1. Introduction
In the theory of curves, one of the important and interesting problems is the characterization of regular curves, in particular, the involute–evolute of a given curve. Evolutes and involutes (also known as evolvents) were studied by C. Huygens [
1]. According to D. Fuchs [
2], an involute of a given curve is a curve to which all tangents of the given curve are normal. He also defined the equation for an enveloping curve of the family of normal planes for a space curve. Suleyman and Seyda [
3] determined the concept of parallel curves, which means that if the evolute exists, then the evolute of the parallel arc will also exist and the involute will coincide with the evolute. Brewster and David [
4] stated that a curve is composed of two arcs with a common evolute, and the common evolute of two arcs must be a curve with only one tangent in each direction. In general, the evolute of a regular curve has singularities, and these points correspond to vertices. Emin and Suha [
5] determined that an evolute Frenet apparatus can be formed by an involute apparatus in four dimensional Euclidean space, so, in this way, another orthonormal of the same space can be obtained. Shyuichi Izumiya [
6] defined evolutes as the loci of singularities of space-like parallels and geometric properties of non-singular space-like hyper surfaces corresponding to the singularities of space-like parallels or evolutes. Takami Sato [
7] investigated the singularities and geometric properties of pseudo-spherical evolutes of curves on a space-like surface in three-dimensional Minkowski-space. Marcos Craizer [
8] stated that the iteration of involutes generates a pair of sequences of curves with respect to the Minkowski metric and its dual.
According to Boaventura Nolasco and Rui Pacheco [
9], correspondence between plane curves and null curves in Minkowski three-space exists. He also described the geometry of null curves in terms of the curvature of the corresponding plane curves. M. Turgut and S. Yilmaz [
10] obtained the Frenet apparatus of a given curve by defining the space-like involute–evolute curve couple in Minkowski space-time. Some researchers have investigated evolute curves and their characterization in Minkowski space [
11,
12,
13,
14,
15,
16] as well as in Euclidean space. Many researchers have dealt with evolute–involute curves, but no research has been carried out on the Cartan null curve. In this study, a special kind of generalized evolute and involute curve is considered in four-dimensional Minkowski space. We obtained necessary and sufficient conditions for the curve possessing a generalized evolute as well as an involute.
2. Preliminaries
Consider the Minkowski space-time, , where and . For any and . We denote . Let I be an open interval in R and be a regular curve in that is parameterized by the arc length parameter, s, and is the moving Frenet frame along , consisting of the tangent vector, T; the principal normal vector, N; the first binormal vector, , and the second binormal vector, , respectively, so that coincides with the standard orientation of . Then,
In particular, the following conditions hold: .
In accordance with reference [
17], the Frenet–Serret formula for
in
is given by
We introduce some methodologies in this paper. At any point of , the plane spanned by is called the (0,2)-tangent plane of . The plane spanned by is called the (1,3)-normal plane of .
Let and be two regular curves in , where s is the arc-length parameter of . Denote to be the arc-length parameters of . For any , if the (0,2)-tangent plane of at coincides with the (1,3)-normal plane at of , then is called the (0,2)-involute curve of in and is called the (1,3)-evolute curve of in .
An arbitrary curve,
in
, can locally be space-like, time-like, or null (light-like) if all of its velocity vectors,
, are respectively space-like, time-like, or null [
18]. A null curve,
, is parametrized by the pseudo-arc
s if
[
19]. On the other hand, a nonnull curve,
, is parametrized by the arc-length parameter,
s, if
. In accordance with references [
19,
20], if
is null Cartan curve, the Cartan Frenet frame is given by
where
if
is a null straight line or
in all other cases. In this case, the next conditions hold:
,
,
.
3. The (0,2)-Involute Curve of a Given Curve in
In this section, we proceed to study the existence and expression of the (0,2)-involute curve of a given curve in .
Theorem 1. Let be a regular curve parameterized by arc-length s so that , and are not zero. If α possesses the (0,2)-involute mate curve, , with , then , and satisfywhere and are given constants. Moreover, the three curvatures of are given bywhere . The associated Frenet frame are given by Proof. Let be a regular curve with arc-length parameter s so that , and are not zero. Suppose that is the (0,2)-involute curve of . is the Frenet frame along and , and are the curvatures of . Then
span , span .
Moreover,
can be expressed as
where
and
are
functions on
I.
