Abstract
We have generalized the involute and evolute curves of the pseudonull curves in ; that is, is a spacelike curve with a null principal normal. Firstly, we have shown that there is no involute of the pseudonull curves in . Secondly, we have found relationships between the evolute curve β and the pseudonull curve in . Finally, some examples concerning these relations are given.
1. Introduction
The general theory of curves in an Euclidean space (or more generally in a Riemannian manifold) was developed a long time ago, so now, we have a deep knowledge of its local geometry as well as its global geometry. In the theory of curves in Euclidean space, one of the important and interesting problems is the characterizations of a regular curve. And, in particular, the involute-evolute of a given curve is a well-known concept in the classical differential geometry (see [1]).
At the beginning of the twentieth century, A. Einstein’s theory opened a door for use of new geometries. One of them is simultaneously the geometry of special relativity and the geometry induced on each fixed tangent space of an arbitrary Lorentzian manifold and was introduced and some of classical differential geometry topics have been treated by the researchers. According to reference (see [2, 3]), the involute and evolute curves of the spacelike curve in with a spacelike binormal or a spacelike principal normal in Minkowski -space have been investigated. Also, Bükcü and Karacan studied the involute and evolute curves of the timelike curve in Minkowski 3-space [4].
In this paper, we have generalized the involute and evolute curves of the pseudonull curves in ; that is, is a spacelike curve with a null principal normal. Firstly, we have shown that there is no involute of the pseudonull curves in . Secondly, we have found relationships between the evolute curve and the pseudonull curve in . Finally, some examples concerning these relations are given.
2. Preliminaries
The Minkowski -space is the Euclidean -space provided with the standard flat metric given by where is a rectangular coordinate system of . Since is an indefinite metric, recall that a vector can have one of three Lorentzian causal characters: it can be spacelike if or , timelike if , and null (lightlike) if and . In particular, the norm (length) of a nonnull vector is given , and two vectors and are said to be orthogonal, if . A lightlike vector is said to be positive (resp., negative) if and only if and a timelike vector is said to be positive (resp., negative) if and only if .
Next, recall that an arbitrary curve in can locally be spacelike, timelike, or null (lightlike), if all of its velocity vectors are, respectively, spacelike, timelike, or null (lightlike). If , the nonnull curve is said to be of unit speed (or parameterized by arc length function ).
We denote by the moving Frenet frame along the curve . Then , and are the tangent, the principal normal, and the binormal vector of the curve , respectively. Depending on the causal character of the curve , we have the following Frenet-Serret formulas.
If is a null space curve with a spacelike principal normal , then the following Frenet formulas hold where For a null curve, can take only two values: when is a null straight line or in all other cases.
If is a pseudonull curve, that is, is a spacelike curve with a null principal normal , then the following Frenet formulas hold: where For a pseudonull curve, can take only two values: when is a straight line or in all other cases.
If is a spacelike space curve with a spacelike principal normal , then the following Frenet formulas hold where
If is a spacelike space curve with a timelike principal normal , then the following Frenet formulas hold where
If is a timelike space curve, then the following Frenet formulas hold where see [5].
If the curve is nonunit speed, then If the curve is unit speed, then see [6].
In this study we are going to have the curve as nonline pseudonull curve or .
3. The Involute of the Pseudonull Curve
Definition 1. Let a pseudonull curve and a curve be given. For all , then the curve is called the involute of the curve , if the tangent vector of the curve at the point passes through the tangent at the point of the curve and where and are Frenet frames of and , respectively.
This definition suffices to define this curve mate as (see Figure 1).
Theorem 2. Let be a pseudonull space curve with curvature functions and . Then, there is no involute of the curve in Minkowski -space .
