1. Introduction
First, some
fundamental definitions and concepts related to the algebra of dual quaternions are given. A real quaternion
q has the form
Let
q and
be two real quaternions. A dual quaternion is defined as
If
then we can write
where
,
,
and
are dual numbers. The dual numbers
D,
A,
B and
C are called dual components of
[
1].
Hence, a quaternion
consists of two parts: the scalar part
and the vector part
. That is,
, where
is a dual number and
is a dual vector [
1]. From now on, we show the set of dual quaternions by
.
The sum of two dual quaternions
and
is defined as
where
and
[
1].
The product of two dual quaternions
and
is defined as
where the operation
x on the right hand side is the real quaternion multiplication, that is,
The conjugation of a dual quaternion
is shown by
and defined by
, [
1].
Let
be the set of dual numbers. The symmetric
valued bilinear form
is defined by
where
h is the inner product on real quaternions defined by
for all real quaternions
p and
q [
2,
3].
The norm of a dual quaternion
is defined by
where
is a dual number. If we show the real part by
and dual part by
, then these are:
If the norm of a dual quaternion is unit, that is the norm of the real part is one and the norm of the dual part is zero, then this is called unit dual quaternion and is shown as
. Furthermore, it can
be expressed as
where
If , then is called a dual spatial-quaternion. A dual spatial-quaternion may be considered as a dual vector in . Dual quaternions and are called -ortogonally if and only if .
Let
and
be two unit dual spatial-quaternions. If
and
are unit dual vectors, then we have
where
is the dual angle between
and
quaternions [
1].
In this study, a dual quaternion valued function of a single real variable is called a dual quaternionic curve. Let I be an open interval in
, then a dual quaternionic curve in
is in the form
Throughout the work, we assume that all curves are given with arc-length parameter. Let
be a spatial-quaternionic curve parameterized by arc-length. With
being the Frenet frame field along
, the Serret–Frenet Formulae of dual spatial-quaternionic curve
is given by
and
is the Frenet Apparatus of the curve
in
, where
and
are the principal curvature and torsion of
, respectively. Moreover,
k and
r are the principal curvature and torsion of the curve in
, which are determined by the real part of
, respectively [
4].
Theorem 1. Let the quaternionic curvebe derived from the dual spatial-quaternionic curve Then, the Serret–Frenet Formulas for the curve in can be derived in the terms of help of the Serret–Frenet vectors of β so we havewhere is the Frenet Apparatus for the curve such that K and R are principal curvature and torsion of the dual spatial-quaternionic curve β, respectively [5]. 2. Dual Spatial-Quaternionic Helixes and Harmonic Curvatures
Definition 1. Let be a spatial-quaternionic curve that is parameterized by arc-length s and u be a constant unit dual vector in . Ifthen β is called a dual spatial-quaternionic helix in . Let be a constant unit dual vector in , be the Frenet frame field along the spatial-quaternionic curveand , . In addition, the conjugations of T and dual spatial-quaternions are and , respectively. Thus, from Equation (
3), we get
where
h represents the inner-product on real quaternions. From Equations (
3) and (
4), we obtain
Corollary 1. Letbe a helix in and u be a constant unit spatial-quaternion. If , then the dual spatial-quaternionic curve β given byis also a dual spatial-quaternionic helix, where is the distance between the lines correspond the dual vectors and u [3]. Definition 2. Let be a regular dual spatial-quaternionic curve with arc-length parameter s and , and be the Frenet vectors of β at . With u being a constant unit dual spatial-quaternion, the function given byis called harmonic curvature function of β and the dual number is called the harmonic curvature at with respect to u, where ϕ is the dual angle between the dual vectors and , [3]. Now, we can give the theorem that gives the harmonic curvature in terms of the curvatures of .
