgυ(t)=
det(γυ(t),γ

′
υ(t),γ

′′
υ(t))

‖γ′υ(t)‖3
.

We obtain

|m(t)+υn(t)|κgυ(t)= (−n(t)−υm(t)).

Thus, this completes the proof. �

Proposition 3.3. For a framed immersion curve (γ,ν) : I −→ S21×S
2
1, the parallel curve γυ : I −→ S

2
1

is a front for each υ 6=±1.

Proof. We take νυ : I −→ S21 by

νυ(t)=
1√
|υ2 −1|

(υγ(t)+ν(t)) ,

since 

γυ(t)=

1√
|υ2 −1|

(γ(t)+υν(t)) ,

γ′υ(t)=
1√
|υ2 −1|

(γ′(t)+υν′(t)) .



Int. J. Anal. Appl. (2022), 20:47 7

If γ′υ(t0)=0 at a point t0 ∈ I, then we have

γ′(t0)+υν
′(t0)=0.

Also, if ν′(t0)=0, then γ′(t0)=0. It is contradicted with the fact that (γ,ν) is a framed immersion

and hence (γυ,νυ) is a framed immersion. By ‖ν(t)‖=1, we have 〈ν(t),ν′(t)〉=0. Then

〈γ′υ(t),νυ(t)〉=
1

(υ2 −1)
〈γ′ +υν′,υγ +ν〉=0,

therefore, it leads to the curve γυ is a front. �

Proposition 3.4. Let (γ,ν) be a framed curve. If γ is a regular curve and υ 6=1/κg, then a parallel
curve γυ is also a regular curve and its evolute is given by

Eγυ(t)=−Eγ(t). (3.4)

Proof. Since

γυ(t)=
1√
|υ2 −1|

(γ(t)+υE(t)) ,

γ′υ(t)=
‖γ′‖√
|υ2 −1|

(1+υκg)T(t),

γ′′υ(t)=
‖γ′‖√
|υ2 −1|

(
−‖γ′‖(1+υκg)γ(t)+υκ′gT(t)+‖γ

′‖κg(1+υκg)E(t)
)
.

By the assumption υ 6=1/κg, γυ is a regular curve. Therefore, we get

γυ ∧γ′υ =
1

(υ2 −1)

∣∣∣∣∣∣∣∣
−γ(t) T(t) E(t)
1 0 υ

0 ‖γ′‖(1+υκg) 0

∣∣∣∣∣∣∣∣
=
‖γ′‖υ(1+υκg)γ(t)+‖γ′‖(1+υκg)E(t)

(υ2 −1)
,

and hence

〈γυ ∧γ′υ,γ
′′
υ〉=

‖γ′‖3υ(1+υκg)2 +‖γ′‖3κg(1+υκg)2

|υ2 −1|
3
2

,

we get

κgυ(t)=
κg +υ

|1+υκg|
,

and

Tυ(t)=
γ′υ
‖γ′υ‖

=
1+υκg
|1+υκg|

T(t).

Since Eυ = γυ ∧Tυ, we obtain

Eυ(t)=
1+υκg
|1+υκg|

1√
|υ2 −1|

(E(t)+υγ(t)) .



8 Int. J. Anal. Appl. (2022), 20:47

Thus, from (2.8) we find

Eγυ(t)=
1√∣∣κ2gυ(t)−1∣∣ (κgυ(t)γυ(t)−Eυ(t))

=
1√√√√∣∣∣∣∣

(
υ +κg
|1+υκg|

)2
−1

∣∣∣∣∣
(

υ +κg
|1+υκg|

(γ(t)+υE(t))√
|υ2 −1|

−
1+υκg
|1+υκg|

(E(t)+υγ(t))√
|υ2 −1|

)

=
1√√√√∣∣∣∣∣

(
υ +κg
|1+υκg|

)2
−1

∣∣∣∣∣
1

|1+υκg|
√
|υ2 −1|

(
(1−υ2)κgγ(t)− (1−υ2)E(t)

)

=
1√∣∣(υ2 −1)−κ2g(υ2 −1)∣∣

(1−υ2)√
|υ2 −1|

(κgγ(t)−E(t))

=−
1√∣∣κ2g(t)−1∣∣ (κg(t)γ(t)−E(t))

=−Eγ(t).