By differentiating
with respect to
s and using the Frenet formula (
1), we get
Taking the inner product on both sides of (4) with
T and
, respectively, we get
and
, which implies that
is constant and
, where
is the integration constant. So, (
4) turns into
If we denote
then (
5) turns into
Case 1:. In this case,
.
implies that
and
By differentiating (
7) with respect to
s and using the Frenet formula (
1), we get
By taking the inner product from both sides of (
9) with
N and
, respectively, we get
and
, which implies that
and
are constants. So, (
9) turns into
Denote
then (
10) turns into
implies that
and
From Equations (
8) and (
13), we have
implies that
. From (
11), we get
By differentiating (
12) with respect to
s using the Frenet formula (
1), we get
By taking inner product on both side of (
16) by
T and
respectively, we get
and
, which implies that
f and
g are constants. In this case, (
16) turns into
By substituting (
7) and (
15) into (
17), we get
From (
18), we may choose that
By differentiating (
19) about
s and using the Frenet formula (
1), we get
from which we obtain
From (
21), we may choose that
From Equations (
14), (
15), (
18) and (
22), we can easily acquire our theorem. ☐
Case 2: If , we have the following theorem.
Theorem 2. Let be a regular curve with arc-length parameter s so that , and are not zero. If α possesses the (0,2)-involute mate curve , then and satisfywhere , f, and g are given constants. Moreover, the three curvatures of
are given by
The associated Frenet frames are given by
In this case, (
4) turns into
By differentiating (
24) with respect to
s and using the Frenet Formula (
1), we get
from which we may assume that
By differentiating the second equation of (
26) about
s and using the Frenet Formula (
1), we get
By differentiating (
27) about
s, we obtain that
f and
g are constants:
By differentiating (
30) about
s, we obtain
From Equations (
27), (
30) and (
33), we have achieved the desired theorem.
Remark 1. Theorems 1 and 2 are quite different.
4. The (1,3)-Evolute Curve of a Given Curve in
In this section, we want to study the (1,3)-evolute curve of a given curve in .
Theorem 3. Let be a regular curve with arc length parameter s so that , and are not zero, If α possesses the (1,3)-evolute mate curve, , then , and satisfy , where i and j are given constants. Three curvatures of are given byThe associated Frenet frames are given by Proof. Let
be a regular curve with arc-length parameter
s so that
,
and
are not zero. Let
be the (1,3)-evolute curve of
.
is the Frenet frame along
and
,
and
are the curvatures of
. Then,
Moreover,
can be expressed as
where
and
are
functions on
I.
Differentiating (
35) with respect to
s using Frenet Formula (
1), we get
By taking the inner product from both sides of (
36) with
T and
, respectively, we get
Denote
then (
37) turns into
By differentiating (
39) with respect to
s and using the Frenet formula (
1), we get
By taking inner product on both sides of (
40) with
N and
respectively, we get
and
, which implies that
i and
j are constants.
Moreover, (
40) turns into
Denote
then (
42) turns into
Case 1:. By differentiating (
44) about
s and using the Frenet Formula (
1), we get
By taking inner product on both sides of (
46) with
T and
respectively, we get
and
, which implies that
r and
t are constants. In this case, (
46) turns into
Denote
then (
47) turns into
Since
, it follows from (
40) and (
50) that
, which implies that
From (
45) and (
50), we can see that
Since
, it follows from (
51) that
. Hence, (
49) turns into
By differentiating (
52) about
s using (
1), we get
from which we obtain
It follows from (
54) that
From (
43), (
52) and (
55), we can easily acquire our desired theorem. ☐
Case 2: If , we have the following theorem.
Theorem 4. Let be a regular curve parameterized by arc-length s so that , and are not zero. If α possesses the (1,3)-evolute mate curve, , then and satisfy , where i and j are given constants. Moreover, the three curvatures of are given by The associated Frenet frames are given by Proof. For this case, we may suppose that
From (
41) and the third equation of (
58), we acquire
By differentiating (
58) about
s and using (
1), we get
It follows that we may choose
By differentiating (
61) about
s using the Frenet Formula (
1) and third equation of (
58), we get
From (
58), (
61) and (
62), we can easily acquire our desired theorem. ☐
Remark 2. Theorems 3 and 4 are quite different.
5. The (1,3)-Evolute Curve of a Cartan Null Curve in
In this section, we proceed to study the existence and expression of the (1,3)-evolute curve of a given Cartan null curve in . At any point of , the plane spanned by is called the (1,3)-normal plane of .
Let and be two regular curves in , where s is the arc-length parameter of . Denote to be the arc-length parameters of . For any , if the (0,2)-tangent plane of at coincides with the (1,3)-normal plane at of , then is called the (0,2)-involute curve of in and is called the (1,3)-evolute curve of in .