Theorem 2. Let be a dual spatial-quaternionic helix with arc-length parameter s. The harmonic curvature of β can be given bywhere K and R the principal curvature and the torsion of β, respectively. Proof. Let
T,
and
be the Frenet vector fields of the spatial-quaternionic helix
and
u be a constant unit dual vector. With
being the dual angle between
and
u, we have
By differentiating from the last equation, we get
From Equations (
1) and (
8), we obtain
By differentiating from Equation (
9), we get
Taking account of Equations (
1) in (
10), we obtain
Using Equations (
4) and (
5) in Equation (
11), we get
Thus, the proof is completed. □
Theorem 3. Let be a dual spatial-quaternionic helix with arc-length parameter s. If the harmonic curvature at the point is and is Frenet frame field for this curve, then β is a dual space-quaternionic helix if and only if constant.
Proof. If
is a dual space-quaternionic helix, then there is a unit constant
u dual vector satisfying the equality
This dual vector expressed in the terms of
bases of dual space quaternionic
curve at the point
is given by
If the considered
,
, with the use of Equations (
4) and (
5), we obtain,
Since
,
and
, we obtain
Conversely, let us consider
for
dual space-quaternionic curve. In this case, there is a
angle that satisfies
. Then, we define a dual space quaternion
u as
According to this equality,
- (1)
Dual space-quaternionic u is constant.
- (2)
u is unit.
(1) We show that dual space
u is constant: By taking derivative of Equation (
15) with respect to
s, we obtain
Here, if we consider the equalities
,
and
from
, then we obtain,
Furthermore, u is a constant dual space-quaternion.
(2) We show that dual space-quaternion
u is unit:
Thus,
. Moreover, we obtain
Thus, it is shown that is a dual space-quaternionic helix. □
Here,
3. Dual Quaternionic Helixes in and The Harmonic Curvatures of Them
Definition 3. Let be a dual quaternionic curve such that the tangent vector of has unit length along and u be a constant unit dual spatial-quaternion. Ifthen is called a dual quaternionic helix [3]. Theorem 4. Let be a dual space quaternionic helix. On the condition,each dual quaternionic helix derived from β,is also a dual quaternionic helix with the same axis of β [2]. Definition 4. Let be a regular dual quaternionic curve parameterized by arc-length s and u be constant unit dual spatial-quaternion. Let be the Frenet frame field along . With being the angle between and u, the function , defined byis called the harmonic curvature function of , of order i. We define also . We can give the theorem as follows:
Theorem 5. Let be a dual quaternionic curve with the arc-length parameter s. Then, there are the following relations between the curvatures and harmonic curvatures: Proof. Let
u be a constant unit dual spatial-quaternion and
and
be the Frenet vectors of
at the point
. Since
is a helix, we have
By differentiating from the last equation, we obtain that
From Equation (
2),
and Equation (
20), we obtain
. By differentiating the last equation, we have
By using Equation (
2) in Equation (
21), we obtain
Since
from Equation (
22) and the last equation, we obtain
where
and
are the first and second curvatures of
, respectively. By differentiating Equation (
23), we get
From Equations (
2) and (
24), we obtain
Considering Equations (
19) from (
25), we get
or
Thus, the proof is completed. □
Theorem 6. Let be a dual quaternionic helix with arc-length parameter s, be Frenet frame field for this curve at the point and , be the harmonic curvature. Then, is a quaternionic helix
Proof. Let us consider
is a helix. Then, there is a unit and constant dual space-quaternion
u for curve
, which satisfies
This dual space quaternion is expressed as
in terms of the base
. From Equations (
18) and (
19), since
,
,
,
and
u are units, we obtain
: Let us receive
for
dual quaternionic curve. In this case, there is a dual angle
satisfying
. According to this, the dual space quaternion defined as
is a constant and unit quaternion.
(1) Let us once again show
u is constant: By taking derivative of Equation (
30) with respect to
s, we obtain
On the other hand, if we rewrite for
i = 2 Equation (
19),
and we re-derive with respect to
s, we obtain
By using Equations (
2) and (
24) in Equation (
31) with
,
Hence, u is constant space quaternion.
(2) Let us show
u is unit: We can write,
thus,
.
On the other hand, using the definition of
u, we can find
Thus, we obtain is a helix and hence the proof is completed. □
Corollary 2. The equations of derivative of the harmonic curvatures given by Equations (26) and (33) for dual quaternionic curve in matrix form are expressed as