Thus, the proof is completed. �

Definition 3.3. Let (γ,ν) : I −→ S21 × S
2
1 be a framed immersion curve. We define an evolute

Eγ : I −→ S21 of γ as follows:
If t is a regular point, then

Eγ(t)=
1√∣∣κ2g(t)−1∣∣ (κg(t)γ(t)−E(t)) . (3.5)

If t0 is a singular point, for any t ∈ (t0 −δ,t0 +δ), we get

Eγυ(t)=
−1√∣∣κ2gυ(t)−1∣∣ (κgυ(t)γυ(t)−Eυ(t)) , (3.6)

where δ is a sufficiently small positive real number and υ ∈R satisfies the condition υ 6=1/κg(t).

Now, we give another representation of the evolute by using the moving frame {γ(t),ν(t),µ(t)}
and its geodesic curvature {m(t),n(t)}.

Theorem 3.1. Under the condition |m(t)| 6= |n(t)|, the evolute of a front curve Eγ(t) : I −→ S21 is
represented by

Eγ(t)=
1√

|n2(t)−m2(t)|
(−n(t)γ(t)+m(t)ν(t)) , (3.7)

and Eγ(t) is a front curve.



Int. J. Anal. Appl. (2022), 20:47 9

Proof. (i) Suppose that γ is a regular curve. Since γ′(t)= m(t)µ(t), we have |m(t)| 6=0 and

T(t)=
m(t)

|m(t)|
µ(t), E(t)=−

m(t)

|m(t)|
ν(t).

From Eqs.(3.1) and (3.5), we get

Eγ(t)=
1√∣∣κ2g(t)−1∣∣ (κg(t)γ(t)−E(t))

=
1√√√√∣∣∣∣∣

(
−
n(t)

|m(t)|

)2
−1

∣∣∣∣∣
(
−
n(t)

|m(t)|
γ(t)+

m(t)

|m(t)|
ν(t)

)

=
1√

|n2(t)−m2(t)|
(−n(t)γ(t)+m(t)ν(t)) .

(ii) Suppose that t0 is a singular point of γ and consider γυ in de Sitter sphere, also we know that γυ

is a regular curve around the neighbourhood of t0 with υ 6=1/κg(t).
From Eq.(3.2), we get

γ′υ(t)=
m+υn√
|υ2 −1|

µ(t),

then, |m+υn| 6=0 and

Tυ(t)=
m+υn

|m+υn|
µ(t),

where Eυ(t)= γυ(t)∧Tυ(t), we have

Eυ(t)=
m+υn

|m+υn|
1√
|υ2 −1|

(−υγ(t)−ν(t)) .

Therefore, from Eq.(3.3), we find

κgυ =−
n+υm

|m+υn|
,

and from Eqs.(3.4) and (3.6), we get

Eγ(t)=−Eγυ(t)

=
1√∣∣κ2gυ(t)−1∣∣ (κgυ(t)γυ(t)−Eυ(t))

=
1√√√√∣∣∣∣∣

(
−
(n+υm)

|m+υn|

)2
−1

∣∣∣∣∣
((
−
(n+υm)

|m+υn|

)
(γ(t)+υν(t))√
|υ2 −1|

+
m+υn

|m+υn|
(υγ(t)+ν(t))√
|υ2 −1|

)

=
1√

|(n+υm)2 − (m+υn)2|

(
nγ(t)+υ2mν(t)−υ2nγ(t)−mν(t)

)√
|υ2 −1|

=
1√

|(υ2 −1)(n2 −m2)|
(υ2 −1)(mν(t)−nγ(t))√

|υ2 −1|



10 Int. J. Anal. Appl. (2022), 20:47

=
1√

|n2 −m2|
(−nγ(t)+mν(t))

= Eγ(t).