Theorem 5. Let be a null Cartan curve with arc length parameter s so that , and are not zero, if α possesses the (1,3)-evolute mate curve, , then , and satisfy , where i and j are given constants. Three curvatures of are given byMoreover, the associated Frenet frames are given by Proof. Let
be a Cartan null curve parameterized by the pseudo-arc parameter
s with curvatures
, and
and
are not zero. Let
be the (1,3)-evolute curve of
. Denote
as the Frenet frame along
and
,
and
as the curvatures of
. Then
Moreover,
can be expressed as
where
and
are
functions on
I. By differentiating (
66) with respect to
s using the Frenet Formula (
2), we get
By taking the inner product on both sides of (
67) with
T and
, respectively, we get
Denote
then (
68) turns into
By differentiating (
70) with respect to
s and using the Frenet formula (
2), we get
By taking the inner product on both sides of (
71) with
N and
respectively, we get
and
which implies that
i and
j are constants. From (
69), we get
Moreover, (
71) turns into
Denote
then (
73) turns into
Case 1:. By differentiating (
75) about
s and using the Frenet Formula (
2), we get
By taking the inner product from both sides of (
77) with
T and
respectively, we get
and
, which implies that
r and
t are constants. In this case, (
77) turns into
Denote
then (
78) turns into
Since
, it follows from (
70) and (
80) that
, which implies that
From (
76) and (
81), we can see that
Since
, it follows from (
82) that
.
By differentiating (
83) about
s using (
2), we get
It follows from (
85) that
From (
74), (
83) and (
86), we easily acquire our desired theorem. ☐
Case 2: For , we have the following theorem.
Theorem 6. Let be a null Cartan curve with arc-length parameter s so that , and are not zero. If α possesses the (1,3)-evolute mate curve, , then and satisfy , where i and j are given constants. Moreover, the three curvatures of are given by The associated Frenet frames are given by Proof. For this case, we may suppose that
Moreover, from (
72) and the third equation of (
89), we get
By differentiating (
89) about
s and using (
2), we get
It follows that we can choose
By differentiating (
92) about
s using the Frenet Formula (
2) and third equation of (
89), we get
From (
89), (
92) and (
93), we can easily acquire our desired theorem. ☐
Remark 3. Theorems 5 and 6 are quite different.
Condition 2:
Theorem 7. Let be a null Cartan curve with arc length parameter s so that , and are not zero if α possesses the (1,3)-evolute mate curve, . Then, , and satisfy , where i and j are given constants. Three curvatures of are given, as follows: The associated Frenet Frame are given by Proof. Let
be a Cartan null curve parametrized by pseudo-arc parameter
s with curvatures
, and
and
are not zero. Let
be the (1,3)-evolute curve of
. Denote
as the Frenet frame along
and
,
and
as the curvatures of
. Then,
Moreover,
can be expressed as
where
and
are
functions on
I. By differentiating (
97) with respect to
s using the Frenet Formula (
2), we get
By taking the inner product from both sides of (
98) with
T and
respectively, we get
Denote
then (
99) turns into
By differentiating (
101) with respect to
s and using the Frenet formula (
2), we get
By taking the inner product from both sides of (
102) with
N and
, respectively, we get
and
which implies that
i and
j are constants. From (
100), we get
Moreover, (
102) turns into
Denote
then (
104) turns into
Case 1:
. By differentiating (
106) about
s and using the Frenet Formula (
2), we get
By taking the inner product on both sides of (
108) with
T and
, respectively, we get
and
, which implies that
r and
t are constants. In this case, (
108) turns into
Denote
then (
109) turns into
Since
, it follows from (
101) and (
111) that
, which implies that
Since
, it follows from (
113) that
. Hence, (
111) turns into
By differentiating (
114) about
s using (
2), we get
It follows from (
116) that
From (
106), (
114) and (
117), we can easily acquire our desired theorem. ☐
Case 2: For , we have the following theorem.
Theorem 8. Let be a null Cartan curve with arc-length parameter s so that , and are not zero. If α possesses the (1,3)-evolute mate curve , then and satisfy , where i and j are given constants. Moreover, the three curvatures of are given by The associated Frenet frames are given by Proof. In this case, we may suppose that
Moreover, from (
112) and the third equation of (
120), we get
By differentiating (
120) about
s and using (
2), we get
It follows that we can choose
By differentiating (
123) about
s using the Frenet Formula (
2) and using the third equation of (
120), we get
From (
120), (
123) and (
124), we can easily acquire our desired theorem. ☐
Remark 4. Theorems 7 and 8 are quite different.
6. Conclusions
This paper established new kinds of generalized evolute and involute curves in four-dimensional Minkowski space by providing the necessary and sufficient conditions for the curves possessing generalized evolute and involute curves. Furthermore, the study invoked a new type of (1,3)-evolute and (0,2)-evolute curve in four-dimensional Minkowski space. The study also provided a new kind of generalized null Cartan curve in four-dimensional Minkowski space. For this new type of curve, the study provided several theorems with necessary and sufficient conditions and obtained significant results. The understanding of evolute curves with this new type evolute curve in four-dimensional Minkowski space will be beneficial for researchers in future studies. The designing of a framework for the involutes of order k of a null Cartan curve in Minkowski spaces will be considered in future work.