Proof. Let be involute of in . We assume that is distinct from . Then we can write
where is a -function on . Differentiating (17) and by using Frenet formulas given in (4), we get
If we take the inner product with on both sides of (18), we have
Recalling definition of the involute curve couple, and, by using (5), we get
for all . Thus, since , we get ; that is, is a nonzero constant. In this case, we can write (17) as follows:
and from (18) we get
Here we notice that
and by using (5) we get
Thus, since , we get that is a null vector; that is, curve is a null curve.
Differentiating (24) with respect to and by using Frenet formulas given in (4), we have
and by using (5) we get
Since for all , we get
Thus, we obtain that is a null vector. This is a contradiction with null curve . Then, there is no involute of the curve in Minkowski -space .
4. The Evolute of the Pseudonull Curve
Definition 3. Let the pseudonull curve and a curve with the same interval be given. For all , then the curve is called the evolute of the curve , if the tangent vector of the curve at the point passes through the tangent at the point of the curve and where and are Frenet frames and curvatures of and , respectively.
This definition suffices to define this curve mate as
Theorem 4. Let a space curve be the evolute of the pseudonull curve ; then for all where and are Frenet frames and curvatures of and , respectively.
Proof. The tangent of the curve at the point is the line constructed by the vector . Then we can write where and are a -function on . Differentiating (31) and by using Frenet formulas given in (4), we get and, from , If we take the inner product with on both sides of (33), we have Recalling definition of the evolute curve couple, , and by using (5), we get for all . In this case, we can write (31) as follows: and from (35) we get
Let and be Frenet frames and curvatures of and , respectively. If the space curve is the evolute of the pseudonull curve , then, from Theorem 4, we can write where is a -function on .
Since this line passes through the point , the vector is perpendicular to the vector . From (36) and (37), the vector field is parallel to the vector field . Then, we have
4.1. Special Cases
If for all , then (39) is not defined. But, from (36) and (37), we get Here we notice that and by using (5) we get Thus, since , we get that is a null vector; that is, the curve is a null curve. Differentiating (41) and by using Frenet formulas given in (4), we have and from (5) we get Since the space curve is a null curve, then from and we get
Case 1. If from (46), then we can write (41) and (44) as follows:
Differentiating (48) and by using Frenet formulas given in (2) and (4), we get
By using Frenet formulas given in (2), we get
Conclusion 1. Let a space curve be the evolute of the pseudo null curve in and let and are the curvatures of the curve . If the torsion of the curve is
for all , then and are helices and the curvatures of the curve are
for all .
Case 2. If from (46), then we can write (41) and (44) as follows:
Differentiating (54) and by using Frenet formulas given in (2) and (4), we get
By using Frenet formulas given in (2), we get
Conclusion 2. Let a space curve be the evolute of the pseudo null curve in and let and are the curvatures of the curve . If the torsion of the curve is
for all , then and are helices and the curvatures of the curve are
for all .
If , then from (39) we get and if for all , then the curve is not defined.
4.2. Special Cases
If , then from (39) we get From (4), (5), and (37), we get and by using (60) we get
Case 3. If is a unit spacelike curve, then from (62) we have Solving the differential equation (63), we get where and are constant. Then, from (36) we can write
Case 4. If is a unit timelike curve, then from (62) we have Solving the differential equation (65), we get where and are constant. Then, from (36) we can write
Case 5. If is a null curve, then from (62) we have Solving of the differential equation (65), we get where is constant. This is a contraction with . Then, the curve is not defined.
5. Example
Let be a unit speed pseudonull curve in the Minkowski -space with parameter equation (see Figure 2)
Then we have the Frenet frame vectors as follows: where is spacelike vector and and are null vectors for all . Moreover the curvatures and of are as follows: Let the space curve with the same interval be the evolute of the pseudonull curve . Then we can write
(1) From (40), the curve is given by (see Figure 3)
(2) From (65), the curve is given by (see Figure 4)
(3) From (68), the curve is given by (see Figure 5)
Acknowledgment
The authors would like to thank the referee for his/her helpful suggestions and comments which significantly improved the first version of the paper.