If we take ν̃(t)= µ(t), then (Eγ(t), ν̃(t)) is a framed immersion. In fact, we have

E′γ(t)=
mm′ −nn′

|n2 −m2|
3
2

(−nγ(t)+mν(t))+
1√

|n2 −m2|
(
−n′γ(t)+m′ν(t)

)
=
m′n−mn′

|n2 −m2|
3
2

(−mγ(t)+nν(t))

=
d

dt

(m
n

) n2
|n2 −m2|

3
2

(−mγ(t)+nν(t)) ,

we have 〈Eγ(t), ν̃(t)〉= 〈E′γ(t), ν̃(t)〉=0. And (γ,ν) is a framed immersion satisfying (m(t),n(t)) 6=
(0,0). Since, ν̃(t) = µ(t), we get ν̃′(t) = −mγ(t)+ nν(t) 6= 0. It follows that Eγ(t) is a front.
Hence, this completes the proof. �

4. Evolutes of fronts in hyperbolic sphere H20

In the hyperbolic sphere H20, if β has a singular point, we can not construct a moving frame of β in
a traditional way. However, we could define a moving frame of a front curve. So, we give the following

definitions.

Definition 4.1. We say that (β,ν) : I −→ H20 × H
2
0 is a framed curve, if 〈β(t),ν(t)〉 = 0 and

〈β′(t),ν(t)〉=0 for all t ∈ I. Moreover, if (β,ν) is an immersion, namely, (β′(t),ν′(t)) 6=(0,0), we
call (β,ν) a framed immersion curve.

Definition 4.2. We say that β : I −→ H20 is a frontal curve if there exists a smooth mapping
ν : I −→ H20 such that (β,ν) is a framed curve. We also say that β : I −→ H

2
0 is a front curve if

there exists a smooth mapping ν : I −→H20 such that (β,ν) is a framed immersion curve.

Let (β,ν) : I −→ H20 ×H
2
0 be a framed curve. If β is singular at t0, we can’t define a frame in a

traditional way. However, ν always exists even if t is a singular point of β. We take µ = ν ∧β. We
call the pair {β,ν,µ} is a moving frame of β and the hyperbolic Frenet-Serret matrix is given by


β′(t)

ν′(t)

µ′(t)


 =



0 0 m(t)

0 0 n(t)

m(t) −n(t) 0





β(t)

ν(t)

µ(t)


 ,

where n(t) = 〈ν′(t),µ(t)〉, ν(t) and µ(t) are both unit spacelike vectors. We declare that (β,−ν)
is also a framed curve. In this case, n(t) dose not change, but m(t) changes to −m(t). If (β,ν) is
a framed immersion, we have (m(t),n(t)) 6= (0,0) for each t ∈ I and call the pair (m,n) geodesic
curvature of the framed curve (for more details see [8,10]).

In what follows, some important properties of the meant curves are introduced.



Int. J. Anal. Appl. (2022), 20:47 11

Proposition 4.1. Let β be a regular curve and (β,ν) : I −→ H20 ×H
2
0 be its framed curve, then the

relationship between their geodesic curvatures is given by:

κg(t)=
n(t)

|m(t)|
. (4.1)

Proof. Direct calculations lead to

β′(t)= m(t)µ(t),

β′′(t)= m′(t)µ(t)+m(t)µ′(t)

= m′(t)µ(t)+m(t)(m(t)β −n(t)ν(t)),

where

κg(t)=
det(β(t),β′(t),β′′(t))

‖β′(t)‖3

=
det
(
β(t),mµ(t),m2β(t)−mnν(t)+m′µ(t)

)
|m3|

,

then, we get

κg(t)=
n(t)

|m(t)|
.

For the framed immersion (β,ν), we say that t0 is an inflection point of the front β if n(t0)=0. And

then the condition m(t0) 6=0 and n(t0)=0, is equivalent to the condition κg(t0)=0. �

If β is not a regular curve, then we can not define the evolute as in Eq. (2.8). Since the geodesic

curvature may be divergence at a singular point. Then, we can give the definition of the evolute of a

front and focus on its properties.

