Abstract
We expressed the higher-order velocities, accelerations, and poles under the one-parameter planar hyperbolic motions and their inverse motions. The higher-order accelerations and poles are also presented by considering the rotation angle as a parameter of the motion and its inverse motion.
1. Introduction
First of all, we need to define the set of hyperbolic numbers. For this reason let us recall complex numbers, which are the extension of real numbers that include the imaginary unit and can be written in terms of the standard basis as , where . In this expression, is the real part and is the imaginary part of complex numbers.
Similarly, hyperbolic numbers can also be defined as an extension of the real numbers. This extension is performed by adjoining the unipotent (hyperbolic imaginary) , where but . In this case, the hyperbolic numbers set can be written as follows: Similar to complex numbers, hyperbolic numbers also have the standard basis and every hyperbolic number can be written in the form below: where . In this expression, the real and the unipotent parts are and , respectively.
The collection of all hyperbolic numbers is called the hyperbolic plane (or the split-complex plane). A complex number, , is the point in the plane with the coordinates . A hyperbolic number can also be represented by the point with the coordinates and the conjugate of is defined by
It has been useful to identify the hyperbolic numbers which have also been called perplex numbers, split-complex numbers, or double numbers with 2-dimensional Minkowski space-time. As the complex numbers are naturally related to the Euclidean plane, the hyperbolic numbers are related to the Lorentzian plane (or the 2-dimensional Minkowski space-time).
Let and ; then the addition and multiplication of hyperbolic numbers can defined as follows: This multiplication is commutative, associative, and distributive over addition.
Since the sets of hyperbolic numbers and complex numbers are two-dimensional vector spaces over the real numbers field, we can identify with a point or vector . The hyperbolic inner product is indicated as below: where and . Hyperbolic numbers are said to be hyperbolic (Lorentzian) orthogonal if . The vectors are classified by the inner product as follows: The hyperbolic modulus of is defined by the following: and is considered to be the hyperbolic distance to the point from the origin. Note that the points on the lines are isotropic because they are nonzero vectors with . Thus, the hyperbolic distance yields a geometry, which is the Lorentzian geometry, on , quite unlike the usual Euclidean geometry of the complex plane, where only if . Hyperbolic numbers are also the universal Clifford algebra for ; see [1–4].
Every nonisotropic hyperbolic number can be written in one of the forms below: or where and or , respectively, which corresponds to 4 branches of the unit hyperbole . In this case, the points of these branches and are in quadrants, , , , or , respectively. Just like the area of the region of the unit circle with a central angle is , the area of the unit hyperbolic region with a central angle is also determined by . A hyperbolic rotation by corresponds to the multiplication by the matrix Finally, if a vector is multiplied by , it gives a hyperbolic orthogonal vector similar to multiplications by in the complex plane.
As seen above, the complex numbers plane and the hyperbolic numbers plane are quite similar. Hence, in analogy to one-parameter planar complex motion which was introduced by Blaschke and Müller [5], Yüce and Kuruoǧlu [6] also introduced the one-parameter planar hyperbolic motion and they obtained the velocities, accelerations, and poles accordingly.
2. The Planar Hyperbolic Motion
Let and be moving and fixed hyperbolic planes and let and be their orthonormal coordinate systems, respectively. If the vector is represented by the hyperbolic number , then the motion can be defined with the aid of (8) and (10) by the transformation below: This transformation is called a one-parameter planar hyperbolic motion and denoted by , where is the rotation angle of the motion ; that is, the hyperbolic angle between the vectors and and the hyperbolic numbers and represent the point with respect to the moving and the fixed rectangular coordinate systems, respectively (Figure 1). Besides, the rotation angle and , , and are continuously differentiable functions of a time parameter and at an initial time , the coordinate systems are coincident.
Let the hyperbolic number represent the origin of the fixed point system with respect to the moving system. Then, if we take , we obtain and . Thus, we can have from (11):
If is a moving point of , then the velocity of with respect to is known as the relative velocity of the motion and is denoted by . This vector can be written as . This vector can be expressed with respect to by the equation below: If we differentiate (11) with respect to , we obtain the absolute velocity of the motion as follows: where is the sliding velocity of the motion . By differentiating (12) with respect to , we also obtain the following equation: Hence, we can rewrite the sliding velocity as follows:
For a general one-parameter planar hyperbolic motion, there is a point that does not move, which means that its coordinates are the same in both reference coordinate systems and . This point is called the pole point or the instantaneous rotation pole center. In this case, we obtain and use (17); then, for the pole point of the motion, we get so, from (17) and (18), we can rewrite the sliding velocity with the aid of the pole point as below:
During the motion , the loci of the pole points , which are fixed on both planes for all , are called the moving and fixed pole curves and will be denoted by and , respectively. Because the velocity of and is the same for each , and roll upon each other without sliding.