The notion of a parallel curve of β can be presented as follows:

Let (β,ν) : I −→ H20 ×H
2
0 be a framed curve which has the geodesic curvature (m,n). Then, we

define a parallel curve βλ : I −→H20 of β as

βλ(t)=
1√
|λ2 −1|

(β(t)+λν(t)) , (4.2)

where λ ∈R and λ 6=±1.

Proposition 4.2. If βλ is a regular curve, then

κgλ(t)=
n(t)+λm(t)

|m(t)+λn(t)|
. (4.3)

Proof. From Eq.(4.2), we have

β′λ(t)=
m+λn√
|λ2 −1|

µ(t),

β′′λ(t)=
m2 +λmn√
|λ2 −1|

β(t)+
−mn−λn2√
|λ2 −1|

ν(t)+
m′ +λn′√
|λ2 −1|

µ(t),



12 Int. J. Anal. Appl. (2022), 20:47

which implies that

βλ ∧β′λ =
1

(λ2 −1)

∣∣∣∣∣∣∣∣
−β(t) ν(t) µ(t)
1 λ 0

0 0 m+λn

∣∣∣∣∣∣∣∣
=
−λ(m+λn)β(t)− (m+λn)ν(t)

(λ2 −1)
.

Since

κgλ(t)=
det
(
βλ(t),β

′
λ(t),β

′′
λ(t)

)
‖β′
λ
(t)‖3

,

then, we get

|m(t)+λn(t)|κgλ(t)= (n(t)+λm(t)),

which completes the proof. �

Proposition 4.3. For a framed immersion curve (β,ν) : I −→H20×H
2
0, the parallel curve βλ : I −→H

2
0

is a front for each λ 6=±1.

Proof. We take νλ : I −→H20 as follows:

νλ(t)=
1√
|λ2 −1|

(λβ(t)+ν(t)) ,

since 

βλ(t)=

1√
|λ2 −1|

(β(t)+λν(t)) ,

β′λ(t)=
1√
|λ2 −1|

(β′(t)+λν′(t)) .

If β′λ(t0)=0 at a point t0 ∈ I, then we have

β′(t0)+λν
′(t0)=0.

If ν′(t0)= 0, then β′(t0)= 0. It is contradicted with the fact that (β,ν) is a framed immersion and

hence (βλ,νλ) is a framed immersion. By ‖ν(t)‖=−1, we have 〈ν(t),ν′(t)〉=0. Then

〈β′λ(t),νλ(t)〉=
1

(λ2 −1)
〈β′ +λν′,λβ +ν〉=0,

and from this, the curve βλ is a front. �

Proposition 4.4. Let (β,ν) be a framed curve. If β is a regular curve and λ 6=1/κg, then a parallel
curve βλ is also a regular curve and its evolute is given by

Eβλ(t)= Eβ(t). (4.4)



Int. J. Anal. Appl. (2022), 20:47 13

Proof. Since

βλ(t)=
1√
|λ2 −1|

(β(t)+λE(t)) ,

β′λ(t)=
‖β′‖√
|λ2 −1|

(1−λκg)T(t),

β′′λ(t)=
‖β′‖√
|λ2 −1|

(
‖β′‖(1−λκg)β(t)−λκ′gT(t)+‖β

′‖κg(1−λκg)E(t)
)
.

By the assumption λ 6=1/κg, βλ is a regular curve. By direct calculations, we obtain

βλ ∧β′λ =
1

(λ2 −1)

∣∣∣∣∣∣∣∣
−β(t) T(t) E(t)
1 0 λ

0 ‖β′‖(1−λκg) 0

∣∣∣∣∣∣∣∣
=
‖β′‖λ(1−λκg)β(t)+‖β′‖(1−λκg)E(t)

(λ2 −1)
,

and hence

〈βλ ∧β′λ,β
′′
λ〉=

−‖β′‖3λ(1−λκg)2 +‖β′‖3κg(1−λκg)2

|λ2 −1|
3
2

,

then, we get

κgλ(t)=
κg −λ
|1−λκg|

,

and

Tλ(t)=
β′λ
‖β′
λ
‖
=
1−λκg
|1−λκg|

T(t),

where Eλ = βλ ∧Tλ, we have

Eλ(t)=
1−λκg
|1−λκg|

1√
|λ2 −1|

(E(t)+λβ(t)) .

From (2.8) we get

Eβλ(t)=
1√∣∣∣κ2gλ(t)−1∣∣∣

(
κgλ(t)βλ(t)+Eλ(t)

)

=
1√√√√∣∣∣∣∣

(
κg −λ
|1−λκg|

)2
−1

∣∣∣∣∣
(

κg −λ
|1−λκg|

(β(t)+λE(t))
√
λ2 −1

+
1−λκg
|1−λκg|

(E(t)+λβ(t))√
|λ2 −1|

)

=
1√√√√∣∣∣∣∣

(
κg −λ
|1−λκg|

)2
−1

∣∣∣∣∣
1

|1−λκg|
√
|λ2 −1|

(
(1−λ2)κgβ(t)+(1−λ2)E(t)

)

=
1√∣∣κ2g(1−λ2)− (1−λ2)∣∣

(1−λ2)√
|λ2 −1|

(κgβ(t)+E(t))



14 Int. J. Anal. Appl. (2022), 20:47

=
1√∣∣κ2g(t)−1∣∣ (κg(t)β(t)+E(t))

= Eβ(t).

In the light of the above calculations, the proof is completed. �

Definition 4.3. Let (β,ν) : I −→ H20 ×H
2
0 be a framed immersion curve. We define an evolute

Eβ : I −→H20 of β as follows:
If t is a regular point, then we find

Eβ(t)=
1√∣∣κ2g(t)−1∣∣ (κg(t)β(t)+E(t)) . (4.5)

If t0 is a singular point, for any t ∈ (t0 −δ,t0 +δ), we get

Eβλ(t)=
1√∣∣∣κ2gλ(t)−1∣∣∣

(
κgλ(t)βλ(t)+Eλ(t)

)
, (4.6)

where δ is a sufficiently small positive real number and λ ∈R satisfies the condition λ 6=1/κg(t).

Now, we give another representation of the evolute by using the moving frame {β(t),ν(t),µ(t)}
and its geodesic curvature {m(t),n(t)}.

Theorem 4.1. Under the condition of |m(t)| 6= |n(t)|, the evolute of a front curve Eβ(t) : I −→ H20
is represented by

Eβ(t)=
1√

|n2(t)−m2(t)|
(n(t)β(t)−m(t)ν(t)) , (4.7)

and Eβ(t) is a front curve.

Proof. (i) Suppose that β is a regular curve. Since β′(t)= m(t)µ(t), we have |m(t)| 6=0 and

T(t)=
m(t)

|m(t)|
µ(t), E(t)=−

m(t)

|m(t)|
ν(t).

From Eqs.(4.1) and (4.5), we get

Eβ(t)=
1√∣∣κ2g(t)−1∣∣ (κg(t)β(t)+E(t))

=
1√√√√∣∣∣∣∣

(
n(t)

|m(t)|

)2
−1

∣∣∣∣∣
(
n(t)

|m(t)|
β(t)−

m(t)

|m(t)|
ν(t)

)

=
1√

|n2(t)−m2(t)|
(n(t)β(t)−m(t)ν(t)) .



Int. J. Anal. Appl. (2022), 20:47 15

(ii) Suppose that t0 is a singular point of β and consider βλ in a hyperbolic sphere, also we know that

βλ is a regular curve around the neighbourhood of t0 with λ 6=1/κg(t).
From Eq.(4.2), so we get

β′λ(t)=
m+λn√
|λ2 −1|

µ(t),

then, |m+λn| 6=0 and

Tλ(t)=
m+λn

|m+λn|
µ(t),

where Eλ(t)= βλ(t)∧Tλ(t), we find

Eλ(t)=
m+λn

|m+λn|
1√
|λ2 −1|

(−λβ(t)−ν(t)) .