If we differentiate the relative velocity with respect to , we obtain the relative acceleration vector as below: This vector is expressed by the following: in the fixed coordinate system. During the motion , if we differentiate the absolute velocity vector with respect to , then we get the absolute acceleration vector as follows: where are called the sliding acceleration vector and the coriolis acceleration vector of the motion , respectively. From (22) and (23), we have the following equation:
Under the motion , the acceleration pole is characterized by eliminating the sliding acceleration vector . Therefore, if we take and assume that , then for the acceleration pole point of the motion is as follows:
3. Higher-Order Accelerations and Poles under the One-Parameter Planar Hyperbolic Motions
3.1. Higher-Order Accelerations under the One-Parameter Planar Hyperbolic Motions
Let the motion be the one-parameter planar hyperbolic motion and let be a fixed point. Then, the absolute velocity and the sliding velocity are equal to each other and this velocity is given as below: If we differentiate (26) with respect to , then we obtain the second-order velocity (absolute velocity or sliding velocity) as below: From (26), let ’s coefficient be , and from (27), let ’s coefficient be . Thus, we obtain in terms of as . For this reason, from (26) and (27), we can rewrite and as below: In this way, if we differentiate (29) with respect to , then we obtain the third-order velocity (or the second-order acceleration) as below: If we use (28) in the latter equation, we have the following: or where . If we differentiate (32) with respect to , then we obtain the fourth-order velocity (or the third-order acceleration) as below: So, from (28) and the latter equation, we have the following: or where .
If we continue to subsequent differentiations, we can get higher-order velocities and accelerations. As we can clearly see in the following equations, are true for , , , and (where we suppose that ). If we assume that (36) and (37) are also true, then we can get the following: by differentiation (36). Furthermore, we can write from (37). Finally, (36) has been proved by the induction method. Thus, we may give the following theorem.
Theorem 1. During the one-parameter planar hyperbolic motion , if is a fixed point, then we get for the velocities from -order and the acceleration from order for each , where
3.2. Higher-Order Poles under the One-Parameter Planar Hyperbolic Motions
During the one-parameter planar hyperbolic motion , if is a fixed point, then from (28), we can calculate the first-order pole point as follows: By using (29), we obtain the second-order pole point as below: In a similar way, from (37), we can get the -order acceleration pole as shown below:
Also, from (39), we have the following: So, if we use (37) and the latter equation, we can rewrite in terms of as follows: Similarly, we can obtain the higher-order pole points with respect to moving coordinate system as follows: If the absolute value of both sides of (45) is taken, then one can find that the hyperbolic number is independent from the point . So we can give the following theorems.
Theorem 2. The absolute value of higher-order accelerations under the one-parameter planar hyperbolic motion is rational with distance between the point and corresponding pole point and it takes the same value above the circles which have this pole point as a center.
Theorem 3. If we consider higher-order acceleration vectors corresponding order pole rays of all points of the moving plane for each , we can obtain the same angle between these vectors in the case of both of the space-like and time-like vectors and we can also obtain the same angle between these vectors in case one of these vectors is space-like and the other is time-like.
For more properties of hyperbolic angle, the reader is referred to [7–9].
4. The Inverse Motion of the One-Parameter Planar Hyperbolic Motion and Higher-Order Accelerations and Poles under This Motion
Let and be fixed and moving hyperbolic planes and and their orthonormal coordinate systems, respectively. Then we obtain the inverse motion of the motion , denoted by . In this motion, all of the velocities, accelerations, and poles are analyzed in , which is the fixed plane during the inverse motion of the one-parameter planar hyperbolic motion, instead of . This is the main difference from the one-parameter planar hyperbolic motion. So, from (11), we can obtain the transformation of the one-parameter planar inverse hyperbolic motion with the below solution way: and form (12) can be written as follows: where is the rotation angle of the motion . In analogy to the motion , we can get the velocities, accelerations, and poles for the motion . But we want to focus on higher-order accelerations and poles.
4.1. Higher-Order Accelerations under the One-Parameter Planar Inverse Hyperbolic Motions
During the motion , is a fixed point, then the absolute velocity and the sliding velocity are equal to each other, and this velocity is given as below: It is convenient to write the latter equation as follows: If we differentiate (50) with respect to , then we obtain the second-order velocity (absolute velocity or sliding velocity) as below: or from (50) and the latter equation, we obtain the second-order velocity (or the first-order acceleration) as below:
From (50) and (52), we can write that the coefficients of are and , respectively. So we can obtain in terms of as . It is convenient to rewrite (50) and (52) as below: Similarly, if we differentiate (54) with respect to , then we obtain the third-order velocity (or the second-order acceleration) such as If we use (50) in the latter equation, we get one of the following equations: or where . If we differentiate (57) with respect to , then we obtain the fourth-order velocity (or the third-order acceleration) as below: So, from (50) and the latter equation, we have the following: or where .