Therefore, from Eq.(4.3), we obtain

κgλ =
(n+λm)

|m+λn|
,

and from Eqs.(4.4) and (4.6) we have

Eβ(t)= Eβλ(t)

=
1√∣∣∣κ2gλ(t)−1∣∣∣

(
κgλ(t)βλ(t)+Eλ(t)

)

=
1√√√√∣∣∣∣∣

(
(n+λm)

|m+λn|

)2
−1

∣∣∣∣∣
((
(n+λm)

|m+λn|

)
(β(t)+λν(t))
√
λ2 −1

+
m+λn

|m+λn|
(−λβ(t)−ν(t))√

|λ2 −1|

)

=
1√

|(n+λm)2 − (m+λn)2|

(
nβ(t)+λ2mν(t)−λ2nβ(t)−mν(t)

)√
|λ2 −1|

=
1√

|(1−λ2)(n2 −m2)|
(1−λ2)(nβ(t)−mν(t))√

|λ2 −1|

=
1√

|n2 −m2|
(nβ(t)−mν(t)) .

If we take ν̃(t)= µ(t), then (Eβ(t), ν̃(t)) is a framed immersion. In fact, we have

E′β(t)=
mm′ −nn′

(n2 −m2)
3
2

(nβ(t)−mν(t))+
1

√
n2 −m2

(
n′β(t)−m′ν(t)

)
=
m′n−mn′

(n2 −m2)
3
2

(mβ(t)−nν(t))

=
d

dt

(m
n

) n2
(n2 −m2)

3
2

(mβ(t)−nν(t)) ,

therefore, we find 〈Eβ(t), ν̃(t)〉 = 〈E′β(t), ν̃(t)〉 = 0. And (β,ν) is a framed immersion satisfying
(m(t),n(t)) 6= (0,0). Since, ν̃(t) = µ(t), we get ν̃′(t) = mβ(t)−nν(t) 6= 0. It follows that Eβ(t)
is a front and thus, the proof is completed. �



16 Int. J. Anal. Appl. (2022), 20:47

5. Computational examples

Example 5.1. Let γ : I −→ S21 be a regular curve given by

γ(t)=
(
sinh(t3),cos(t2)cosh(t3),sin(t2)cosh(t3)

)
,

then, we get

γ′(t)=
(
3t2cosh(t3),−2t sin(t2)cosh(t3)+3t2cos(t2)sinh(t3),2t cos(t2)cosh(t3)+3t2 sin(t2)sinh(t3)

)
.

Since, t =0 is a singular point on γ. If we take ν =(ν1,ν2,ν3), where

ν1 =

1
P
(
2cosh2(t3)

)
ν2 =

1
P
(
2sinh(t3)cos(t2)cosh(t3)−3t sin(t2)

)
ν3 =

1
P
(
2sin(t2)sinh(t3)cosh(t3)+3t cos(t2)

)
,

(5.1)

and P =
√∣∣9t2 −4cosh2(t3)∣∣, then we have 〈γ(t),ν(t)〉 = 〈γ′(t),ν(t)〉 = 0 and 〈ν(t),ν(t)〉 = 1.

Hence, (γ,ν) is a framed curve. Also, from the relation µ = ν ∧γ, we get (see Fig.(1a)).

µ(t)=
1

P

∣∣∣∣∣∣∣∣
−i j k

2cosh2(t3) 2sinh(t3)cos(t2)cosh(t3)−3t sin(t2) 2sin(t2)sinh(t3)cosh(t3)+3t cos(t2)
sinh(t3) cos(t2)cosh(t3) sin(t2)cosh(t3)

∣∣∣∣∣∣∣∣ ,
and we get

µ =
1

P
(
3t cosh(t3),3t cos(t2)sinh(t3)−2sin(t2)cosh(t3),3t sin(t2)sinh(t3)+2cos(t2)cosh(t3)

)
.

(5.2)

From the fact that 〈µ(t),µ(t)〉=1, one can obtain

m(t)= ‖γ′(t)‖= t
√∣∣9t2 −4cosh2(t3)∣∣. (5.3)

Also, from Eq.(5.1) we have

ν′(t)=
−(9t −4cosh(t3)sinh(t3))∣∣9t2 −4cosh2(t3)∣∣32 (2cosh2(t3),2sinh(t3)cos(t2)cosh(t3)−3t sin(t2),
2sin(t2)sinh(t3)cosh(t3)+3t cos(t2))

+
1√∣∣9t2 −4cosh2(t3)∣∣(12t2cosh(t3)sinh(t3),6t2cos(t2)(cosh2(t3)+sinh2(t3))

−4t sinh(t3)sin(t2)cosh(t3)−3sin(t2)−6t2cos(t2),6t2 sin(t2)(cosh2(t3)+sinh2(t3))

+4t sinh(t3)cos(t2)cosh(t3)−3cos(t2)−6t2 sin(t2)),



Int. J. Anal. Appl. (2022), 20:47 17

which leads to

n(t)= 〈ν′(t),µ(t)〉

=
2(−18t3 sinh(t3)+4t sinh(t3)cosh(t3)+3cosh(t3))

9t2 −4cosh2(t3)
,

and we have (m(0),n(0)) 6=(0,0), thus, γ is a frontal curve.
On the other hand, the evolute curve of γ is given as

Eγ(t)= (Eγ1,Eγ2,Eγ3) ,

where

Eγ1(t)=
1

P
√
|4F2 − t2P4|

(
2F sinh(t3)−2tP2cosh2(t3)

)
,

Eγ2(t)=
1

P
√
|4F2 − t2P4|

(
2F cos(t2)cosh(t3)−2tP22sinh(t3)cos(t2)cosh(t3)−3t2P2 sin(t2)

)
,

Eγ3(t)=
1

P
√
|4F2 − t2P4|

(
2F sin(t2)cosh(t3)− tP2 sin(t2)sinh(t3)cosh(t3)−3t2P2cos(t2)

)
,

keep in mind that F =−18t3 sinh(t3)+4t sinh(t3)cosh(t3)+3cosh(t3).

(a) (b)

Figure 1. (A) The curve γ(t) with singular point at t = 0. (B) The curve ν(t) =

(ν1,ν2,ν3).

Example 5.2. Consider the de Sitter asteroid curve β : I −→ S21, β(t)= (β1,β2,β3) expressed as [8]

β1 =

√
|cos6(t)+sin6(t)−1|

β2 =cos
3(t)

β3 =sin
3(t),

(5.4)



18 Int. J. Anal. Appl. (2022), 20:47

then, we get

β′(t)=3sin(t)cos(t)

(
sin4(t)−cos4(t)√
|cos6(t)+sin6(t)−1|

,−cos(t),sin(t)

)
.

It is obvious that β is singular at t =0, π/2, π and 3π/2.

If we take ν =(ν1,ν2,ν3), where

ν1 =

1
Q

(
sin(t)cos(t)

√
|cos6(t)+sin6(t)−1|

)
ν2 =

1
Q
(
sin(t)

(
cos4(t)−1

))
ν3 =

1
Q
(
cos(t)

(
sin4(t)−1

))
,

(5.5)

with the knowledge that Q=
√∣∣1− sin2(t)cos2(t)∣∣,

then by a straightforward calculations, we have 〈β(t),ν(t)〉= 〈β′(t),ν(t)〉=0 and 〈ν(t),ν(t)〉=1.
Hence, (β,ν) is a framed curve. From the equation µ = ν ∧β, we have (see Fig.(2a) and Fig.(2b)).