If we continue to subsequent differentiations, we can get the higher-order velocities and accelerations. So, we can clearly see that the following equations are true for , , , and (we suppose that ). If we assume that (61) and (62) are also true, then we can get the following: by differentiation (61). Furthermore, we can write from (62). Finally, (61) has been proved by the induction method. Thus, we may give the following theorem.
Theorem 4. During the one-parameter planar inverse hyperbolic motion , if is a fixed point, then we get for the velocities from -order and the acceleration from -order for each , where
4.2. Higher-Order Poles under the One-Parameter Planar Inverse Hyperbolic Motion
During the one-parameter planar inverse hyperbolic motion , from (50), we can calculate the first-order pole point for the fixed point as below: By using (54), we obtain the second-order pole point as below: In a similar way, from (57), we can get the -order acceleration pole as follows:
Also, from (70), we can write the following equation: So, if we use (65) and the latter equation, we can rewrite in terms of as follows:
Similarly, we can obtain the higher-order pole points under the one-parameter planar inverse hyperbolic motion , with respect to moving coordinate system as follows:
5. The Relationships between and
5.1. The Relationship between Functions and
We are going to obtain two relationships between functions and , during the motions and . We are going to use the induction method for the first one. For this reason, we can see that the following equation is true for , , and . Now, let us assume that (74) is true for and prove that it is true for . Hence, if we assume that is true, from (67) we can obtain the following: Then, because of and we get the below equation: So, we can give the following theorem.
Theorem 5. During the motions and , the following relationship between functions and holds.
For the other relationship between functions and , if we differentiate functions and with respect to , we can get the following: If we continue with consecutive differentiations, we obtain the following: or Similarly, we get the following: From and Leibniz formula , we can give the following theorem.
Theorem 6. During the motions and , the following relationship between functions and holds.
5.2. The Relationship between Poles and
From (43), (46), (70), and (73), we can rewrite the , and , as follows: Furthermore, we can write the -order derivative of (12) as below: with the aid of (80) and Leibniz formula. Then, from (84) part , we can obtain . Also, if we use this expression we can rewrite as follows: We know this from (84) and . Hence, we can write another expression for as follows: Ultimately, if we use (87) and (83) part , we can give the following theorem.
Theorem 7. During the motions and , the following equation between and holds.
Similarly, we can obtain the following results: Result 1 Result 2 Result 3
6. Rotation Angle as a Parameter
In this section, we choose the rotation angle as a parameter of the motion instead of the parameter . The purpose of this choice is to get the formulation in Sections 2, 3, and 4 that are more simple. So, if we use the rotation angle as a parameter, then we obtain and as follows: Now, with the aid of the last equation, we can rewrite (40) and (41) as below: Furthermore, from (94), we can write as below: Similarly, from (72), we can obtain and as below: Hence, it is known that the multiplication with a hyperbolic number like means rotating as its argument and translates as its module to the multiplicand vector. But the isotropic lines divide the hyperbolic plane into four separate regions because of the fact that we must be careful if the multiplication with hyperbolic numbers changes the region or not. For instance, if we multiplicate and , we obtain , where and are the hyperbolic quadrants of the hyperbolic plane; see [10]. Hence, we can give the following theorem.
Theorem 8. During the one-parameter planar hyperbolic motion , if we choose the rotation angle as a parameter of the motion instead of the parameter , then adding to the vector, which we have obtained by multiplication with the hyperbolic number and , is giving the -order pole point (or -order acceleration pole point) of the planar hyperbolic motion, . Also, we add to the vector which we have obtained by multiplication with the hyperbolic number and is giving the -order pole point (or -order acceleration pole point) of the inverse planar hyperbolic motion, .
Example 9. Let us consider the one-parameter planar hyperbolic motion with and . In this situation we can show the curve of the point by using Maple Programming for on the plane (Figure 2).
Then, we calculate by using Maple Programming also the higher-order accelerations of the point under this motion for as follows:
where . Similarly, we can calculate the higher-order poles of the point under this motion for as follows:
where . If we consider the rotation angle as a parameter of the motion, it is clear that the relation holds for .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This research has been supported by Yildiz Technical University Scientific Research Projects Coordination Department, Project no. BAPK 2013-01-03-DOP02.