µ =
1

Q

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

−i j k

sin(t)cos(t)

(|cos6(t)+sin6(t)−1|)−
1
2

sin(t)
(
cos4(t)−1

)
cos(t)

(
sin4(t)−1

)
√
|cos6(t)+sin6(t)−1| cos3(t) sin3(t)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣
,

which gives

µ(t)=

√
|cos6(t)+sin6(t)−1|√∣∣1− sin2(t)cos2(t)∣∣

(
sin4(t)−cos4(t)√
|cos6(t)+sin6(t)−1|

,−cos(t),sin(t)

)
. (5.6)

From the previous equation, we have 〈µ(t),µ(t)〉=1. Then, we obtain

m(t)= ‖β′(t)‖=3sin(t)cos(t)

√
cos6(t)+sin6(t)

|cos6(t)+sin6(t)−1|
. (5.7)

Also, from Eq.(5.1) we have

ν′(t)=
sin(t)cos(t)(cos2(t)− sin2(t))∣∣1− sin2(t)cos2(t)∣∣32 (sin(t)cos(t)

√
|cos6(t)+sin6(t)−1|,

sin(t)(cos4(t)−1),cos(t)(sin4(t)−1))

+
1√∣∣1− sin2(t)cos2(t)∣∣(

√
|cos6(t)+sin6(t)−1|(cos2(t)− sin2(t)

+
3sin2(t)cos2(t)(sin4(t)−cos4(t))

cos6(t)+sin6(t)−1
),cos3(t)(cos2(t)−4sin2(t))

−cos(t),−sin3(t)(sin2(t)−4cos2(t))+sin(t)).



Int. J. Anal. Appl. (2022), 20:47 19

Hence, we get

n(t)= 〈ν′(t),µ(t)〉

=

√
|cos6(t)+sin6(t)−1|
1− sin2(t)cos2(t)

(
3sin2(t)cos2(t)

(
1−

(cos2(t)− sin2(t))2

cos6(t)+sin6(t)−1

)
+1

)
. (5.8)

In the light of the above calculations, we have (m(0),n(0)) 6=(0,0), thus, β is a frontal curve.
Also, from Eqs.(3.7), (5.4), (5.5), (5.7), and (5.8), we get the evolute curve of β as

Eβ(t)=
(
Eβ1,Eβ2,Eβ3

)
, where

Eβ1(t)=
−Qn+m√
|n2 −m2|

(√
|cos6(t)+sin6(t)−1|(1+sin(t)cos(t))

)
,

Eβ2(t)=
−Qn+m√
|n2 −m2|

(
cos3(t)(1+sin(t)cos(t))− sin(t)

)
,

Eβ3(t)=
−Qn+m√
|n2 −m2|

(
sin3(t)(1+sin(t)cos(t))−cos(t)

)
.

(a) (b)

Figure 2. (A) The de Sitter astroid curve β. (B) The curve ν(t)= (ν1,ν2,ν3).

6. Conclusion

In 2-dimensional de Sitter and hyperbolic spaces, some types of curves such as framed curves,

framed immersion curves, frontal curves and front curves are studied. Also, the evolutes and some

of their properties of fronts at singular points under some conditions are investigated. Finally, two

computational examples in support of our main results are given and plotted.

Acknowledgment: The researchers wish to extend their sincere gratitude to the Deanship of Scientific

Research at the Islamic University of Madinah for the support provided to the Post-Publishing Program

1.



20 Int. J. Anal. Appl. (2022), 20:47

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

References

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[10] X. Cui, D. Pei, H. Yu, Evolutes of Null Torus Fronts, J. Nonlinear Sci. Appl. 8 (2015), 866-876.

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https://doi.org/10.1007/s00022-013-0162-6
https://doi.org/10.5427/jsing.2014.10f
https://doi.org/10.5427/jsing.2014.10f
https://doi.org/10.1016/j.jmaa.2015.12.029

	1. Introduction
	2. Geometric meanings and basis concepts
	3. Evolutes of fronts in de Sitter sphere S12 
	4. Evolutes of fronts in hyperbolic sphere H02 
	5. Computational examples
	6. Conclusion
	References